Combinatorics 1984: Finite Geometries and Combinatorial Structures: Colloquium Proceedings: Finite Geometries and Combinatorial Structures

COMBINATORICS '84

NORTH-HOLLAND MATHEMATICS STUDIES Annals of Discrete Mathematics(30)

General Editor: Peter L. HAMMER Rutgers' University, New Brunswick, NJ, U.S.A.

Advisory Editors C. BERGE, Universite de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U.S.A. V. KLEE, University of Washington, Seattle, WA, U.S.A. J.-H. VAN LINT CaliforniaInstitute of Technology,Pasadena, CA, U.S.A. G . 4 . ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.

NORTH-HOLLAND-AMSTERDAM

NEW YORK OXFORD .TOKYO

123

COMBINATORICS '84 Proceedings of the International Conference on Finite Geometries and Combinatorial Structures Barl; ItalK 24-29September, 1984

edited by

A. BARLOlTI Universita di Firenze, Firenze, Italy

M. BILIOITI Universitadi Lecce, Lecce, Italy

A. COSSU Universitadi Bart Bari, Italy

G. KORCHMAROS Universitadelta Basilicata, Potenza, Italy

G.TALLINI Universita 'La Sapienza: Rome, Italy

1986

NORTH-HOLLAND -AMSTERDAM

0

NEW YOAK

QXFORD .TOKYO

@

ELSEVIER SCIENCE PUBLISHERS B.V., 1986

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ISBN: 0 444 87962 5

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Library of Congress Catalogingin-PublicationData

International Conference on Finite Geometries and Combinatorial Structures (1984 : Bari, Italy) Combinatorics '84 : proceedings of the International Conferenco on Finite Geometries and Combinatorial Structures, Bari, Italy, 24-29 September 1984. (Annals of discrete mathematics ; 30) (North-Holland mathematics studies ; 123) Includes bibliographies. 1. Combinatorial geometry--Congresses. I. Barlotti, A. (Adriano), 1923. 11. Title. 1 1 1 . Title: Combinatorics eighty-four. IV. Series. V. Series: North-Holland mathematics studies ; 1 2 3 . QA167.158 1984 511l.6 85-3 1 121 ISBN 0-444-87962-5

PRINTED I N THE NETHERLANDS

V

PREFACE Every year, since 1980, an International Combinatoric Conference has been held in Italy: Trento, October '80; Rome, June '81 ; La Mendola, July '82; Rome, at the Istituto Nazionale di Aka Matematica, May '83. The International Conference Combinatorics '84, held in Giovanazm (Bari) in September '84 is part of the well established tradition of annual conferences of Combinatorics in Italy. Like the previous ones, this Conference was really successful owing to the number of participants and the level of results. The present volume contains a large part of these scientific contributions. We are indebted to the University of Bari and to the Consiglio Nazionule delle Ricerche for fmancial support. We are pmfoundly grateful to the referees for their assistance. A. BARLOTTI M. BILIOTTI A. COSSU G. KORCHMAROS G. TALLINI

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vii

Intervento di Apertura del Prof. G. Tallini al Convegno “COMBINATORICS 84” Vorrei dare, a nome del Comitato scientific0 e mio,ilbenvenuto aimoltipartecipanti che dall’Italia e dall‘estero sono qui convenuti per prendere parte a questo convegno. Esso si ricollega e fa seguito ai congressi internazionali di combinatoria tenuti a Roma nel giugno del 1981, a La Mendola nel luglio del 1982, a Roma presso I’Istituto Nazionale di Alta Matematica nel maggio del 1983. Questi incontri, ormai annuali in Italia e che spero possanc continuare, s’inquadrano nell’ampio sviluppo che la combinatoria va acquistando a livello internazionale. Come 8 noto il mondo modern0 si va indirizzando ed evolvendo sempre di pib verso la programmazione e l’informatica, al punto che un paese oggi B tanto pih progredito, importante e all’avanguardia quanto pib B avanzato n e b scienza dei computers. I1 ram0 della Matematica che B pi^ vicino a questi indirizzi e che ne I!la base teorica B proprio la combinatoria. Essa a1 gusto astratto del ricercatore, del matematico, associa appunto le applicazioni pib concrete. Cib spiega il prepotente affermarsi di questa scienza nel mondo e ne prova il fervore di studi e di ricerca che si effettuano in quest’ambito, le pubblicazione dei molti periodici specializzati, i numerosi convegni internazionali a1 riguardo. Vorrei ringraziare gli Enti che hanno permesso la realizzazione di questo convegno, tutti i partecipanti, in particolare gli ospiti stranieri che numerosi hanno accolto il nostro invito e tra i quali sono presenti insigni scienziati. Concludo con I’augurio che questo convegno segni una tappa da ricordare nello sviluppo della nostra scienza.

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ix

CONTENTS Preface G. TALLINI, Intervento di apertura a1 Convegno

V Vii

V. ABATANGELO and B. LARATO, Translation planes with an autornorphism group isomorphic to SL(2,5)

1

L. BENETEAU, Symplectic geometry, quasigroups, and Steiner systems

9

W. BENZ, On a test of dominance, a strategic decomposition and structures T(t,q,r,n)

15

A. BEUTELSPACHER and F. EUGENI, On n-fold blocking sets

31

A. BEUTELSPACHER and K. METSCH, Embedding finite linear spaces in projective planes

39

A. BICHARA, Veronese quadruples

57

M.BILIOTTI, S-partitions of groups and Steiner systems

69

M. BILIOTTI and G. KORCHMAROS, Collineation groups strongly irreducible on an oval

85

B. BIONDI and N. MELONE, On sets of Plucker class two in PG(3,q)

99

F. BONETTI and N. CIVOLANI, A free extension process yielding a projective geometry

105

F. BONETTI, G . 4 . ROTA, D. SENATO and A.M. VENEZIA, Symmetric functions and symmetric species

107

R. CAPODAGLIO DI COCCO, On thick (Qi2)sets

115

P.V. CECCHERINI and N. VENANZANGELI, On a generalization of injection geometries

125

P.V. CECCHERINI and A. SAF'PA, A new characterization of hypercubes

137

Contents

X

P.V. CECCHERINI and A. SAPPA, F-binomial coefficients and related oombinatorial topics: perfect matroid designs posets of full binomial type and F-geodetic graphs

143

L. CERLIENCO, G. NICOLETTI and F. PIRAS, Polynomial sequences associated with a class of incidence coalgebras

159

M.DE SOETE and J.A. THAS, R-regularity and characterizations of the generalized quadrangle P(W(s), (-))

171

M.DEZA and T. IHRINGER, On permutation arrays, transversal seminets and related structures

185

G. FAINA, Pascalian configurations in projective planes

203

P. FILIP and W. HEISE, Monomial code-isomorphisms

217

S. FIORINI, On the crossing number of generalized Petersen graphs

225

J.C. FISHER, J.W.P. HIRSCHFELD and J.A. THAS, Complete arcs in planes of square order

243

M.CIONFRIDDO, A. LIZZIO and M.C. MARINO, On the maximum number of SQS(v) having a prescribed PQS in common

25 1

A. HERZER, On finite translation structures with proper dilatations

263

M. HILLE and H. WEFELSCHEID, Sharply 3-transitive groups generated by involutions

269

F. KRAMER and H. KRAMER, On the generalized chromatic number

275

P. LANCELLOTTI and C. PELLECRINO, A construction of sets of pairwise orthogonal F-squares of composite order

285

D. LENZI, Right S-n-partitions for a group and representation of geometrical spaces of type “n-Steiner”

29 I

G. LO FARO, On block sharing Steiner quadruple systems

297

G. MENICHETTI, Roots of affine polynomials

303

S. MILICI, On the parameter D(v,tv. 13) for Steiner triple systems

311

C. PELLECRINO and P. LANCELLOTTI, A new construction of doubly diagonal orthogonal latin squares

331

Contents

xi

C. PELLEGRINO and N.A. MALARA, On the maximal number of mutually orthogonal F-squares

335

C. PICA, T. PISANSKI and A.G.S. VENTRE, Cartesian products of graphs and their crossing numbers

339

G. TALLINI, Ovoids and caps in planar spaces

347

B.J. WILSON, (k,n#arcs and caps in finite projective spaces

355

N. ZAGAGLIA SALVI, Combinatorial structures corresponding to reflective matrices circulant (0,l)-

363

H. ZEITLER, Ovals in Steiner triple systems

373

PARTICIPANTS

383

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Annals of Discrete Mathematics 30(1986) 1-8 0 Elsevier Science Publishers B.V.(NorthHolland)

1

TRANSLATION PLANES WITH AN AUTOMORPHISM GROUP ISOMORPHIC TO SL (2,5) Vito

Abatangelo Bambina Universith di Bari Italia

Larato

q , In this paper translation planes of odd order , are constructed. Their main interest consists in the fact that their translation complement contains At first these planes a group isomorphic to SL(2,5) were obtained in other ways by 0. Prohaska in the case 51q+l ( [lo] ,1977) and by G. Pellegrino and G. Korchm5ros in the case 51q-1 ( [9] ,1982), but in both papers the Authors did not establish the previous group property. Moreover we show that Pellegrino and Korchmhros plane 2 is not a near-field plane of order 11

2 5 1 q -1

.

.

1. ORDER

AN AUTOMORPHISM GROUP OF THE AFFINE 2 2 q , 51q -1 , ISOMORPHIC TO SL(2,5)

DESARGUESIAN PLANE OF

2 Set K = GF(q J , q odd. We may assume that the elements of K can be 2 written in the form g + t T with 5 , V € F = GF(q) and t = s , where s is a non-square element of F

.

Let a be the affine Desarguesian plane coordinatized by K : points are pairs (x,y) of elements of K and lines are sets of points satisfying equations of x = c with m,b,c elements of K The affine the form y = mx+b or no coordinatiz@d by F is an affine Baer subplane; the image of subplane no under a composition of a linear transformation with a translation of 3c is also taken to be an affine Baer subplane. The lines at infinity of Baer subplanes are called Baer sublines at infinity. By standard arguments (similar to those of [7]. p. 80-91) one can show the following facts: Baer sublines at infinity are sets of elements of K u { oo} of the form:

.

(1.1)

[

ap+b

I

a,b€K , p runs over F u { m }

j

or (1.2)

.

Let v and r be any two elements of K F o r any d E K such that gq+l = 1 , the set of all points (x,y) for which (1.3)

and

g

EK ,

y = xv + rxqg + d

Research partially supported by M.P.I. (Research project "Strutture Geometriche Combinatorie e l o r o Applicazioni'').

2

V. Abatangelo and B. Larato

i s t h e p o i n t - s e t o f an a f f i n e Baer subplane. Its Baer s u b l i n e a t i n f i n i t y h a s e q u a t i o n ( 1 . 2 ) as t h e l i n e y = xmtd i n t e r s e c t s (1.3) e i t h e r i n q p o i n t s o r i n t h e o n l y p o i n t ( 0 , d ) a c c o r d i n g as ( 1 . 2 ) h o l d s o r does n o t hold. The a f f i n e Baer s u b p l a n e s with e q u a t i o n s of t h e form ( 1 . 3 ) c o n s t i t u t e a n e t on t h e a f f i n e points of n

.

The l i n e a t i n f i n i t y becomes t h e Miquelian i n v e r s i v e p l a n e M(q) t a k e a s c i r c l e s t h e sets (1.1) and ( 1 . 2 ) o f elements o f K ~ { c o } . isomorphic t o I n o r d e r t o c o n s i d e r a n automorphism group o f $C d i s t i n g u i s h two c a s e s , a c c o r d i n g a s 51q+l o r 51q-1

.

CASE

51qtl

a

t h e r e e x i s t s an element

of

( c f . [4]). W e p u t b = (a-aq)-' and c such t h a t l e t u s c o n s i d e r t h e f o l l o w i n g a f f i n e mappings o f a

fl: and t h e i d e n t i t y Note t h a t < a ,

fl>

a x ' = bx

t

x , SL(2,5) , (cf.

& : x' = 2

,

: x ' = ax

,

cy

we

y' = y

141,

-

1. and

: mqtl

1 and

Ci+2

: (m-2a

t

. >

on t h e l i n e a t i n f i n i t y , r.l(q).

...

}

i(1-q) q 2 -1 q t l 2 -2 b c ( 2 b +1) ) = (2b +1)

-

: (C1)(C2C3C4C5C6)

, fl

,

a

( i = 0,

and

: (C1C2)(C3C6)(C4)(C5)

PROPOSITION 2 . - The group I' a c t s on t w o - t r a n s i t i v e r e p r e s e n t a t i o n on s i x o b j e c t s .

-

bqy

p. 1 9 9 ) .

PROOF Some l o n g and e a s y c a l c u l a t i o n s prove t h a t i t s e l f and a c t on t h e s e t $f as f o l l o w s

a

b q + l = 1. After t h i s

q+1 I n M(q) t h e group I' maps t h e c i r c l e C 0 :. = -l l e a v e s t h e s e t V = { C1 , C 2 , ,C6 i n v a r i a n t , where

PROPOSITION onto i t s e l f =

t

a5 = 1

y t = aqy

,p

PROOF (1.4) and (1.5)

SL(2,5)

such t h a t

K

cq+l

y ' = -cqx

Now o u r purpose is t o s t u d y t h e a c t i o n o f < a i . e . t h e a c t i o n o f I' = ZPSL(2,5) on

1

we

5(q+l

By assumption

C

if

as

PSL(2,5)

fl

...

,4).

send

C

0

onto

. a c t s i n its usual

Assuming B. H u p p e r t ' s t e r m i n o l o g y , [41 , l e t d = GF(5) u { m ) be t h e s e t c o n s i s t i n g of s i x o b j e c t s ; let

PSL(2,5) be t h e p e r m u t a t i o n group on d o f t h e form x' = ( a x + b ) ( c x + d ) - ' , ad-bc € { 1 , 4 ) , a , b , c , d E G F ( 5 )

.

w i t h t h e symbol w , C2 w i t h 0 , Cg w i t h 3 , with 1 , c1 c41 C w i t h 4 and C with 2 ; then a': x' = x+3 and x' = x a c t on 6 , r e s p e c t i v e l y , a c t on V AS < a ' , N P S L ( ~ , ~, ) we A5 as a and g e t o u r assert.

We i d e n t i f y

fl

.

8': p'>

Translation Planes

3

PROPOSITION 3. - Any two circles of have two o r zero common points according as q E 3 (mod 4 ) o r q 5 1 (mod 4). PROOF - By Prop. 2 it is sufficient to prove that the circles satisfy our assert.

C

1

and

C

2

STEP 1 If C1 and C 2 have any common point, its coordinates would satisfy both their equations which form a system which is equivalent to the following =: 1

1 1

(1.6)

As

{

m = ch

I

h EGF(q)

}

c q m -c mq

=

o

,

is the set of solutions of the second equation of

(1.61, the whole system (1.6) has any solution if and only if the equation

cq+l h2 = 1

is solvable, i.e.

if and only if

cq+l

is a square in

F

.

STEP 2 - cq+l is a square of F or not according as q 5 3 (mod 4 ) o r q 1 (mod 4) In order to prove this, let x and x be elements of F 1 2 such that a = x + tx * note that 1 2 ' 2 2 aq+l = x sx E F (1.7) 1 2 q+l 5 and (a ) = 1 , so

.

-

= 1

'+'a

(1.8)

(because in F 4 2 x s = 0 , 2 +.r/;;= 8x2

+

there is no element of order 5 ) and therefore 4

16x1

By means of (1.7) and (1.81, we get

-

2

12x

.

1

5x

+ 1

=

4

2 2

+ lox x s + 1 2 0 , i.e.

1

- 3 : this proves that 5 is a square in F Now 1 2 2 -1 2 2 -1 = p(l+sx ) - 3 7 ( 4 x 2 s ) = ( 4 x - 3 ) ( 4 x s) cq+l = c(a-aq)2 + 13la-a' 2 1 2

-

-1

(1 2 m 2 ( 4 x 2 ) q+l. and therefore c is a square in F if and only if F , i.e. if and only if q 3 (mod 4). = -1 s

PROPOSITION 4 . -

Co

is disjoint from any circle of

PROOF - It is easy to check that follows by means of Prop. 2.

CASE

=

-2

C

0

and

C

1

-1

%

is not a square in

.

are disjoint. Our assertion

5)q-1

By assumption 51q-1 there exists an element a of F such that a5 = J. -1 -1 Let b,c,d be elements of K such that b = (a-a ) , cd = -1-b let us consider the following affine mappings of 7C -1 -1 y : x' z a x , y' = (2+t)b x + ay 2 6 : X' = [-b+(Z+t)d]x + dy , y' = [2(2+t)b+c-(2+t) dJx + [b-(2+t)ay

and note t h a t < y , d > - - S L ( 2 , 5 )

(cf.[4],

.

. Now

p . 198).

on the line at infinity, i.e. the We shall now look at the action of< 7,6> action of the group A = < Y , ~ > / < - E >on M(q).

V. Abatangelo and B. Larato

4 PROPOSITION 5.

-

In M(q)

A

the group

maps the circle Do : 2tmq+l-m+mq =O

,D6 } invariant where D :4mq+1+m+mq=0 onto itself and leaves the set { D1e 1 and 2 2i 2 -1 a-icd-ll .q+1 + C(2tt)-a 2i (1+2b2)(2bd)-3m+ : C(4-t ) - 4a (1+2b )(2bd) D i+2 2i 2 ,4). + [(2-t) - a (1+2b )(2bd)-Y mq + 1 = 0 , (i = 0.1, -

0

.

-

...

PROOF - Long and easy calculations prove that y and act on the set { D1 ,D2, , D ~ ] as follows

...

y

:

(D1)(D2D3D4D5Ds)

PROPOSITION 6. way as PSL(2,5) objects

.

and

6

and

6 map D0 onto itself

: (D1D2)(D3D6)(D4)(D5)

.

...

The group d acts on { Dl,D2, ,D6 } in the same acts in its usual two-transitive representation on six

PROOF - The proof is similar to that one of Prop. 2 , provided we identify D 1 with the symbol 00 , D2 with 0 , D3 with 1 , D4 with 2 , D5 with 3 and D6 with 4 and moreover y and 6 with 7' and 6 ' , respectively, where y ' : x' = x + 1 and 6' : x ' = 4x

.

Assume now 9 = { Do ,D1 , we get the following

.. .

,D6

}

; by direct calculations and by Prop.6

-

7. (i) Any two circles of 9 PROPOSITION have a common point. (ii) no three circles of 9

2.

-

DERIVED AND

R-DERIVED

have two common points;

PLANES

In this section we discuss briefly the processes of derivation and multiple derivation due to T.G. Ostrom and of R-derivation and multiple &derivation due to A.A. Bruen. The reader can find some details in Hughes-Piper c37, Ostrom c7,8] and Bruen Ll]. Let us assume q > 3 ; every circle C of M(q) is a derivation set, i.e. every pair of affine points lies in a unique Baer affine subplane, whose From now on we will consider circles C with subline at infinity is C equation (1.2). An affine plane can be obtained by replacing lines whose equation is y = mx+b , f o r each m satisfying ( l . Z ) , with the q+l Baer affine subplanes, called also components, with equation y = xv + rxq g

.

where g runs over A = { g E K 1 gq+' = 1 } , together with their translates. This construction is said derivation by C and the obtained plane is a trans2 lation plane of order q Derivation can be repeated many times, when there is a set of circles C l , C a , ,C each disjoint from the other; in this case the process is said C3 , k. multiple derivation with respect to the circles C1'C2' * . * 'Ck

.

. ..

admits a chain of circles, i.e. a About R-derivation, suppose that M(q) family % of circles satisfying the following properties: (i) any two circles of W have two common points; (ii) no three circles of % have a common point; (iii) % consists of (q+3)/2 circles.

Translation Planes

5

.

Let I denote the point set covered by the circles of V It follows that I contains exactly (q+l)(q+3)/4 points and each point of I belongs exactly to two circles of % Hence any two points on a line with its ideal I belong exactly to two affine Baer subplanes whose sublines at point on infinity represent circles of V

.

.

The affine Baer subplanes with Baer sublines at infinity representing circles However, for small q of %' do not form a net on the affine points of 7c ( 5 1 qS 13), several Authors have constructed a net N on the affine points of by taking a suitable half of such affine Baer subplanes; namely, 2 for each circle C of %? , (q+l)q /2 affine Baer subplanes with the same If such a net N exists, we can obtain a transBaer subline representing C lation plane on the affine points of JZ by replacing the affine lines of R whose ideal pointslie on I with N The resulting translation plane is said to be R-derived from 7c

.

.

.

.

Of course if there is a set of disjoint chains of circles on M(q) and the corresponding nets exist, then the above method can be repeated: the resulting translation plane is said to be multiple R-derived from 7c

.

2 3. A CLASS OF TRANSLATION PLANES OF ORDER q , 51q+l , CONTAINING IN THEIR TRANSLATION COMPLEMENT AN AUTOMORPHISM GROUP ISOMORPHIC TO SL(2,5)

-

In this section we assume that q - 1 (mod 4 ) . By Propositions 3 and 4 we get three planes n . (j = 1,2,3): 7c by derivation from 7c with respect to the J 1 ( s o 7c is the well-known Hall plane of order 81, cf. c23, circle C 1 p. 225), by derivation with respect to the circles C1'C2' 'C6 t

a2

R3 by derivation with respect to the circles

Co'cl*

-.- ,C6

; each of

.

them contains in its translation complement the group?SL(2,5) Now we prove the following propositions

8. The group < a,P > leaves invariant each of the PROPOSITION components corresponding with the derivation set

co

PROOF

B

-

The

q+l

*

components are the Baer subplanes

.

9

B

with equations g A straigh forward cal-

: y = rx g where rqtl = -1 and g runs over A g culation shows that U as well as p leaves each B invariant. g PROPOSITION 9. - The group < U , p >splits the set of the 6(q+l corresponding with the multiple derivation set C I U C 2 U (q+1)/2 orbits each of length 12.

components into

...

...

q+l

u c6

PROOF - Let H , (i = 1,2, ,6) be the set consisting of the q+l components which corrispond with the derivation set Then < a , p > acts on the Ci set { H 1 , ,H6 } in the same way as on the set { C1, . ,C6 } By Prcp. 2,

.

...

...

a,p>acts transitively on { H1, 3 2 is< a , l >with A = p a p a p , i.e. 2 A : x ' = c(2b -acq+l-aqb2)y

<

,H6}

,

..

. The stabilizer of

.

H 1

in

.

< a$ >

Y' = - ~ ~ ( 2 b ~ - a ~ c ~ + ~ - a b ~ ) x

V,Abatangelo and B. Larato

6

The q+l components belonging to H are the Baer subplanes E with 1 equations y = xqg where g runs over A A straightforward calsulation invariant and 1 maps E onto E shows that a leaves each E g g -g PROPOSITION 10.- The line-orbit of < a.8 > containing the line joining the origin to Ym has length 12.

.

-

.

-

The vertical line through 0 is left invariant by a and & but Since each subgroup of < a ,fl > containing properly contains also 1 , it follows that the stabilizer of the vertical line through 0 has order 5. This proves our assert. PROOF not by

.

3,

We point out that for q = 9 tions 8,9 and 10 yield:

,

which is the first non trivial case, Proposi-

PROPOSITION 11.- (i) The group r =/<-&> splits the line at infinity of fC into one orbit of length ten and six orbits of length twelve; into six (ii) tie group r = / < - E > splits the line at infinity of f~ 3 orbits of length twelve and ten orbits of length one.

8

Now we state the following PROPOSITION 12.2,3). If SL(2,5) R or f C 3 . 2

Let (I be a plane obtained by derivation from a . (j = is an automorphism group of (I , then (I coincided with

-

PROOF As it is well known, the number of disjoint circles of M ( q ) is q-1 and when it occurs they form a linear flock by a theorem due to W.J. Orr (cf. 161). In our situation C1 ,C2, , ,C6 belong to no linear flock. So if C

..

is any circle which determines a derivation of J C , , C must coincide with some circle C (i = 0,1, ,6) or C must not intersect each of them. If i C E{Co ,C1, ,C6 } , then necessarily C = C ; on the other hand C cannot

...

...

0

stay on Hg because, by Prop. 8 , no orbit of than 10 in H 3 '

SL(2,5)

is long 10 or less

2 q , 5(q-1 , CONTAINING 4. - A TRANSLATION PLANE OF ORDER TRANSLATION COMPLEMENT AN AUTOMORPHISM GROUP ISOMORPHIC TO

IN ITS SL(2,5)

In the previous section 1 we determine the set of circles 9 which is a family satisfying properties (i) and (ii) of chains of circles. Moreover, when q = 11 , 9 satisfies property (iii) and, therefore, is a chain of circles. By means of the automorphism w : X' = (2tt)x of M(11)

,

,

y' =

c ~ m :- m l 1 = 0 , C; : (m

-

2

(8+7t)x + (2tt)y

9 is equivalent to the following chain:

we can check that

-2i 11 )

--

C' : m

P

1+1

(-2)

t

, i

ml1=0 =

1,2,

,

... ,5

which was studied by G. Pellegrino and G. KorchmBros, So translation plane (cf. C9-J 1.

, 9

determines a

Pellegrino and Korchmiros used a geometrical construction and so they cannot notice that the translation plane associated to the chain 9 admits SL(2,5) as automorphism group.

Translation Planes

7

Finally we want to remark that Pellegrino and Korchmlros plane surely is not 2 a near-field plane of order 11 (cf. [q, p. 8 8 ) , though it satisfies the same group property. The near-field planes have only tdo orbits on the line at infinity: the first has length 2 and the other consists of all the remaining points. In the present case the orbit length are 2 and 120, while the Pellegrino and Korchmlros plane has an orbit of length 42 = ( H 1 on its line at 2 infinity

.

REFERENCES Bruen, Inversive geometry and some new translation planes I, Geom. Dedic., 7 (19771, 81-98.

A.A.

P. Dembowski, Finite geometries (Springer-Verlag, Berlin-Heidelberg-New York. 1968). D.R. Hughes-F. Piper, Projective planes (Springer-Verlag, Berlin-Heidelberg-New York, 1973).

B. Huppert, Endliche gruppen I (Springer-Verlag, Berlin-Heidelberg-New York, 1967). H. Luneburg, Translation planes (Springer-Verlag, Berlin-Heidelberg-New York, 1980). W.J. O r r , A characterization of subregular spreads in finite 3-space, Geom. Dedic., 5 (19761, 43-50. T.G. Ostrom, Finite translation planes (Springer-Verlag, Berlin-Heidelberg-New York, 1970).

T.G. Ostrom, Lectures on finite translation planes, Conf. Sem. Mat. Univ. Bari, n. 191, 1983. G. Pellegrino-G. KorchmSros, Translation planes of order of Discrete Math., 14 (19821, 249-264.

112, Annals

0. Prohaska, Konfigurationen einander meidender kreise in Miquelschen Mobiusebenen ungerader ordnung, Arch. Math. (Basel), 28 (1977), n. 5, 550-556.

V. Abatangelo-B. Larato Dipartimento di Matematica Via Giustino Fortunato Universitl degli Studi 70125 - B A R I

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Annals of Discrete Mathematics 30 (1986) 9-14

0 Elsevier Science Publishers B.V. (North-Holland)

SYMPLECTIC GEOMETRY,

QUASIGROUPS,

9

A N D STEINER SYSTEMS

L u c i e n Beneteau UER-M.I.G. Universite Paul Sabatier 3 1 0 6 2 TOULOUSE - C E D E X FRANCE

Zassenhaus's process o f c o n s t r u c t i o n o f H a l l T r i p l e Systems can be g e n e r a l i z e d . I t t u r n s o u t t h a t t h e r e i s a c a n o n i c a l correspondence between e q u i v a l e n c e c l a s s e s o f non z e r o a l t e r n a t e t r i l i n e a r forms o f V(n,3) anf isomorphism c l a s s e s o f r a n k ( n t l ) HTSs whose o r d e r i s 3(" 1, Thus t h e problem o f c l a s s i f y i n g t h e s e designs and t h e r e l a t e d S t e i n e r q u a s i groups may be p r e s e n t e d as a s p e c i a l case o f a more g e n e r a l c l a s s i f i c a t i o n problem o f e x t e r i o r a l g e b r a . As an i l l u s t r a t i o n o f these i d e a s we s h a l l d e a l c o m p l e t e l y w i t h t h e case ns6. F o r n=6 one o b t a i n s e x a c t l y 5 isomorphism c l a s s e s o f HTSs

.

1-INTRODUCTION

-

S e c t i o n 2 g i v e s a b r i e f i n t r o d u c t i o n t o t h e H a l l T r i p l e Systems (HTSs) and t o t h e r e l a t e d groups and quasigroups. There a r e two statements g i v i n g p r e c i s i o n s about t h e correspondence between t h e HTSs on one s i d e , and t h e c u b i c h y p e r s u r f a c e quasigroups and t h e F i s c h e r groups on t h e o t h e r s i d e . We r e f e r t h e r e a d e r t o t h e l i t e r a t u r e f o r t h e c o n n e c t i o n s w i t h o t h e r p a r t s o f a l g e b r a and d e s i g n t h e o r y (C7,10,11). F u r t h e r on a process o f e x p l i c i t c o n s t r u c t i o n o f HTSs i s r e c a l l e d ( s e c t i o n 3 ) . T h i s process i s n o t c a n o n i c a l . B u t i t a l l o w s t o g e t a l l t h e non a f f i n e HTSs whose 3 - o r d e r s e q u a l s t h e r a n k p . As u s u a l t h e r a n k i s t o be understood as t h e minimum p o s s i b l e c a r d i n a l number o f a g e n e r a t o r subset. The e q u a l i t y s = p corresponds t o an e x t r e m a l s i t u a t i o n , t h e non a f f i n e HTSs o b e y i n g sdp, w h i l e t h e a f f i n e ones It i s t h e c l a s s i f i c a t i o n o f non a f f i n e HTSs o f g i v e n r a n k obey s=p-1 (see [ll). whose o r d e r i s minimal t h a t l e d us t o a problem o f s y m p l e c t i c geometry. Given some v e c t o r space V, t h e r e i s a n a t u r a l a c t i o n o f GL(V) on t h e s e t o f symp l e c t i c t r i l i n e a r forms o f V. We s h a l l be c o u n t i n g o r b i t s i n some s p e c i a l cases. F o r f u r t h e r i n v e s t i g a t i o n s t h e most i m p o r t a n t r e s u l t i s some process o f t r a n s l a t i o n i n case t h e f i e l d i s GF(3) : t h e r e i s t h e n a one-to-one correspondence between t h e o r b i t s o f t h e non-zero forms and t h e isomorphism c l a s s e s o f some HTSs. T h i s w i l l be used h e r e t o o b t a i n an e x h a u s t i v e l i s t o f t h e HTSs o f o r d e r g2187 whose r a n k s a r e #6. We s h a l l a l s o c l a s s i f y t h e non a f f i n e HTSs a d m i t t i n g a c o d i mension 1 a f f i n e subsystem. 2-HALL TRIPLE SYSTEMS, MANIN QUASIGROUPS AND FISCHER GROUPS

-

A S t e i n e r T r i p l e System i s a 2-(v,3,1) design, namely i t i s a p a i r (E,L) where E i s a s e t o f " p o i n t s " and L a c o l l e c t i o n o f 3-subsets o f E, c a l l e d " l i n e s " , such t h a t ony two d i s t i n c t p o i n t s l i e i n e x a c t l y one l i n e 11 c L. The c o r r e s o n d i n g S t e i n e r q u a s i g r o u p c o n s i s t s o f t h e same s e t E under t h e b i n a r y l a w : Ef +E ; x,y-xoy d e f i n e d by xox=x and, whenever x#y, xoy=z, t h e t h i r d p o i n t o f t h e l i n e

10

L. Be'ne'reau

through x and y. The Steiner quasigroups can be a l g e b r a i c a l l y characterized by the f a c t t h a t the law i s idempotent and symmetric. Recall t h a t a law i s said t o be symmetric when any e q u a l i t y o f the form xoy=z i s i n v a r i a n t under any permut a t i o n of x,y,z ; t h i s i s equivalent t o the conjunction o f the commutativity and the i d e n t i t y xo(xoy)=y. For a f i x e d set E, t o endow E w i t h a f a m i l y o f l i n e s L such t h a t (E,L) be a Steiner T r i p l e System i s equivalent t o provide E w i t h a s t r u c t u r e o f Steiner quasigroup. So i n what f o l l o w s we s h a l l i d e n t i f y (E,L) w i t h (E,o). A H a l l T r i p l e System (HTS) i s a Steiner T r i p l e System i n which any subsystem t h a t i s generated by three non c o l l i n e a r p o i n t s i s an a f f i n e plane =AG(2,3). This a d d i t i o n a l assumption i s equivalent t o the f a c t t h a t the corresponding S t e i n e r quasigroup i s d i s t r i b u t i v e (a o (xoy)=(aox)o(aoy) i d e n t i c a l l y ; see Marshall H a l l J r . [ 6 ] ) . Therefore the HTSs are i d e n t i f i e d w i t h the d i s t r i b u t i v e Steiner quasi groups.

Let K be a commutative f i e l d . Consider an absolutely i r r e d u c i b l e cubic hypersurface V o f the p r o j e c t i v e space Pn(K). Let E be t h e set o f i t s non-singular K-points. Three p o i n t s x,y,z o f V w i l l be s a i d t o be c o l l i n e a r ( n o t a t i o n : L(x,y,z)) i f there e x i s t s a l i n e L containing x,y,z such t h a t e i t h e r l l c V o r xtytz=R.V ( i n t e r s e c t i o n c y c l e ) . The best known case i s when dim V = l , and n=2 : V i s then a plane curve, i t does not contain any l i n e and o v e r a l l f o r any x,y i n E, there i s e x a c t l y one p o i n t z i n E such t h a t L(x,y,z). The corresponding law x , y ~xoy=z i s obviously symmet r i c . The set of the idempotent p o i n t s o f (E,o) i s the set o f f l e x e s ; i t i s isomorphic t o AG(t,3) w i t h t<2 (endowed w i t h the mid-point law). L a s t l y f o r any f i x e d u i n E, x,yx*y=uo(xoy) makes E i n t o an abelian group. Let us now consider the case dim V>1. Assume t h a t K i s i n f i n i t e . We have the f o l l o w i n g f a c t t h a t we mention here without a l l t h e required d e f i n i t i o n s ( f o r a more complete account see Manin [9] pp. 46-57, e s p e c i a l l y theorems 13.1 and 13.2): Theorem o f Manin : I f V admits a p o i n t o f "general type", then i n a s u i t a b l e f a c t o r s e t E o f E, the three-place r e l a t i o n o f c l l i n e r i t y gives r i s e t o a symmetric law obeying (aox)o(aoy)=a2o( xoy) and xjox2=xq i d e n t i c a l l y . As a r e l a t i v e l y easy consequence we have :

2

C o r o l l a r y : The square mapping xcf x =p(x) i s an endomorphism. The set o f t h e idempotent elements o f (E,o) i s I=Im p ; i t i s a d i s t r i b u t i v e Steiner quasigroup. A l l the f i b r e s A=p-l(e) o f p are isomorphic elementary abelian 2-groupsY and

(E,o) = . I x A ( d i r e c t product). Let us say t h a t a Fischer qroup i s a group o f the form G=<S> where S i s a conjugacy class o f i n v o l u t i o n s o f G such t h a t O(xy)h3 f o r any two elements x and y from S ( i n other terms the dihedral group generated by any two elements o f S has order ~ 6 ) I. n case we have O(xy)=3 f o r any x,y S, x#y, G i s , say a special Fischer qroup. I n any special Fischer group G t h e r e i s j u s t one class o f i n v o l u t i o n s S (namely, the s e t o f a l l the i n v o l u t i o n s from G ) , and S may be provided w i t h a g t r u c t u r e o f HTS by s e t t i n g xoy=xY=yxy(=xyx). We c a l l ,o) the HTS corresponding t o 6. This group-theoretic construction o f HTSs canonical. More precisely :

(5

i4

Theorem : Given any HTS E, the (non-empty) family 7 o f special Fischer groups whose corresponding HTS i s E admits : (i) a universal o b j e c t U ; any G i n i s o f the form G=U/C where CcZ(U). ( i i ) a smallest object I = U / Z ( U ) , which i s a l s o t h e unique centerless element o f F .

11

Symplectic Geometry, Quasigroups, and Steiner Systems

3-A PROCESS OF EXPLICIT CONSTRUCTION : L e t E be a v e c t o r space over GF(3) w i t h dim E=*+l. P i k u some basis e ,e )...en:entl. Besides choose a non-zero sequence o f eyements from G$(3f, say

(i)

.

) i j k lGi<j
u=(A

i=ntl

Define a binary law i n E by s e t t i n g that, i f x=

. .

xi,yJ6

C

i=l

.

j=n+l x ei and y= C yJe. where j=1 J

GF(3), then :

x 0 Y = -x-Y+( Proposition : (E,ou) i n t h i s manner.

1

l$i<j
i i j k - k j A i j k ( x -Y ) ( x Y x Y 1) en+l

i s a HTS of rank n t l . Any rank n+l HTS o f order 3"'

arises

For n=3, u i s j u s t a non-zero s c a l a r : ?1 ; up t o isomorphism, one gets o n l y one HTS which i s the order 81 d i s t r i b u t i v e Steiner quasigroup discovered by Zassenhaus (see Bol [53). The e q u a l i t i e s : t(e.,e.,ek)

= A.. l s i < j < k s n determine a unique a l t e r n a t e 1 J 1Jk' t r i l i n e a r form t : V3 * GF(3) w i t h V=<e ,e2,. .e >. Since several d i f f e r e n t sequences u'=(Aijk) y i e l d isomorphic H h s , ther@ are several such forms r e l a t e d

.

t o the same HTS (E, o ) . I n f a c t a l l these t r i l i n e a r forms are "equivalent" i n a sense t h a t we are goigg t o precise. 4-THE PROBLEM OF CLASSIFICATION OF ALTERNATE TRILINEAR FORMS : Consider a n-dimensional vect r space V=V(n,K) over a commutative f i e l d K. Recall t h a t a t r i l i n e a r form t : V +K, (x.y,z)t-+ t(x,y,z) i s a l t e r n a t e ( o r : symplecfic, skewt(x,y,z) f o r every permutation IT o f e t r i c ) i f t(a(x),n(ty),a(z))=sign(a) ny two t r i l i n e a r forms t and u are equivalent i f there e x i s t s L i n GL (V) i d e n t i c a l l y . Desig a t e as A(n,K) the f a h i l y such t h a t u(x,y,z)=t(!L(x),a(y),!L(z)) o f the non-vanishing a l t e r n a t e t r i l i n e a r forms o f V and (n,K) the corresponding s e t of equivalence classes. We s h a l l be concerned w i t h the c l a s s i f i c a t i o n o f the t r i l i n e a r forms up t o equivalence i n two special cases : f i r s t when t h e r e i s a t o t a l l y i s o t r o p i c vector space, second when ns6. I n the f i r s t s i t u a t i o n , n can be any i n t e g e r >3 b u t we r e s t r i c t a t t e n t i o n t o t h e forms t h a t vanish i d e n t i c a l l y on M3 where M i s a (n-1)-dimensional subspace. When ns6 we o b t a i n an almost complete c l a s s i f i c a t i o n , and i n f a c t a complete one i n case K=GF(3). Now any r e s u l t concerning the special case K=GF(3) can be i n t e r p r e t e d i n terms o f HTSs i n view o f the following r e s u l t :

-:*

9

.

#

Theorem 4.1 - ( t r a n s l a t i o n theorem) : There i s a one-to-one correspondence b@yeen X(n, GF(3)) and the isomorphism classes o f HTSs o f rank n + l whose order i s 3

.

This t r a n s l a t e s the problem o f the c l a s s i f i c a t i o n o f non a f f i n e HTSs o f maximal rank i n t o an e x t e r i o r algebraic c l a s s i f i c a t i o n problem. We s h a l l study two special cases where the number o f classes can be s p e c i f i e d . As examples l e t us g i v e a couple o f statements c o n s i s t i n g f i r s t o f c l a s s i f i c a t i o n theorems o f symplectic geometry and second o f an a p p l i c a t i o n s t o t h e case K=GF(3) through t h e t r a n s l a t i o n theorem. Theorem 4.2 - For any commutative f i e l d K the elements o f A(n,K), n23, a d m i t t i n g complete equivalence a t o t a l l y i s o t r o p i c codimension 1 subspace form ["l/2] classes. I f n i s odd there i s o n l y one such class f o r which V i s non s i n g u l a r ; i f n i s even t h e r e i s none. C o r o l l a r y - The rank n t l HTSs o f order 3"', 1123, a d m i t t i n g an a f f i n e subsystem o f order 3n form e x a c t l y ["l/2] complete isomorphism classes. I f n i s odd there i s e x a c t l y one such system without non t r i v i a l a f f i n e d i r e c t f a c t o r ; i f n i s even there i s none.

L. Bkniteau

12

For instance ifn=9, t h e r e are 4 rank 10 HTSs o f order 31° a d m i t t i n g an a f f i n e order 39 subspace ; they can be described i n the process o f section 3 by s e t t i n g X123=1y X145=ay X167=B, X189=y and, f o r (i, j . k l 4 {{1,2,31, ~1,4,51, {1,6,7},{1,8,9)f Xijk=Oy where ct,B,y are 0 or 1. The non s i n g u l a r case a r i s e s when one takes

a

a=B=y=l. Theorem 4.3

-

For any commutative f i e l d K, 11(3,K)I=l=llk(4,K)I

and 1%(5,K)I=2.

Besides 1%(6, GF(3))1=5. C o r o l l a r y - Up t isomorphism there are e x a c t l y (i) one rank 4 (resp. 5) HTS o f m r g s p . 3 ) , ( i i ) two rank 6 HTSs o f order 36, and ( i i i ) f i v e rank 7 HTSs o f order 3

.

5l

The l i n k between theorem 4.3 and t h e c o r o l l a r y f o l l o w s from t h e t r a n s l a t i o n theorem. I t ives a quick proof o f (i) and (ii)h a t had been previously established q6,8,1,2,3,4]. The determination o f (6, GF(3)) was obtained i n c o l l a b o r a t i o n w i t h J. Lacaze.

i

The f o l l o w i n g t a b l e gives the number o f isomorphism classes o f HTSs corresponding t o a given 3-order ~ $ 1 3and a given rank ~ $ 8 . x

3 4

5

4

5

7

6

9

8

10

11

1 lJ1/////// 1

1

1

1 2J2

1

4

?

?

?

?

1

J3 ?

?

? ?

12

16-55 1

-

?

/

The l a s t c o r o l l a r y s e t t l e s t h e case (s,p)=(7,7). I t turns out t h a t there are as many HTSs corresponding t o t h i s case as the t o t a l number o f non a f f i n e HTSs w i t h ss6. I n the c e l l s corresponding t o s=p>7 the complete c l a s s i f i c a t i o n seems d i f f i c u l t . But the HTSs w i t h an a f f i n e maximal sub-system are now determined, indecomposable. and f o r s=2nt2=p t h e r e i s j u s t one such system Jn

BIBLIOGRAPHY

C11 L. Beneteau ; Topics about Moufang loops and H a l l t r i p l e systems, Simon Stevin 54 (1980) 107-124. C21 L. Beneteau ; Une classe p a r t i c u l i e r e de matro'ides p a r f a i t s , i n "Combinatorics 79", Annals o f Discrete Math. 8 (1980) 229-232. C31 L. Beneteau ; Les Systemes T r i p l e s de H a l l de dimension 4, European J. Combinatorics (1981) 2, 205-212. [41 L. Beneteau ; H a l l T r i p l e systems and r e l a t e d t o p i c s . Proc. Interna. Conf. on Combinatorial Geometries and t h e i r applications, 1981 ; Annals o f Discrete Math. 18 (1983) 55-60. C51 G. Bol ; Gewebe und Gruppen, Math. Ann. 114 (1937) 414-431 ; Zbl. 16, 226. C61 Marshall HALL Jr. ; Automorphisms o f S t e i n e r T r i p l e systems, IBM J. Res. Develop. (1960), 406-472. MR 23AX1282 ; Zbl. 100, p. 18.

Symplectic Geometry, Quasigroups, and Steiner Systems

C7 Marshall H a l l J r . ; Group theory and block designs, i n Proc. I n t e r n . Conf. on t h e Theory o f Groups, Camberra 1965, Gordon and Breach, New-York. MR 36/2514 ; Zbl. 323.2011. C8 T. Kepka ; D i s t r i b u t i v e E i n e r quasigroups o f order 35, Comment. Math. Univ. Carolinae 19,2 (1978) ; MR 58 X 6032. [ 9 Yu. I. Manin ; Cubics forms, North-Holland Publishing Company, AmsterdamLondon ; Amercian E l s e v i e r Publishing Company, Inc. New-York (1974).

[I0 H.P. Young ; A f f i n e t r i p l e systems and matroid designs, Math. Z. 132 (1973) 343-366. MU. 50 # 142.

13

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Annals of Discrete Mathematics 30(1986) 15-30 0 Elsevier Science Publishers B.V. (North-Holland)

IS

ON A TEST OF D O M I N A N C E ,

A STRATEGIC DECOMPOSITION AND STRUCTURES T ( t , q , r , n )

Walter Benz Mathemtisches S e m i n a r der Universitat Hamburg

1. A t a s k i n psychology or economics i s t o compare d i f f e r e n t g r o u p s g s G1, Gr of o b j e c t s ( p e o p l e , g o o d s ) c o n c e r n i n g a p r o p e r t y E ( o r p r o p e r t i e s E l , ..., E n ) and t o o r d e r them c o n c e r n i n g E s t a r t i n g w i t h t h e dominant g r o u p i n g . T h i s c a n be done by c o n s i d e r i n g i n t e r a c tions ( i . e . r-sets i n t e r s e c t i n g e v e r y g r o u p i n g i n one e l e m e n t ) and by j u d g i n g a l l t h e p o s s i b l e i n t e r a c t i o n s : I f I = [ g l , . . . , g r ) , giEGi, i s an i n t e r a c t i o n one l o o k s f o r an o r d e r i n g g i 1 2 . . t g i r , where 'I >_ s t a n d s for " i s E - b e t t e r t h a n " o r " i s E-equal t o " . Hence e v e r y i n t e r = t i o n l e a d s t o a p o s i t i o n number f o r a c e r t a i n g r o u p i n g and one h a s t o add up t h e s e numbers i n o r d e r t o g e t t h e E - o r d e r i n g f o r a l l t h e groupings.

...,

.

U s u a l l y t h e r e are t o o many i n t e r a c t i o n s , and t h e problem i s t o f i n d a b a l a n c e d s u b c l a s s o f i n t e r a c t i o n s on which t h e judgement s h o u l d b e based. When I became c o n f r o n t e d w i t h t h i s problem i n a s p e c i a l c a s e , my p r o p o s a l w a s t o u s e g e n e r a l i z e d L a g u e r r e g e o m e t r i e s i n c a s e o f one s i n p l e n r o p e r t y E ( a n d g e n e r a l i z e d Minkowski g e o m e t r i e s for s e v e r a l p r q erties E l , . E n ) : T h e p a r a l l e l c l a s s e s o f p o i n t s o f a p l a n e La-guerre geometry may f o r i n s t a n c e r e p r e s e n t t h e g r o u p i n g s , and t h e s e l e c t ed i n t e r a c t i o n s c o u l d be g i v e n by t h e b l o c k s o f t h e geometry. The f a c t t h a t t h r o u g h t h r e e F a i r w i s e non p a r a l l e l p o i n t s t h e r e i s e x a c t l y one b l o c k c o u l d s e r v e as p r o p e r t y o f b a l a n c e c o n c e r n i n g t h e chosen subclass of i n t e r a c t i o n s . A s a matter of f a c t those generalized L a g u e r r e g e o m e t r i e s are s t u d i e d i n t h e l i t e r a t u r e u n d e r d i f f e r e n t names l i k e o r t h o g o n a l a r r a y s (Bush [ 2 ] ) , o p t i m a l g e o m e t r i e s ( % l c k r , & h [4])and t h e y p l a y an i m p o r t a n t r81e i n c o n n e c t i o n w i t h optir,ial c o d e s i n c o d i n g t h e o r y ( H a l d e r , Heise [ 4 1 ) .

..,

I n s e c t i o n 2 w e l i k e t o s o l v e a p r a c t i c a l problem which comes up i n c a r r y i n g o u t a t e s t u n d e r c o n s i d e r a t i o r , : T o f i n d a d i s j o i n t decompos i t i o n of t h e s e t o f b l o c k s o f an o p t i m a l geometry s u c h t h a t a l l t h e components o f t h e d e c o m p o s i t i o n are p a r t i t i o n s o f t h e s e t o f p o i n t s . By u s i n g s u c h a d e c o m p o s i t i o n i n a p r a c t i c a l c a s e one c a n d i v i d e t h e whole t e s t i n a number o f s u b t e s t s s u c h t h a t a l l o b j e c t s are i n v o l v e d i n a subtest. I n s e c t i o n 3 w e l i k e t o d e e l w i t h a simultaneous t e s t o f a s e t o f okj e c t s c o n c e r n i n g p r o p e r t i e s El E n . The c o m b i n a t o r i a l s t r u c t u r e s T ( t , q , r , n ) which we o f f e r i n t h i s c o n n e c t i o n a r e g e n e r a l i z a t i o n s o f

,...,

W.Benz

16

c e r t a i n c h a i n g e o m e t r i e s ( [ l ] ) . The c l a s s o f o p t i m a l g e o m e t r i e s can be i d e n t i f i e d w i t h t h e c l a s s o f T ( t , q , r , l ) . The Minkowski-m-strucm s a r e t h e s t r u c t u r e s T ( t , q , q , 2 ) w i t h t = m+2. - I n Theorem 3 we show t h a t a n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e of T ( t , q , r , n ) is t h a t rn-1 i s a d i v i s o r of q . I n Theorem 4 we d e t e r m i n e t h e number of b l o c k s o f a T ( t , q , r , n ) and a l s o t h e number of t h e s o c a l l e d g l o b a l i n t e r a c t i o n s . I n Theorem 5 , 6 we c h a r a c t e r i z e t h e T ( t , X r n - l , r , n ) X > 1 ) by a p p l y i n g p e r m u t a t i o n s e t s and s p e c i a l c l a s s e s (cases X = l , of f u n c t i o n s . 2 . Let q , r , t be i n t e g e r s s u c h that q > 1 and 2 I t 5 r. C o n s i d e r t h e matrix

where t h e o r d e r e d p a i r s ( i , j ) a r e c a l l e d p o i n t s ( o r o b j e c t s ) , and where we p u t ( i , j ) = ( i l , j l ) i f f i = i l and j = j r . The columns of M a r e a l s o c a l l e d g r o u p i n g s . An i n t e r a c t i o n of M is an r - s e t c o n t a i n i n g one element of e v e r y column. There a r e qr i n t e r a c t i o n s of M . By I ( M ) we denote t h e s e t o f a l l i n t e r a c t i o n s of M . Consider now a s u b s e t B ( t ) of I ( M ) and c a l l t h e i n t e r a c t i o n s o f B ( t ) b l o c k s . We a r e t h e n i n t e r e s t e d i n t h e f o l l o w i n g p r o p e r t y of b a l a n c e S h a v i n g a non-empty i n t e r s e c t i o n w i t h t d i s t i n c t g r o u p i n g s o f M t h e r e is e x a c t l y one b l o c k c o n t a i n i n g S.

(*) To every t - s e t

By T ( t , q , r ) ( o r T ( t , q , r , l ) w i t h r e s p e c t t o s e c t i o n 3 ) we denote a s t r u c t u r e ( M , B ( t ) ) s a t i s f y i n g ( * ) . Many examples of s t r u c t u r e s T ( t , q , r ) f o r c e r t a i n t , q , r and a l s o non e x i s t e n c e s t a t e m e n t s f o r c e r t a i n t , q , r a r e known ( s . f o r i n s t a n c e H a l d e r , Heise [ 4 1 , Heise 151, H e i s e , Karzel [ 6 ] ) . Two s t r u c t u r e s T ( t , q , r ) , T ( t l , q l , r l )a r e c a l l e d isomorphic i f f t h e r e i s a b i j e c t i o n ( c a l l e d isomorphism) o f t h e s e t o f p o i n t s of T ( t , q , r ) o n t o t h e s e t o f p o i n t s o f T ( t l , q B , r t )s u c h t h a t t h e b l o c k s o f t h e first s t r u c t u r e are mapped o n t o b l o c k s o f t h e second s t r u c t u r e . S i n c e two d i s t i n c t p o i n t s of T ( t , q , r ) a r e i n t h e samegmupi n g i f f t h e r e i s no b l o c k j o i n i n g them ( n o t e t L 2 ) isomorphisms map columns o n t o columns. Obviously, isomorphic T ( t , q , r ) , T ( t l , q t , r l ) c o i n c i d e i n t h e p a r a m e t e r s , 1 . e . t = t l ,q = q ' , r = r ' . I n [ l ] we have s t u d i e d c h a i n g e o m e t r i e s , The f i n i t e c h a i n g e o m e t r i e s of Laguerre t y p e ( [ l ] , p . 144) C ( K , L ) are s t r u c t u r e s T ( t , q , r ) . Here K i s a G a l o i s f i e l d GF(Y) and L > K is a f i n i t e l o c a l r i n g w i t h L / N 2 K , where N d e n o t e s t h e maximal i d e a l o f L. The p a r a m e t e r s a r e Ni, r = ~ + 1 . The c l a s s o f c h a i n g e o m e t r i e s of g i v e n by t = 3 4 = != Laguerre type Z ( K , L ) , L = K [ E ] / < ~ ~ , , c o i n c i d e s w i t h t h e c l a s s o f m i q u e l i a n Laguerre p l a n e s . Two Laguerre g e o m e t r i e s C ( K , L ) , C ' ( K ' , L ' ) w i t h c h a r K $: 2 4 char K ' a r e isomorphic i f f t h e r e i s an isomorphism 0 : L -t L ' such t h a t o l K is an isomorphism o f K o n t o K 1 ( [ l ] , p . 176,

17

On a Test of Dominance

S a t 2 3 . 1 ) . Consider a G a l o i s f i e l d G F ( y ) w i t h 2 ,/ y a n d p u t K = G F ( y ) = K , . Let n 2 3 be an i n t e g e r and V be t h e v e c t o r space of dimension n-1 o v e r K. Define l o c a l r i n g s L : = K [ ' ] / < , n , and L' : = I ( k , v ) ( k E ,~v r v 1 with

(kl

vl) + (k2

, v2)

:=

(kl

V1)

.

, v2)

:=

+

(k2

+

Hence L L , because of N n - l 0 , ( N 1 ) 2 = 0 , where r e s p e c t i v e l y N , N ' a r e t h e maximal i d e a l s of L , L , . We t h u s g e t two non isomorphic s t r u c y +, l ) , T I ( 3 , y n - 1 , y + l ) . tures ~ ( 3 , ~ - 1 The s t r a t e g i c decomposition we have announced i n s e c t i o n 1 c o n c e r n s t h e f o l l o w i n g c l a s s of s t r u c t u r e s T ( t , q , r ) which i s a s u b c l a s s o f t h o s e s t r u c t u r e s d e f i n e d i n H a l d e r , Heise [ 4 ] on pages 268, 269 by u s i n g l i n e a r forms. L e t K be a G a l o i s f i e l d GF(Y) and l e t V be a vector space o v e r K w i t h 1 < dim V < For an i n t e g e r t such t h a t 3 2 t < y + 1 now d e f i n e T ( t , # V , u + l ) as follows: The s e t o f p o i n t s i s given by K ' x V w i t h K ' : = K u : - l and t h e b l o c k s a r e given by

-.

By u s i n g Vandermonde's d e t e r m i n a n t i t i s e a s y t o check t h a t t h e p r o p e r t y of balance ( * ) is s a t i s f i e d f o r t . t-1)

Theorem 1: L e t A be t h e G a l o i s f i e l d G F ( y and l e t f E K[x] be t h e minimal polynomial of a p r i m i t i v e element 6 of A o v e r K . Assume V ~ , . . . , V ~ - ~ V . Then t h e s e t B ( v l , v t - 1 ) of b l o c k s

...,

r(a,vf(a)

t-1 t-1-v E v = l vva

+

) l a E

K ) U I(m,v)l

, v

E

V,

i s a p a r t i t i o n of t h e s e t of p o i n t s and

is a d i s j o i n t decomposition o f B ( t ) .

( N o t i c e t h a t t h e degree o f f is

t-1).

P r o o f . Let v,w be two d i s t i n c t elements of V. We l i k e t o show t h a t t h e two b l o c k s t(a,vf(a) + t(r:,wf(6)

+

o f B(v l s . . . , v t - l ) have no p o i n t i n common. Assume t o t h e c o n t r a r y t h a t (C,x),C E K ' , x E V , is a p o i n t i n b o t h b l o c k s . T h i s i m p l i e s 5$ because of v w. Hence

-

W.Benz

18

i . e . f ( 6 ) = 0 which i s n o t t r u e s i n c e f i s i r r e d u c i b l e o v e r K.‘Ihe s e t E(vl,. ,vtdl) c o n t a i n s a s many b l o c k s as t h e r e a r e e l e m e n t s i n V. The number of p o i n t s on a b l o c k i s k K ’ = ~ + 1 .Hence B(v1, vt-3 c o n t a i n s ( y + l ) .# V many p o i n t s and i s t h u s a p a r t i t i o r . o f t h e s e t of points.

..

...,

Now w e l i k e t o show B(vlI..

.

,V

t-1

)

fl

B(wl,.

. ., W t-1

=

8

i n c a s e t h a t t h e two o r d e r e d ( t - 1 ) - p l e t s ( V ~ , . . . , V ~ a r e d i s t i n c t . Assume t o t h e c o n t r a r y t h a t t h e b l o c k s

- ~ ) (,

~ ~ , . . . , w ~ - ~ )

a r e e q u a l . T h i s i m p l i e s v=w and hence t,l

v=l

(wu

-vu)a

t-1-u

= o

for a l l a E K.Because of t < Y + 1 t h e r e e x i s t p a i r w i s e d i s t i n c t e l e ments a l , . . . , a s - l i n K . We hence have i n m a t r i x n o t a t i o n

...,

The Vandermonde m a t r i x P h e r e i s r e g u l a r b e c a u s e of # ( a l , a t:l} = = t-1 and by m u l t i p l y i n g t h e m a t r i x e q u a t i o n w i t h P - 1 from t h e r i g h t w e g e t (wl-vl...wt-l-vt-l) = 0 which i s n o t t r u e . - There are 6 c f V ) t - 1 many s e t s B ( v l , v t - 1 ) . Every B ( v l , . . . , v t - l ) c o n t a i n s #V many b l o c k s . S i n c e t h e r e are qt many b l o c k s i n a s t r u c t u r e T ( t , q , r ) t h e s e t

...,

U

B(vl,

...,v t-1 1

contains a l l the blocks. Example. C o n s i d e r K = GF(3). V = K and t = 3. A r e q u i r e d decomposit i o n h e r e i s g i v e n by

On a Test of Dominance

19

aApP bBqQ cCrR

aBrP bCpQ cAqR

aCqP bArQ cBpR

aArR bBpP cCqQ

aBqR bCrP CAPQ

aCpR bAqP cBrQ

aAqQ bBrR cCpP

aBpQ bC qR cArP

aCrQ bApR cBqP

3. Let q,r,t,nbe integers greater than 1 and such that 2 5 t r. Eet P be a set of cardinality qr. The elements of P are called points ( o r objects). Consider moreover n matrices

such that

...,(r.l,i), ...,(l,q,i), ...,(r,q,i)] f o r all i=l,.,.,n. A global interaction of MI ,...,Mn is a r-set which is an interaction for all the matrices MI, ...Mn. Two points are called competitors if they are not in a common column for all the matrices M1. ...,Mn. By G(M1, ...,Mn) we denote the set of all global interactions of Ml, ...,M , . We are now interested in a set B(t)c G(M1, P

=

t(l,l,i),

...,

Mn) such that the following two conditions are satisfied (the elements of B(t) are called blocks) (i)

Through t distinct points which are parwise competitors there is exactly one block (ii) For every integer j with 1 5 j 5 n the following holds true: If D1 is the point intersection of j distinct columns (of MI, ),+I such that no two of them belong to the same M, and if D2 is another such intersection of j columns then # D1 = fc D2.

.

...,

...,

We denote a structure (MI, Mn; B(t)) by T(t,q,r,n). Conditions (i), (ii) serve as properties of balance. Example. Assume that four firms are offering each a comparable collection of four wines and that four other firms are offering each a comparable collection of four bottles manufactured to be filled up with wine. The question is to test sinultaneously the quality of the wine collections and that of the bottle collections. We like to do this with a T(3,4,4,2):

W.Benz

20

M1

=

M2

(2J.

=

The columns o f M I r e p r e s e n t t h e f o u r wine c o l l e c t i o n s and t h e columns o f M 2 t h e b o t t l e c o l l e c t i o n s . Now t h e b o t t l e c o l l e c t i o n A , q , c , R f o r i n s t a n c e is f i l l e d up i n A , q , c , R w i t h wine of t h e 2 . , 3 . , 1 . , 4 . wine p r o d u c e r r e s p e c t i v e l y . Now c h e c k b o t h q u a l i t i e s a l o n g t h e f o l l o w i n g s e t of b l o c k s : bBqQ Ap aQ rcCQ brAQ qCaQ BcpQ

bqDS cpSD cBPs aAP s pBdR rCdR

rAPd BqPd brRD paRD bAsS ccss

bBRs CaRs CqSd ApSd aqDP rcDP.

( T h i s example is t h e m i q u e l i a n Minkowski p l a n e o f o r d e r 3 . ) Theorem 2 : Let K b e t h e G a l o i s f i e l d G F ( r ) and l e t n > 1 be an i n t e w i t h n f a c t o r s . The c h a i n geomeg e r . Denote by Ln t h e r i n g K x . . . x K t r y Z(K, Ln) is then a s t r u c t u r e T ( 3 , ( u + l ) n - l I ~ + l , n ) . P r o o f . a ) C o n s i d e r t h e f o l l o w i n g maximal i d e a l s

J.

:=

{(kl,

...,kn)

of Ln f o r i = l , . . . , n . A p o i n t of R ( p l , P,)

:=

E

Ln

1 ki

= 01

E(K,Ln) i s g i v e n by

{(rpl, rp2)

I r

E

R},

where R d e n o t e s t h e group of u n i t s of Ln and where p l r p 2 a r e e l e m e n t s o f Ln s u c h t h a t t h e i d e a l g e n e r a t e d by p l , p 2 is t h e whole r i n g Ln. For t w o p o i n t s P = R(p1,p2) Q = R(q1,q2) w e d e f i n e

b ) The p o i n t s R ( a , b ) w i t h a = ( a l , . . . , a n ) , b = ( b l l a . . , b n ) K(an,bn)) can be i d e n t i f i e d w i t h t h e o r d e r e d n - p l e t s ( K ( a l l b l ) , of p o i n t s K ( a i , b i ) o f t h e p r o j e c t i v e l i n e n o v e r K . Moreover: R ( a , b ) I l i R ( c , d ) i f f K ( a i , b i ) = K ( c i , d i ) . Hence11 i i s an e q u i v a l e n c e r e l a t i o n on t h e s e t o f p o i n t s and t h e r e are ( y + l ) " - l many o r d e r e d ,Pn), Pl,..,,Pn e n , s u c h t h a t P i = c o n s t . Thus t h e n-plets (PI, is (y +l)n-I. number of p o i n t s i n an e q u i v a l e n c e c l a s s c o n c e r n i n g 1 1 The number o f e q u i v a l e n c e c l a s s e s c o n c e r n i n g I l i is u + l s i n c e t h e r e a r e r+l p o i n t s K(ai,bi) i n n , n-1 c l We now d e f i n e t h e m a t r i x M i ( i { l l . . . l n l ) with (y+l) rows and y t 1 c01um.s. T k matrix Mi can be chosen a r b i t r a r i l y up t o t h e f a c t t h a t t h e columns a r e supposed t o b e t h e 1 1 - e q u i v a l e n c e c l a s s e s .

...,

...

i

21

On a Test of Dominance d ) Two p o i n t s R ( a 1 , a 2 ) , R ( b l r b 2 ) a r e o b v i o u s l y c o m p e t i t o r s

iff

- I n c h a i n geometry [ l ] i t i s proved t h a t t h r o u g h p o i n t s A , B , C such t h a t

lal a 2 1 c R .

three b1 b2

t h e r e i s e x a c t l y one c h a i n and t h a t f o r two d i s t i n c t p o i n t s P . Q t h e element o f R . Chains a r e hence g l o b a l

...&.

( n o t e t h a t any c h a i n c o n t a i n s u + l points)ofM,, i n t e r a c tions Define t h e s e t o f c h a i n s t o be t h e s e t B ( 3 ) . Then ( i ) h o l d s t r u e f o r t = 3 . I n o r d e r t o v e r i f y ( i i ) l e t j be an i n t e g e r w i t h 1 2 j 5 n and l e t i l , . , i j ~ t l , . , n i be j d i s t i n c t i n t e g e r s . Consider e q u i v a l e n c e E ( i j ) of the relations ( ( i l , IIij r e s p e c t i v e l y . Then c l a s s e s E(il),

..

..

...,

...,

...,

i j in (Pi, f o r j = l , ...,n . For i f we f i x t h e components i l , 6 n , t h e n t h e number of t h e remaining n - p l e t s i s (Y + l ) n - j .

...,P n ) ,

Pi

Remark: A b i j e c t i o n o f t h e s e t o f p o i n t s o f a s t r u c t u r e T ( t , q , r , n ) is c a l l e d an automorphism i f f images and i n v e r s e images o f b l o c k s a r e b l o c k s . A s a s p e c i a l c a s e of a theorem of S c h a e f f e r [ l o ] t h e automorphism group o f Z ( K , L n ) i s known for Ln s e m i l o c a l , #K > 3 and char K#2(inmse t h a t K i s f i n i t e , o b v i o u s l y , Ln must be s e m i l o c a l ) : T h i s i s t h e group p r L ~ ( 2 . L ~ ) -. TheZ(K,L2) are the m i q u e l i a n Minkowski p l a n e s , K an a r b i t r a r y f i e l d . Theorem 3: Consider a s t r u c t u r e T ( t , q , r , n ) . Then rn-' must be a d i v i s o r o f q . Moreover: If D i s t h e p o i n t i n t e r s e c t i o n of j ( l 5 j 5 n ) M n ) such t h a t no two of them b e l o n g t o d i s t i n c t columns ( o f M1, t h e same M V t h e n

...,

P r o o f : The formula i s t r u e f o r j=1. Assume now 2 2 j 2 n. Let E, be a column o f M, f o r v = l , . . . , j - 1 and l e t C1, Cr be t h e columns o f M j : Observe t h a t t h e C 1 , C r a r e p a i r w i s e d i s j o i n t and t h a t t h e i r union i s t h e whole s e t of p o i n t s . Hence

...,

and t h u s a

j-1

=

r

...,

a.. T h i s p r o v e s t h e theorem.

J

A t t h e b e g i n n i n g of s e c t i o n 3 we r e q u i r e n > l f o r a s t r u c t u r e T ( t , q , r , n ) . But o b v i o u s l y t h e s t r u c t u r e s T ( t , q , r ) of s e c t i o n 2 c a n be c o n s i d e r e d

a s s t r u c t u r e s T ( t , q , r , l ) , s i n c e ( i i ) p l a y s no r61e i n c a s e n=l. The s t r u c t u r e s T ( t , q , r , Z ) have been s t u d i e d e x t e n s i v e l y i n t h e l i t e r a t u r e i n c a s e q = r . See f o r i n s t a n c e t h e r e s u l t s i n C e c c h e r i n i [ 3 1 ,

W.Benz

22

Heise, Karzel [ 61, Heise, Quattrocchi [ 71 , Quattrocchi [ 81 , where Minkowski-m-structures are considered. Concerning the real case z ( I R , IR x IR x IR) compare Samaga [ 9 ] . Theorem 4: Let (M1,

...,Mn;

B(t)) be a structure T(t,q,r,n). Then

and the cardinality of the set of all global interactions of Mn is given by MI,

...,

Proof. If b is a point, denote by [b]i the column of Mi through b. Ct of M1.We now like to define Consider t distinct columns C1, sets D1, Dt.Those sets (but not their cardinalities) will depend on certain points al,a2,. Put D1 := C1. In case Du(l <- v ft-1) is defined, take a E D and put

...,

...,

.. .

v

Dv+l

v

.- %+1 \ p n=u 2

([allp u

* -

...u

[ a 1 1. V

P

We then have

ff B(t)

=

t

,,ilff

Dv)

. ..,bt),

since the number of blocks must be equal to the number of (bl, bi E Ci, such that the b's are pairwise competitors. Put

for

V E ( ~ , . .

.,ti and

...,nl . For

I.IE{~,

4

I J ~we

get

and hence by Theorem 3

since the sumands of the right hand side of joint. Sirnilary we have

for pairwise distinct

p 1'

- *

Dv =

9

...,n},

P s in 12,

cs

( 0 ) are

n p:2

BvP

pairwise dis-

Because of

On a Test of Dominance

=

U - 1 n-1

q . ( l - -) r

for

V=

(This especially implies that D

ff B ( t )

= q.q(l-

2,

23

. .. , t .

f 0.) Hence

1 n-1

7)

-..q(l-

t-i n-1

-) r

I n o r d e r t o g e t t h e c a r d i n a l i t y of t h e s e t of a l l g l o b a l i n t e r a c t i o n s w e determine

with K Remarks: 1) I n case z(K,L,) number o f b l o c k s by Theorem 4

ff B(3) = (6*('i1)

=

=

GF(Y) and. n > 1 w e g e t f o r t h e

3 n-1 ( Y -Y)

.

The number o f g l o b a l i n t e r a c t i o n s i n t h i s c a s e i s g i v e n by

[

(y

+1)!

2 ) If a l , a 2 a r e c o m p e t i t o r s o f a s t r u c t u r e T ( t , q , r , n ) t h e n a c c o r d i n g t o t h e c o n s t r u c t i o n i n t h e p r o o f o f Theorem 4 t h e r e is a b l o c k ( n o t i c e t 1_ 2 ) c o n t a i n i n g a l , a 2 . On t h e o t h e r hand: Two d i s t i n c t p o i n t s o f a b l o c k must be c o m p e t i t o r s . We t h u s have: Two d i s t i n c t p o i n t s a r e c o m p e t i t o r s i f f t h e r e i s a b l o c k j o i n b i g them.

n- 1 I n Theorem 2 we have p r e s e n t e d s t r u c t u r e s T ( 3 , ( y + l ) ,~+l,n), where n > 1 i s an i n t e g e r and where Y is a prime power. W e now l i k e t o define further structures T(t,q,r,n). A.

C o n s i d e r n m a t r i c e s MI,

...,Mn

a l l w i t h q > 1 rows and r > 1 columns P = qr, s u c h that p r o p e r t y

and a l l f i l l e d up w i t h t h e e l e m e n t s of P,#

W.Benz

24

(ii) is satisfied. Taking then the set B(r) of all global interactims VE get a T(r,q,r,n),

Let H = [hl,.,.,hrlbe a set containing r 1. 2 elements. Define P to be the Cartesian product

...

P = H x x H with n factors for a given integer n > 1. In order to define matrices Mi, i E 11,. ,nl , put

..

:= {(xl, ".,X

'iv

n

1

E

p

IXi =

hv)

for the columns Cil,...,Cir of Mi. Then property (ii) is satisfied for the matrices Ml, Mn.- Let t be an integer with 2 5 t 5 r and let r2, rn be sets of permutations of H such that every ri is sharply t-transitive on H. We then like to define a set B(t) of glcbMn : For given permutations n E r , a1 interactions for MI, P P p = 2,. ..,n, call

...,

...,

...,

+

a block, i.e. an element of B(t). Since hl h2 implies TI v (h1 ) = T Iu (hd ( W = 2,...,n) we are sure that two distinct elements of b(v2 , . . . ,nn) are competitors. Consider now t distinct points (hWrx2V,...,xr,y), V= l,...,t, such that no two of them are competitors. There is exactly one permutation TI,,E~,, with r , , (h,) = x,,, for v = l , t. There ,vn), joining the t points. Hence is thus exactly one block, b(n2, we get a T(t,rn-l, r,n). The Mathieu groups M ,M12 for instance lead to structures T(4,11n-l , 11,n) (put r2 = = Mll) and T(5,12"-l, 12,n) for every n > 1.- In case that H is the projective = r n is its projective group we get line over K = GF(r) and r2 = z(K,Ln) up to an isomorphism.

...,

...

...

...

Theorem 5 : Every structures T(t,rn-',r,n), n > 1, can be described on H as was done before by an r-set H and permutation sets r2,,..,rn which are sharply t-transitive on H.

,...,

Proof. Let H : = {hl bribe a block of T(t,rn-l,r,n) and let x be an arbitrary point. Put xi := hV iff x,h are in the same column of Mi. We like to show that x is completely'determined by (XI, xn) and that every (yl,...,yn\yi E H , occurs as a point: This is a consequence of the fact that n columns, no two of them in the same matrix,

...,

intersect in

a

n

+1 = .n-l

=

,...,xpn 1 ,

(xpl

1 points. Consider now a block P=l,...,r,

of ~(t,r~-l,r,n).

Then for a given j c i 2 , ,..,nl TI

j

(x

p1

) E X p j

( P = I , . ..,r)

is a permutation of H. Define r j t o be the set of all ' j stemming from

25

On a Test of Dominance

b l o c k s , j = 2 , . . . , n . I t i s rj3 f o r e v e r y joc{2,..,,rd s h a r p l y t - t r a n s i of H and t i v e on H: Consider t d i s t i n c t elements y11, moreover t d i s t i n c t e l e m e n t s y l j o , . . . , y t j o of H . L e t ( Xpl,...'X

pn

),

P=l,

...,t ,

be t d i s t i n c t p o i n t s such t h a t no two o f them a r e c o m p e t i t o r s and such t h a t x

Pl - Y P 1 '

for

= Y pjo

..

~=l,, ,t,

p j 0

Since there i s e x a c t l y one j o i n i n g b l o c k t h e r e must be e x a c t l y one njo fi0 such t h a t

for

p=l,

...,t .

as w a s d e s c r i b e d a t t h e b e g i n n i n g B. Let u s have m a t r i c e s M 1 , . . . , M n of A . Suppose n > 1. Then a l s o Mi,.. .,Mn-1 s a t i s f y ( i i ) and a s t r u c t u r e T ( r , q , r , n ) thus leads t o a s t r u c t u r e T ( r , q , r , n - l ) . C . D e r i v a t i o n p r o c e s s . Consider a s t r u c t u r e T ( t , q , r , n ) s u c h t h a t t, 3 and r 2 3. Let MI, Mn be t h e u n d e r l y i n g m a t r i c e s , P be t h e p o i n t s e t and B ( t ) be t h e s e t o f b l o c k s . Take a p o i n t p E P . I n e v e r y m a t r i x M u c a n c e l t h e column [ p l V through p g e t t i n g t h e m a t r i x M:

...,

.

Denote

...

PI1

by 4

4

[PIn

and c a n c e l i n every remaining column o f M1,

. Define

Then

Mi,

...,M A

...,Mn

the points of

together w i t h B'(t-I)

:=

E \ [PI

I

B

E

is a structure T(t-l,q',r-l,n) w i t h q' =

Bp(t)I

9* .n- 1

n- 1 (r-1)

.

Remark. S t a r t i n g w i t h t h e s t r u c t u r e T ( 3 , ( ~ + 1 ) " - ~ , r + l , n ) n > 1, of Theorem 2 , we g e t a s t r u c t u r e T ( 2 , v n Y 1 , Y , n ) by t h e d e r i v a t i o n proce s s . T h i s s t r u c t u r e can be d e s c r i b e d as follows: Take t h e K n , K = = GF(Y), and d e f i n e the columns of M i t o be t h e h y p e r p l a n e s X i = = const ( i = l , n ) . The b l o c k s a r e given by t h e l i n e s

...,

D.

For i n t e g e r s t , X , r such t h a t 2

5

t <_ r

and X 2 2 we denote by

W.Benz

26

o(t,x,r) a set of functiom

’$’:

tl, . . . , rl * t l ,

..., A }

satisfying the following condition:

...,A 1 ...,t.

{l, v=l, E

...,

...,

i t € (1,. . . ,rl and f o r given J1, Jt there is exactly one YC @ such that %i,) = j, for

( * ) For given distinct il,

Examples: 1)

o(2,2,3):

1

2

3

4

Remark. Consider a class of functions @(t,h,r) satisfying ( * ) . Then obviously

with the blocks b(9) : = {(i,q(i))li

E

11

,...,1-11

,

Y E o(t,A,r),

is a T(t,A,r,l).- On the other hand consider a T(t,h,r,l). Define to every block

27

On a Test of Dominance

...,

the function : {l,,..,r) U, X I byVb(i) = ji (i=1 , . . . , r ) . Then it is clear that the class of functions vb satisfies ( * ) . +

Let H = {hl,...,hr1 be a set containing r 2 2 elements. Consider integers t,A,n greater than 1 such that 2 5 t 5 r . Let r2, ...,rn be permutation sets which are sharply t-transitive on H and let @(t,X,r) be a class of functions as defined above. We then like to define a structure T(t,Arn-I,r,n). The points are the ordered (n+l)-plets (XI,...,xn,w) such that XI,.. .,xnE H and u E A :={l,...,XI In order to define matrices Mi, i c {1,2, n), put

.

...,

...,

for the columns Cil, Cir of Mi. Then property (ii) is satisfied for the matrices MI,...,Mn. The set B(t) of blocks is defined as follows: F a r given permutations n p c T p , ~ = 2 , n, and f o r a given Q E @ call

...,

a block, i.e. an element of B(t). This construction leads to a T(t,Xn-l,r,n). n- 1 Theorem 6. Every structure T(t,Xr ,r,n) n > 1, A > 1, can be described as was done before by an r-set H, by permutation sets r2,...,Tn on H which are sharply t-transitive on H and by a function set @(t,A. r ) satisfying ( * ) .

-

Proof. F o r given points a,b write a b iff they are in the same colMp. This is an equivalence relation and since umns of M1, # (C1n n C ,) = X,Ci a column of Mi (i=l,...,n) the equivalence hrl be a block classes contain exactly X points. Let H := {hi, of T(t,AP-l,r,n) and let x be an arbitrary point. Put Xi :=hw iff x,hV are in the same column of Mi. The n-plet (XI,...,xn) does not determine the point x. But there are X points equivalent to x. We call them (XI, xn,u), u = 1 , A We construct the permutation sets r 2 , rn the same way it was done in the proof of Theorem 5. To every block

...

...,

...,

...,

...,

...,

.

...,

d * of T(tlAp-l,rtn)we like to associate a function Cp : (1, {l,,..,X}: Put Cp(i) = v p in case xpl = hi. Call 4(t,X,r) the set of it all such functions stemming from blocks. Given now distinct il, E {l, 1-1 and elements jl,...,jt E {I, XI. Let then

...,

...

...,

be t distinct points such that no two of them are competitors. Since there is exactly one joining block there must be hence exactly one function q i n @(t,X,r) such that 9(il) = jl,...,T(it) = jt.

W.Benz

28

R e m a r k s . 1 ) With t h e examples @ ( 2 2 , 3 ) ,4 ( 3 , 2 , 4 ) one c a n c o n s t r u c t s t r u c t u r e s T ( 2 , 2 ~ 3 " - ~ , 3 , n ) ( p u t r2 = = r n =: S3) , = r n =: S 4 ) i n case n > 1. T(3,2.4"-l,4,n) (put r2 =

...

...

2) Because of the comection o f f u n c t i o n c l a s s e s O ( t , X , r ) and g e o m e t r i e s T ( t , X , r , l ) and b e c a u s e o f t h e n o n - e x i s t e n c e o f T ( 3 , X , A + 2 ) , X odd, ( s . H e i s e [ 5 ] ) , t h e r e do n o t e x i s t T ( 3 , A . ( X + 2 ) " - l , A + 2 , n ) f o r A odd a n d n E N a c c o r d i n g t o Theorem 6 . 3 ) By a p p l y i n g t h e d e r i v a t i o n p r o c e s s i t i s e a s y t o v e r i f y t h a t t h e r e do n o t e x i s t T ( t , A , r ) i n c a s e A 2 < ( A - l ) ( r - t + 2 ) . ( F o r i n s t a n c e t h e r e d o e s n o t e x i s t a T ( 3 , 1 0 , 1 3 ) ) . T h i s i m p l i e s by Theorem 6 t h a t t h e r e do n o t e x i s t T ( t , A r n - I , r , n ) i n c a s e nE N and A2 < (A-l)(r-t+2). n- 1 4 ) A s f a r as t h e number o f b l o c k s o f a T ( t , A r , r , n ) is c o n c e r n e d , Theorem 5 , 6 l e a d t o a new p r o o f o f Theorem 4 : A s h a r p l y (r-t+r) = t - t r a n s i t i v e p e r m u t a t i o n s e t on a r - s e t c o n t a i n s r . ( r - I ) = t ! ( r ) many e l e m e n t s . Hence

...

f

b e c a u s e of # O ( t , X,r) = A t . T h i s remark d o e s n o t c o n c e r n t h e number of g l o b a l i n t e r a c t i o n s o f a r b i t r a r y M1,,..,Mn s a t i s f y i n g ( i i ) which i s d e t e r m i n e d by t h e p r o o f of Theorem 4.

References

W.

Benz, V o r l e s u n g e n iiber Geometrie d e r A l g e b r e n . S p r i n g e r - V e r l a g ,

Berlin-New York 1973. K . A . Bush, O r t h o g o n a l a r r a y s o f i n d e x u n i t y . Ann. Math. S t a t . 23 ( 1 9 5 2 ) , 426-434. P.V. C e c c h e r i n i , Alcune o s s e r v a z i o n i s u l l a t e o r i a d e l l e r e t i . Rend. Acc. Naz. L i n c e i , 4 0 ( 1 9 6 6 ) , 218-221.

H.R. H a l d e r , W. H e i s e , K o m b i n a t o r i k . H a n s e r V e r l a g , Munchen Wien 1976.

-

W.

H e i s e , E s g i b t k e i n e n o p t i m a l e n (n+2,3)-Code e i n e r ungeraden Ordnung n . Math. Z . 164 ( 1 9 7 8 ) , 67-68.

W.

Heise, H. K a r z e l , L a g u e r r e und Minkowski-m-Strukturen.

Rend.

1st. Mat. Univ. T r i e s t e I V ( 1 9 7 2 ) . W. H e i s e , P . Q u a t t r o c c h i , S u r v e y on S h a r p l y k - T r a n s i t i v e S e t s o f P e r m u t a t i o n s and Minkowski-m-Structures. A t t i Sem. Mat. F i s . Univ. Modena 27 ( 1 9 7 8 ) , 51-57.

t8l

P . Q u a t t r o c c h i , O n a t h e o r e m of P e d r i n i c o n c e r n i n g t h e non-exis t e n c e o f c e r t a i n f i n i t e M i n k o w s k i - m - s t r u c t u r e s . J o u r n . Geom. 13 ( 1 9 7 9 1 , 108-112.

On a Test of Dominance [ 91 H. -J. Smaga, Dreidimensionale reelle Kettengeometrien. Journ. Geom. 8 (1976), 61-73.

[ l o ] H. Schaeffer, Das von Staudtsche Theorem in der Geometrie der Algebren. J. reine angew. Math. 267 (1974), 133-142.

29

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Annals of Discrete Mathematics 30 (1986) 3 1-38 0 Elsevier Science Publishers B.V. (North-Holland)

ON n-FOLD

31

B L O C K I N G SETS

Albrecht Beutelspacher and Franco Eugeni Fachbereich

Mathematik der Universitat Mainz F e d e r a l R e p u b l i c o f Germany I s t i t u t o Matematica Applicata Facolta' Ingegneria L'Aquila , I t a l i a

An n - f o l d b l o c k i n g s e t i s a s e t o f n - d i s j o i n t b l o c k i n g s e t s . We s h a l l p r o v e u p p e r a n d l o w e r bounds f o r t h e number o f c o m p o n e n t s i n a n nfold blocking set i n p r o j e c t i v e and affine spaces. INTRODUCTION A b l o c k i n g s e t o f a n i n c i d e n c e s t r u c t u r e ,a=( P , . y , I ) i s a s e t B o f p o i n t s s u c h t h a t any element of 9 ' ( a n y " l i n e " o r "block") c o n t a i n s a p o i n t o f B a n d a p o i n t o f f B . An n - f d d b l o c k i n g s e t o f .a i s a s e t o f n m u t u a l l y d i s j o i n t b l o c k i n g sets of P . Any B = { B , ,Bz B, s e t B , i s s a i d t o ~e a c o m p o n e n t o f B While blocking sets have b e e n s t u d i e d f o r a l o n g t i m e ( c f . f o r i s t a n c e [ l ] , 161, [ 1 2 ] ,1151, [ 1 7 ] , [ 1 8 ] ) , t h e r e a r e n o t many p a p e r s d e a l i n g w i t h n - f o l d b l o c k i n g s e t s .

,..., 1

.

G e n e r a l i z i n g a t h e o r e m o f H a r a r y [ 9 ] ( w h i c h w a s a l r e a d y k n o w n t o Von Newmann a n d M o r g e n s t e r n [ 1 9 ] ) , K a b e l l [ 1 4 ] r e c e n t l y , p r o v e d t h e f o l l o w i n g a s s e r t i o n . (We s h a l l u s e o u r a b o v e t e r m i n o l o g y ) .

RESULT.

I f a p r o j e c t i v e p l a n e of o r d e r q h a s a n n - f o l d b l o c k i n g set, t h e n n s q - I . Any n - f o l d b l o c k i n g s e t o f a n a f f i n e p l a n e o f order q s a t i s f i e s n 5 q-2.

I n S e c t i o n 2 we s h a l l p r o v e a t h e o r e m w h i c h u n i f i e s , g e n e r a l i z e s a n d sets i n i m p r o v e s t h i s r e s u l t . Later o n , w e s h a l l c o n s i d e r b l o c k i n g p r o j e c t i v e and a f f i n e spaces. Let B be a projective or affine any t-dimensional space. A set B of 2 i s s a i d a t-blocking set i f I-B. subspace of H c o n t a i n s a t l e a s t a p o i n t of B and a p o i n t of A s e t B = { B, B,} of n mutually disjoint t-blocking sets is s a i d an n - r x d t-blocking set of P

,...,

.

I n S e c t i o n 3 w e s h a l l d e a l w i t h t h e maximal number n of components of an n-fold t-blocking s e t i n PG(r,q) or AG(r,q). We shall prove u p p e r a n d lower b o u n d s f o r t h i s m a x i m a l number. I n S e c t i o n 4 we s h a l l c o n s t r u c t e x a m p l e s o f n - f o l d b l o c k i n g s e t s . I n prove the following fact:Given a positive particular, we shall i n t e g e r n , t h e r e i s a n i n t e g e r q,, s u c h t h a t a n y p r o j e c t i v e o r a f f i n e p l a n e o f o r d e r q zqo h a s a n n - f o l d b l o c k i n g s e t .

We w a n t t o r e m a r k t h a t we u s e t h e w o r d " n - f o l d b l o c k i n g l i g h t l y d i f f e r e n t m e a n i n g a s H i l l a n d Mason [ l o ] ,

set"

in

a

A . Beutelspacher and F. Eugeni

32

2. B L O C K I N G SETS IN STEINER SYSTEMS. We b e g i n w i t h t h e f o l l o w i n g 2 . 1 THEOREM, L e t S b e a n S ( 2 , k , v ) n-fold blocking set, then n s k - 2 .

S t e i n e r system. I f

S

admits

an

,...,

a n n-fold b l o c k i n g set of S .Consider PROOF. D e n o t e b y { B, B,} a p o i n t x o u t s i d e a c l a s s e Bi , S i n c e a n y l i n e t h r o u g h x i s i n c i d e n t with a t l e a s t one point of Bi , i t r e s u l t s IB,I 5 r , where i s t h e n u m b e r o f l i n e s t h r o u g h x . S u p p o s e we h a v e I B i l r=(v-l)/(k-1) = r . T h e n a n y l i n e t h r o u g h a p o i n t x o u t s i d e Bi meets B, i n j u s t one p o i n t . I n o t h e r w o r d s , a n y l i n e j o i n i n g t w o p o i n t s o f Bi i s t o t a l l y cannot c o n t a i n e d i n B i . S i n c e B i h a s a t l e a s t t w o p o i n t s , {B, , , . , , B n } .,n I b e a n n - f o l d b l o c k i n g s e t , S o , IBiI 2 r + l f o r any 1 E 1 , 2 , ( T h i s i s a l s o a c o n s e q u e n c e o f Theorem 1 i n 171). T h e r e f o r e ,

..

.

n

v =

E I Bz, I( r + l ) n . i -1

On t h e o t h e r h a n d , we h a v e v - 1

= r(k-1).

Toghether we get

n(r+l)sv=r(k-l)+l Hence n s k - l - ( k - Z ) / ( r + l ) <

k-1

a n d so n s k - 2 .

with EXAMPLES. T h e r e e x i s t : n - f o l d b l o c k i n g s e t s i n some S ( 2 , k , v ) n=k-2: The p r o j e c t i v e p l a n e o f o r d e r 3 a n d t h e a f f i n e p l a n e of o r d e r 4 have blocking sets; t h e s e b l o c k i n g sets form, t o g e t h e r w i t h their r e s p e c t i v e complements, 2-fold blocking sets. Also, the projective p l a n e o f o r d e r 4 a d m i t s a p a r t i t i o n i n 3 Baer s u b p l a n e s . T h i s i s a 3-fold b l o c k i n g set. We r e m a r k a l s o t h a t n o S ( 2 , 4 , v ) S t e i n e r s y s t e m h a s a 3 - f o l d b l o c k i n g s e t . (Assume t o t h e c o n t r a r y t h a t a n S ( 2 , 4 , v ) h a s a 3 - f o l d blocking s e t ( B , ,B, , B 3 1 . S i n c e a n y Bi i s a b l o c k i n g s e t a n d s i n c e by [18, (2.13)] any b l o c k i n g set i n S ( 2 , 4 , v ) h e s a t l e a s t (v- h ) / 2 points, we h a v e 3 ( v - f i ) / 2 r v , i . e . v 5 9 , a c o n t r a d i c t i o n . ) A similar argum e n t a t i o n h o l d s i n S ( 3 , 4 , v ) s i n c e ( c f . 118)) i f a b l o c k i n g s e t there e x i s t s then it c o n t a i n s e x a c t l y v/2 points.

Now, w e c o n s i d e r p r o j e c t i v e p l a n e s . T h e f o l l o w i n g r e s u l t i s s t a n t i a l i m p r o v e m e n t of K a b e l l ' s r e s u l t .

a

sub-

projective plane of o r d e r q. and l e t THEOREM. D e n o t e b y p a , ,B, } be a n n - f o l d b l o c k i n g s e t of P .Then n s q - S p + l w i t h e q u a l i t y i f a n d o n l y i f a n y Bi i s a B a e r s u b p l a n e . 2.2

B = { B,

. ..

PROOF. Any c l a s s Bi E B i s a b l o c k i n g s e t i n t h e u s u a l s e n s e . H e n c e , t h e t h e o r e m o f B r u e n [ 6 ] i m p l i e s lBil t q + S q + l . Hence n ( q t f i t 1 ) s q*+q+l = (q+G+l)(q-sq+l) t h e n any i m p l y i n g t h e i n e q u a l i t y o f a s s e r t i o n . If n = q - @ + l , p r e c i s e l y q+\rq+l p o i n t s . Again u s i n g B r u e n ' s r e s u l t , Bi is subplane. The other d i r e c t i o n is t r i v i a l ,

Bi a

has Baer

We r e m a r k t h a t a n y c y c l i c ( s o , i n p a r t i c u l a r , a n y d e s a r g u e s i a n ) p r o j e c t i v e p l a n e of s q u a r e o r d e r h a s a p a r t i t i o n i n B a e r s u b p l a n e s ( s e e H i r s c h f e l d 1111 4 . 3 . 6 ) . S o , t h e b o u n d i n T h e o r e m 2 . 2 i s s h a r p ,

On n-Fold Blocking Sets

33

3 . THE PROJECTIVE A N D A F F I N E CASE Let 2 b e a n r - d i m e n s i o n a l p r o j e c t i v e s p a c e of order q. In this S e c t i o n , we a r e i n t e r e s t e d i n t h e q u e s t i o n , how m a n y components an n - f o l d t - b l o c k i n g set o f H c a n h a v e . C l e a r l y , t h e e x i s t e n c e of an n-fold t-blocking set i m p l i e s t h e e x i s t e n c e of a n m-fold blocking s e t f o r a n y m s n . T h e r e f o r e , we m a y d e f i n e n p = n p ( t , r . I ) a s t h e greatest integer with the property t h a t L h a s an n-fold t-blocking set f o r any n s n p ( t , r , 2 ) . I f H is desarguesian of order q, then we w r i t e a l s o 5 ( t , r , q ) i n s t e a d o f n p ( t , r , 2). S i m i l a r l y , t h e f u n c t i o n s na ( t , r , A ) a n d n , ( t , r , q ) f o r a n a f f i n e s p a c e A of dimension r and o r d e r q are d e f i n e d . I n 2.2. Theorem w e h a v e a l r e a d y shown t h a t n p ( 1 , 2 , q ) < q - q + I . Now we s h a l l d e a l w i t h t h e h i g h e r d i m e n s i o n a l case. F i r s t , w e s h a l l s t a t e s o m e e a s y - t o - p r o v e u p p e r b o u n d s f o r n p . By ?(q)=q'+ 1 we denote t h e number o f p o i n t s i n P G ( r , q ) .

...+

o n i y i f rPROOF. Let B = ( B , ( a ) B Y ("2 , 2 . 1 1 ) ,

,...,B n }

be a n n - f o l d t - b l o c k i n g s e t i n P G ( r , q ) . any t-blocking set Bi i n PG(r,q) s a t i s f i e s

Theref ore,

,r I n

fir(q)

2

I

-I

IBi

2

n

[ i + r - t ~ ) + ~i + r - t - l ( q ) ]

*

( b ) S i n c e r < 2 t , by [41 , a n y t - b l o c k i n g s e t B , i n PG(r,q) has a t l e a s t a,-,(q) . p o i n t s , e q u a l i t y h o l d s i f a n d o n l y i f B, i s t h e p o i n t s e t of a n ( r - t ) - d i m e n s i o n a l s u b s p a c e . T h e r e f o r e : n

I f e q u a l i t y h o l d s , t h e n .9. ( 9 ) d i v i d e s r - t - l l r + l , and so r-t+llt.r-t

a r ( q ) , which

implies

that

REMARK. S u p p o s e r < 2 t a n d r - t + l l t . T h e n i n P G ( r , q ) a t o t a l (r-t)s p r e a d ( s e e [8] ) i s a n n - f o l d t - b l o c k i n g s e t w i t h n = # ( q ) / 6 ( q ) . r-t r Similarly,

t h e f i r s t a s s e r t i o n of t h e following theorem follows.

( a ) I f H ( t , r , q ) d e n o t e s t h e maximal c a r d i n a l i t y of a i n PG(r.q), then n p ( t , r , q ) z M(r-t,r,q). Put r=a(r-ttl)tb, where a and b are i n t e g e r s y i t h a > O and b 5 r - t - 1. T h e n

THEOREM.

t i a l t-spread

i=l

PROOF. By ( 1 3 1 , T h e o r . 4 . 2 ) , t h e r e with the c a r d i n a l i t y i n question.

exists

a

partial

(r-t)-spread

Now, we c o n s i d e r t h e c a s e r = 2 t , . 3 . 3 T h e o r e m . D e n o t e by P=PG(Pt,q) t h e p r o j e c t i v e s p a c e of dimens i o n 2 t 2 4 a n d o r d e r q. S u p p o s e t h a t q i s a s q u a r e . D e f i n e t h e p o s i t i v e i n t e g e r s by s = [ t + 2 / 2 ] Then

.

A . Beutelspacher and F. Eugeni

34

nP ( t * 2 t s q )

2

- h qt-s.

#*2(t--s+l)

PROOF. L e t B b e a s u b s p a c e o f d i m e n s i o n 2 ( t - s t l ) o f P. ( N o t e t h a t i n view of t L 2 , t h e d e f i n i t i o n of s i m p l i e s t - s & O . ) D e n o t e by R a R is a subspace of dimension 2 t - 2 ( t - s t l ) - l c o m p l e m e n t - o f B i n €'.Then = 2s-3. L e t B,, B, b e a p a r t i t i o n o f B i n B a e r s u b s p a c e s o f dimension 2(t-s+l). ( I t i s w e l l known t h a t s u c h a p a r t i t i o n e x i s t s ; f o r a p r o o f see f o r e x a m p l e [ l l ] 4 . 3 . 6 T h e o r . ) T h e n

...,

S i n c e a n ( s - 2 ) - s p r e a d of R h a s e x a c t l y qs-' sis s;:(t+2)/2, t h e r e e x i s t subspaces R,, s-2 w h i c h a r e m u t u a l l y skew, C o n s i d e r t h e "Baer cones"

...,e Rl enm e on ft s , Rb y o of u r d hi my pe on tshi eo -n

%i = % ( R , , B i ) =

U

XE~i(X,Ri)t

(i=1,

By [13] i t f o l l o w s i n p a r t i c u l a r t h a t t h e s e t - b l o c k i n g sets. S i n c e B and R are skew, t h e sets d i s j o i n t . Hence n ( t , 2 t , q ) > n.

$f,

. .. , n ) .

Baer cones a r e are m u t u a l l y

P

EXAMPLE.

Theorems 3.1 and 3 . 3 imply f o r i n s t a n c e q-

G t l Inp ( 2 , 4 , q ) 5

q2 -q S q t q - S q + l ,

i f t h e p r i m e power q i s a s q u a r e . Now we c o n s i d e r t h e a f f i n e _c_a_s~. 3.4

THEOREM. n , ( t , r , q ) s q r [ ( t + i ) q r - t - t

PROOF. Any c o m p o n e n t B, o f a n n - f o l d ( w i t h n t 2 ) i s a t - b l o c k i n g s e t o f A.

I-'. t-blocking set of A=AG(r,q) S o , by [ 2 1 , Cor 2 . 2 3 , w e h a v e

IBiI 2 ( t t l ) q r - t - t ,

Hence,

t h e assertion follows.

As a c o n s e q u e n c e w e h a v e

3 . 5 COROLLARY.

na(l,r,q) sq/2.

I t i s well known [15] t h a t t h e r e e x i s t s a f u n c t i o n b p = b ( t , q ) ( a n d a function b,=b,(t,q)) such t h a t t h e r e e x i s t s a t-bloceting set i n respectively). P G ( r , q ) ( o r A G ( r , q ) ) if a n d o n l y i f rib, ( o r r l b , , T h e s e f u n c t i o n s h a v e b e e n c a l l e d t h e Mazzocca-~Ta.llln_i_f.u~nI_nctio_ns_.By [18] we h a v e b , ( t , q ) < _ b p ( t , q ) . If a p r o j e c t i v e or a f f i n e s p a c e contains an n-fold t-blocking s e t (with n > 2 ) , then i t h a s a l s o a f u n c t i o n s bp ( n , t , q ) and t-blocking set. Consequently, t h e r e e x i s t b a ( n , t , q ) such t h a t PG(r,q) ( o r AG(r,q)) c o n t a i n s an n-fold t-block i n g s e t i f a n d o n l y i f r < b p ( n , t , q ) (0: r < b , ( n , t , q ) ) (112.2). Clearly, bp(n'.t,q)n. Now w e p r o v e a g e n e r a l i z a t i o n of t h e m e n t i o n e d t h e o r e m o f T a l l i n i . 3 . 6 THEOREM. b a ( n , t , q ) < _ b p ( n , t , q ) . PROOF. A s s u m e t o t h e c o n t r a r y t h a t b, > _ b p t l . F i x i n P = P G ( b p t 1 , q ) a h y p e r p l a n e H , a n d d e n o t e by B a n n - f o l d t - b l o c k i n g s e t of H . By o u r a s s u m p t i o n , t h e r e e x i s t s a n n - f o l d t - b l o c k i n g s e t B' i n t h e a f f i n e s p a c e P-H. C o n s e q u e n t l y BuB' would b e a n n - f o l d t - b l o c k i n g set

On n-Fold Blocking Sets of

35

P =PG ( bp + 1 , q ) , a c o n t r a d i c t i o n .

W e f i n i s h t h i s S e c t i o n by d e t e r m i n i n g a p a r t i c u l a r c l a s s b p ( n , t , q ) , when q i s a s q u a r e .

3.7

PROPOSITION.

I f t h e prime-power

of

values

q is a square, then

bp ( q- Sq+ 1 , 1 ,9 ) = 2. PROOF. I t i s k n o w n t h a t i n t h e d e s a r g u e s i a n p r o j e c t i v e p l a n e P G ( 2 , q ) t h e r e e x i s t s an (q-Sq+l)-fold b l o c k i n g s e t . Assume t h a t P G ( 3 , q ) cont a i n s a ( q - S q + l ) - f o l d 1 - b l o c k i n g s e t . T h e n , by 3 . l ( a ) , q - q + l = n s (q3 + q 2 + q + l ) . ( q 2 + q f i + q + ~ + l ) - l < q - f i + l , a contradiction.

4. EXAMPLES t-blocI n t h i s S e c t i o n , w e s h a l l c o n s t r u c t some e x a m p l e s o f n - f o l d k i n g s e t s i n p r o j e c t i v e a n d a f f i n e s p a c e s . F i r s t , we s h a l l d e a l w i t h p r o j e c t i v e spaces of small o r d e r , 4.1.

THEOREM, (a) (b) (c) (d)

(el

a p o s i t i v e i n t e g e r w i t h r 2 3.

Let r b e np(l,r,3) np(l,r,4) np(l,r,5) np(l,r,7) np(l,r,8)

= 0 2 2 5 3 2 4 5

5

PROOF. ( a ) B y [ 1 7 ] , t h e r e d o e s n o t e x i s t a 1 - b l o c k i n g s e t i n P G ( r . 3 ) f o r r 5 3 ; ( b ) a n d ( c ) f o l l o w s i m m e d i a t e l y f r o m 3.1 ( a ) . N o w we p r o v e ( d ) . A b l o c k i n g s e t o f P G ( 2 , 7 ) h a s a t l e a s t I2 points ( c f . [ll] 13.4.8 T h . ) . S u p p o s e t h a t S i s a n i r r e d u c i b l e b l o c k i n g set i n PG(r,7).Hence,VxES t h e r e e x i s t s a t l e a s t a l i n e L with I L n S l = l E a c h o f t h e ( 7 r - 1 - 1 ) / 6 p l a n e s t h r o u g h L c o n t a i n s a t l e a s t 11 p o i n t s o f S - i x l . S o , IS1 2 1 + 1 1 ( 7 r - ' - 1 ) / 6 , T h e n np ( l , r , 7 )

5 (7'+'

-1)/(11-7r-1 -5)

< 5.

N o w w e p r o v e ( e ) . A b l o c k i n g s e t of P G ( 2 , 8 ) h a s a t l e a s t 13 p o i n t s ( c f . 1111 1 3 . 4 . 9 T h . ) . By a s i m i l a r a r g u m e n t a t i o n a s a b o v e , i t follows IS1 2 1 + 1 2 ( 8 ' - ' - 1 ) / 7 . So

np(l,r,8)a(8

r t l

-1)/(12

8r-'-5)<6.

nq o f o r d e r

4 . 2 THEOREM. Any p r o j e c t i v e p l a n e blocking set.

q B 7

PROOF. C o n s i d e r t h r e e l i n e s a , b , c through a point p o i n t s A, , A, E a - {C,) a n d B, , B , E b- I C , t Define

.

S:=A, B, f l c

Q,

,

C,

:=A, Bz

:=A, B, fl A, B,

,

fl c T, : = C ,

9

Q,

: = A , B,

Q, fl C, B,

has

a

C,

fl AZB,

3-fold

.Fix

1

.

F i n a l l y , f i x a p o i n t T, o n B,C,- I B,, C,) w h i c h i s n o t o n a . Then J3, := a U b U I Q , , Q z )- ( A l , A, , B l , €3, t

the

A . Beutelspacher and F. Eugeni

36

are d i s j o i n t blocking sets. S i n c e A,,B,# _B, U _B2 and q 27, a l s o Hence and t h e complement of

&,B2

of

_B,

U _Bp f o r m s a b l o c k i n g s e t . form a 3 - f o l d b l o c k i n g set

nq.

Now w e s h a l l d e a l w i t h t h e f o l l o w i n g q u e s t i o n . Given a p o s i t i v e i n t e g e r n , i s t h e r e a n i n t e g e r qo s u c h t h a t a n y p r o j e c t i v e p l a n e o f order q z q , has a n n-fold blocking set?The following theorem answers this question i n the affirmative. 4 . 3 THEOREM, I f 3tq d e n o t e s a p r o j e c t i v e p l a n e o f o r d e r q , t h e n n q h a s n sa.d;q24)/Z. a n n-fold blocking set w i t h I n o t h e r words, n p ( l , 2 , q ) 2 'd(q+24) 4

.

PROOF. L e t n b e a p o s i t i v e i n t e g e r w i t h n s 3 1 / i ; 2 4 ) / 4 ' . Fix a line 2 i n r $ , a n d l e t PI Pn b e n p o i n t s o n I . t h r o u g h P, such S t e p 1. F o r a n y i ( 1 - 1 s n ) t h e r e a r e t w o a , , b , 4 1 t h a t t h r o u g h n o p o i n t o f n, p a s s t h r e e of t h e l i n e s a ,. , a ., an , b, s b z t - . . t b n i-1 I n o r d e r t o c h o o s e a i a n d bi , a t m o s t of i -1 4 ( j - 1 ) = 2 ( i - l ) ( i - Z ) t h e a l i n e s d i f f e r e n t from 1 t h r o u a h P; are forbidden. Bv t h e h y p o i h e s i s 2 ( n - l ) ( n - 2 ) 5 q-2, h e n c e f o r e v e r y i s n o n e c a n f i n d two l i n e s a i , bi s a t i s f y i n g t h e a s s u m p t i o n .

,? !...,

,...,

Now, we d e f i n e f o r a n y i w i t h 1

B; S t e p 2.

a; 2; Namely:

1 . 0

n

Any l i n e a i 0

a i fl a i + ,

= \ a , U bi -

..*bi

sis n

,. .

,...,a i n b,

b , , a i fl b,-l

or b i c o n t a i n s a

, b i n ai-, , . . . , b i n

point

*

9 %

. , a i n a , , bi fl b i + , , . , of

each

of

a, the

I.

sets

.

L e t j b e a n i n t e g e r w i t h 11 j c n I n o r d e r t o show t h a t ai a n d bi h a v e a p o i n t o f B i , we may a s s u m e t h a t j + i. I f j > i , t h e n a i c o n t a i n s t h e p o i n t a i fl a , 5 Bit , a n d bi i s i n c i d e n t , On t h e . o t h e r h a n d , i f j < i , t h e n bj fl a i l i e s in w i t h bi fl bi EE,' g/ n a i , a n d b i p a s s e s t h r o u g h a i n bi E B_! I

.

Now, we e m b e d d _B: i n a b l o c k i n g s e t _ B i . I n v i e w o f S t e p 2 , we m u s t a d j o i n p o i n t s i n o r d e r t o b l o c k e x a c t l y 4(n-1)' l i n e s connecting a contain just p o i n t o f bi w i t h a p o i n t o f b , , Any s u c h l i n e s h a l l o n e p o i n t o f g , . M o r e o v e r , we w a n t t o d o t h i s i n s u c h a w a y t h a t t h e are mutually d i s j o i n t . C l e a r l y , r e s u l t i n g b l o c k i n g s e t s B 1 , B 2 , ...,En El; c a n b e e n l a r g e d t o a b l o c k i n g s e t _Bi i n t h e w a y d e s c r i b e d a b o v e . Now, l e t i be a n i n t e g e r w i t h l < i i n , and suppose t h a t t h e blocking s e t s B, ,_Bz,. .,En h a v e b e e n c o n s t r u c t e d . On a n y l i n e x w h i c h is n o t b l o c k e d b y Flf , t h e r e a r e a t m o s t

.

2i+(i-l)4(n-l)

2

5

2n

points which a r e contained i n number i s a t m o s t q+24-12n2 +14n-4

+

(n-1)

G, ,gt

4(n-1)

2

= 4n3-12n2+4n-4

,...,Bi-, .

By o u r h y p o t h e s i s t h i s

= q-( 1 2 n 2 - 1 4 n - 2 0 ) <- q .

Therefore, it is possible t o adjoin a point X t o b l o c k t h e l i n e x.

in

order

to

The r e m a i n d e r of t h i s S e c t i o n i s d e v o t e d t o t h e a f f i n e p l a n e - c a s e . 4.4

THEOREM. n , ( 1 , 2 ,

a q ) < [np(1,2, nq)-1]/2,

where aq

is an

affine

37

On n-Fold Blocking Sets plane of order q with p r o j e c t i v e c l o s u r e

n,.

PROOF. L e t B = {g, ,B,,...B,j j e c t i v e p l a n e nq. D e f i n e

b l o c k i n g set i n

B' i f n is even,

=

be a n n-fold

{ Ei u Bz ,B,u -B,

*...s_Bn-l

the

pro-

u -B,}

and

A

*

i f n i s o d d . T h e n B' i s a A s e t o f m ~ ( n - 1 ) / 2 s e t s El , g , ,.,.,En with t h e p r o p e r t y t h a t a n y s e t Bj h a s a t l e a s t 2 p o i n t s o n a n y l i n e o f JCs. C o n s e q u e n t l y , B' i n d u c e s a n n - f o l d b l o c k i n g s e t i n a, ,

A s c o r o l l a r i e s w e have t h e f o l l o w i n g two theorems, 4 . 5 THEOREM. L e t a , b e t h e d e s a r g u e s i a n a f f i n e p l a n e o f o r d e r q . I f q is a s q u a r e , t h e n n,(1,2,aq) 2 (9- Sq)/2. PROOF.

By t h e r e m a r k a f t e r 2 . 2 T h e o r e m ,

np(1,2,

n,)

= (q-G+l)/Z.

4 . 6 THEOREM. Any a f f i n e p l a n e o f o r d e r q h a s a n n - f o l d w i t h n z ,f( q / 2 4 + 1)' -1 / 2 . The proof

f o l l o w s by 4 . 4

and 4.3

blocking

set

.

REFERENCES

[I] B e r a r d i , L . , a n d E u g e n i , F . , On b l o c k i n g s e t s i n

affine planes, J.Geom. 2 2 ( 1 9 8 4 ) 1 6 7 - 1 7 7 . [ 2 ] Berardi,L.,Beutelspacher, A . , a n d E u g e n i , F . , On (s,t;h)-blocking sets i n f i n i t e p r o j e c t i v e and affine spaces, A t t i Sem. Mat. F i s . M o d e n a 31 ( 1 9 8 3 ) 1 3 0 - 1 5 7 . [3] B e u t e l s p a c h e r , A., P a r t i a l s p r e a d s i n f i n i t e p r o j e c t i v e s p a c e s a n d p a r t i a l d e s i g n s , M a t h . 2. 145 ( 1 9 7 5 ) 2 1 1 - 2 3 0 . [ 4 ] Beutelspacher, A.,Blocking sets and p a r t i a l spreads i n finite p r o j e c t i v e s p a c e s , Geom. D e d i c a t a 9 ( 1 9 8 0 ) 4 2 5 - 4 4 9 . [ 5 ] B o s e , R.C.,and B u r t o n , R.C.. A characterization of flat spaces i n a f i n i t e g e o m e t r y a n d t h e u n i q u e n e s s o f t h e Hamming a n d the Mac D o n a l d c o d e s , J . C o m b i n . T h e o r y 1 ( 1 9 6 6 ) 9 6 - 1 0 4 . [6] Bruen, A., Blocking sets i n f i n i t e p r o j e c t i v e planes, SIAM J . Appl. Math. 21 ( 1 9 7 1 ) 380-392. [ 7 ] B r u e n , A . , a n d T h a s , J . A . , B l o c k i n g s e t s , Geom. D e d i c a t a 6 ( 1 9 7 7 ) 192-203. [ 8 ] Dembowsky, P . , F i n i t e g e o m e t r i e s ( S p r i n g e r , B e r l i n , 1 9 6 8 ) . [9] Harary, F., Generalized Ramsey t h e o r y XII: Achievement and a v o i d a n c e games on f i n i t e g e o m e t r i e s and configurations. ( U n p u b l i s h e d ma nu s c r i p t ) [ l O ] H i l l , R . , a n d M a s o n , J.R.M., On ( k ; n ) - a r c s a n d t h e f a l s i t y o f t h e L u n e l l i - S c e c o n j e c t u r e , i n : F i n i t e G e o m e t r i e s a n d D e s i g n , L.M.S. L e c t u r e s n o t e s 4 9 ( C a m b r i d g e U n i v . P r e s s , 1 9 8 1 ) 52-61. [ l l ] H i r s c h f e l d , J.W.P., Projective geometries over f i n i t e f i e l d s (Clarendon Press, Oxford 1979). B l o c k d e s i g n games, Canad. J. [12] H o f f m a n n , A . J . , a n d R i c h a r d s o n , M . , M a t h . 13 ( 1 9 6 1 ) 1 1 0 - 1 2 8 . [13]Huber, M., A c a r a c t e r i z a t i o n o f Baer c o n e s i n f i n i t e p r o j e c t i v e s p a c e s , U n i v e r s i t y of Mainz, P r e p r i n t 1984. [ 1 4 ] K a b e l l , J . , A n o t e on c o l o r i n g s o f f i n i t e p l a n e , Discrete Math. 44 ( 1 9 8 3 ) 319-320.

.

38

A . Beutelspacher and F. Eugeni

[15] M a z z o c c a , F . , a n d T a l l i n i , G., On t h e n o n e x i s t e n c e of blocking sets i n P G ( n , q ) and AG(n,q) for a l l l a r g e enough n , Simon Stevin (to appear). [16] R i c h a r d s o n , M . , On f i n i t e p r o j e c t i v e g a m e s , P r o c . Amer. Math, SOC. 7 ( 1 9 5 6 ) 456-465. [17] T a l l i n i , G . , k - i n s i e m i e b l o c k i n g sets i n P G ( r , q ) e i n A G ( r , q ) , Q u a d e r n o n . 1 1 s t . M a t . A p p l i c a t a U n i v . L' A q u i l a ( 1 9 8 2 ) . [I81 T a l l i n i , G., B l o c k i n g sets n e i sistemi d i S t e i n e r e d - b l o c k i n g sets i n P G ( r , q ) e AG(r.q), Quaderno n.3 1 s t . M a t . A p p l i c a t a Univ. L' A q u i l a ( 1 9 8 3 ) . [19] Von Newmann, J . , a n d M o r g e s t e r n , O . , T h e o r y o f g a m e s a n d e c o n o mic behaviour ( U n i v e r s i t y P r e s s , P r i n c e t o n 1972).

Annals of Discrete Mathematics 30 (1986) 39-56 0 Elsevier Science Publishers B.V. (North-Holland)

39

EMBEDDING FINITE LINEAR SPACES IN PROJECTIVE PLANES Albrecht Beutelspacher

and

Klaus Metsch

Fachbereich Mathematik der Universitat Saarstr. 21 D-6500 Mainz Federal Republic of Germany

It is shown that a finite linear space in which all points have degree n+l can be embedded in a projective plane of order n, provided that the line sizes are big enough. INTRODUCTION A l i n e a r s p a c e is an incidence structure S = (p,L,I) of points, lines and incidences such that any two distinct points are on a unique line and any line contains at least two points. A great variety of linear spaces can be obtained as follows. Let P be a projective plane and denote by x a set of points of P with the property that outside x there are some non-collinear points. Then the incidence structure S = P-x whose p o i n t s are the points of P outside x and whose lines are the lines of P which have at least two points outside x is a linear space. Any l i n e x space isomorphic to such an P-x is called e m b e d d a b l e in the projective plane P. An old result of M. HALL [ 4 ] asserts that any linear space can be embedded in a projective plane, which is usually infinite. There is a conjecture dating back to the time of M. HALL’S paper which says that any f i n i t e linear space can be embedded in a f i n i t e projective plane. This seems to be an extremely difficult problem. We want to deal, however, with an even more difficult question. For a finite linear space S denote by n+l the maximal number of lines which pass through a common point; the so-defined integer n is called the o r d a r of S. Clearly, the order of any projective plane in which S is embedded is at least n. The question we are interested in is the following: When can a finite linear space of order n be embedded in a projective plane of the same order n ? One of our answers can be formulated as follows (cf. corollaries 3 . 3 and 3 . 4 below). Let S be a finite linear space in which any point has degree n+l. Denote by n+l-a the minimal number of points on a line. If 1 s(

n > a’ -1) ( 5aZ-3a+20), then S is embeddable in a projective plane of order n. In other words: For a given a, there is at most a finite number of such linear spaces, which cannot be embedded in a projective plane of order n. As corollaries we obtain the theorem of THAS and DE CLERCK [6] on the embedding of the pseudo-complement of a maximal arc and the theorem of MULLIN and VANSTOPJE [ 5 ] on the embedding of the pseudo-complement of a pencil of lines. This paper contains the main results of the second author’s Diplomarbeit.

40

A . Beutelspacher and K.Metsch

1. THE MAIN IDEA Throughout this section, S = ( p r L , I ) denotes a finite incidence structure consisting of points, lines and incidences such that any two distinct points are incident with at most one common line. For a point p (or a line L ) , its d e g r e e is the number r (or kL) of P lines through p (or points on L, respectively.)

Two lines are called p a r a Z l e 1 , if they have no point in common. For h(L ,L2) denotes the number of litwo distinct lines L1 and nes parallel to both, ani2'L2. hhroughout, we denote the number of points of S by v,L&he number of lines of S by b. Also, S has maximal point degree n+l, and z is the integer defined by z = b - (n'+n+l). The theorems which we are going to prove in this section will play a crucial role in the embedding of linear spaces in projective planes. Our method generalizes the method introduced by BRUCK [ 3 ] and BOSE [21. THEOREM 1.1. Fix a line H of S , and let a, c, d, e and x be integers with the following properties: (1) The degree of H is n+l-d < n+l. (2) The number of lines parallel to H is nd+x > 0. ( 3 ) For every parallel L of H we have n+a 5 h(L,H) 5 n+c. ( 4 ) For any two intersecting lines Llf L2 parallel to H we have h(L1,L2) 5 e. (5) 2n > (d+l)(de-d-2a-2) + 2x. (6) n > (2d-l)(c+l) + e - 1 - 2x. (7) At least one of the following assertions is true: (i) On every parallel of H there is a point of degree n+l, or (ii) n(d-s) > s(c+l) - x for every integer s with 0 5 s 5 d-1. Then we have: (a)There is an integer m with the following property: If M1,. . I Mt are all maximal sets of mutually parallel lines with H € hi and lmil m, then t = d and every line parallel to H is contained in exactly one Mi. (If d 2, we can take m = n - (d-l)(c+l) + x + 1.) ( b ) If S is a linear space such that any point outside H has degree n+l, then the sets M . are parallel classes (i.e., every M . induces a partition of the point set of S ) . The proof of this theorem will be prepared by several Lemmas. From now on, we suppose that S fulfilles the hypotheses of Theorem 1.1. By a c l a w we mean a set S of lines with the following properties: (a) H { s f and H is parallel to every line of 5, (b) any two lines of S intersect. A claw S is called n o r m a l , if ( s l = d. The o r d e r of a claw is the number of its elements. A c l i q u e is a set of mutually parallel lines of S. The clique h is called m a x i m a l , if (a) H € M , ( b ) h is a maximal set of mutually parallel lines, and (c) Ihl > n - (d-l)(c+l) + x. Now we can state our first Lemma.

Embedding Finite Linear Spaces

41

LEMMA 1. Let S be a claw of order s. Denote by the set of lines parallel to H which do not lie in 5. For 0 2 y 5 s let f(y) be the number of lines in 7 which have exactly y parallels in 5 . Then s 5 d. Moreover, f(0)

-

f

f(y)(y-l) 2 n(d-s) + x

-

s(l+c), and

y=l

5

f(y) = nd + x - s. y= 0 PROOF. From our assumption ( 2 ) we get (i) By double couiting we have

Using hypothesis ( 3 ) we obtain therefore

2

f(y)y 5 s(n+c) and (ii)' y=O Now let L1 and L be two distinct lines parallel to L an& L there are at most which are parailel to and L2. Hence (ii)

2;

s(n+a). H is lines in

i

y=l Together with (i) and (ii)' we conclude (iii)

0

5 f(O)

-< n(d-s) If we assume s = d+l, then by (iii) we would obtain the fol lowing contradiction to ( 5 ) : 1 n = n(s-d) 5 x - s(l+a) + ~s(s-l)(e-l) 1 Z(d+l)(de-d-2a-2) + x. Hence there exists no claw of order d+l, therefore any claw has at most d elements.0 =

LEMMA 2. For every line L parallel to H, there exists a normal claw containing L. PROOF. Consider first the case that there is a point p of degree n+l on L. Let 3 be the set of lines through p parallel to H. Since H h a s degree n+l-d, we have 1st = d. Obviously, 5 is a claw containing L. Thus, we may suppose that hypothesis (7)(ii) is true. Let 5 be a claw of maximal order with L E 5 . With the nota-

A . Beutelspacher and K. Metsch

42

tion of Lemma 1 we get f(0) 2 n(d-s) + x

- s(l+c).

Assume now s < d. Then (7)(ii) implies f(0) > 0. Therefore, there is a line L' in which intersects every line of 5. But then 5 ' = .S [L'I is a claw with L € 5 ' and ( 5 ' 1 > 1x1. This contradiction proves Lemma 2.0 LEMMA 3 . Let L be a line parallel to HI and denote by 5 a normal claw containing L. Moreover, let m be the set of all lines L' 5-tLI which are parallel to H and intersect every line of 5-{LI. Then M u (HI is contained in a maximal clique through L. 5-ILl is a claw of order d-1. If L1, L2 E f i , then 1 5 ' ~[L1,L211 = d+l, and therefore 5'u {L1,L21 is not a claw. This shows.that L1 and L2 are parallel and that f l u {HI is a clique. Lemma 1 applied to 5 ' gives

PROOF. Clearly, 5'

=

l f l l = f(0)

- (d-l)(c+l) + x.

n

Therefore, m u [HI is contained in a maximal clique. Since kl u {HI, Lemma 3 is proved.0 LEMMA 4 . If

f11nf12

=

fll

and

fi2

L

€

are different maximal cliques, then

(Hl.

PROOF. Because /)Il secting lines L1

f12

and €

M1

and

are maximal cliques, there exist interL2 € m 2 , From our hypothesis ( 4 ) we get

IM1n f121 5 h(L1,L2) 5 e.

Assume now Ih1nh21 2 2 . Then there is a line From hypothesis ( 3 ) we obtain therefore

If11CJfi21 Ifill

flll

H

in

h1nfi2.

5 h(L,H) + ItL,HII 5 n+c+2, and /112 1 =

-k

On the other hand

I

L

+

I"1uf12

+

IA1n M 2 1

Q

n+c+2

+ e.

we have IM2

->

2(n -

d-l)(c+l) + x + 1);

together we get a con radiction to our hypothesis ( 6 ) . 0 Now we are ready for the proof of theorem 1.1. (a) If d = 1, then the statement is obvious. Therefore we may suppose d ? 2 . Since there are lines parallel to H, there exists a normal claw 5 (cf. Lemma 2 ) . We shall use the notation of Lemma 1

Embedding Finite Linear Spaces

43

for 3 . Because there exists no claw of order d+l, we have f(0) = 0. From Lemma 1 we set d y ) = nd + x d, and f(y)(y-1) x - d l+c). y=l y=2 so I d d f(y) 2 nd + x - d f(y (y-1 f(1) = nd + x - d y=2 y=2 > nd + x - d + x d(l+c) = nd - d(c+2) + 2x. Put S = {L1, L d f , and define I?,’ as the set of all lines L E S-{L. 1 which are parallel to H ahd intersect every line of SILill Obviously, the sets I?; are distinct; therefore,

if

-

-

-

-

...,

-

-

lfil-[HII

+...+

(fid-IHII

1fi;I

+...+

IfiiI

Ihl-IL1ll +...+ Ihd-ILdll = f(1) + d. Assume that there is another maximal clique fld+l. Then f(1) + d + n - (d-l)(c+l) + x -< f(1) + d + Ihd+l-IH)I 5 Ihl-IHII +...+ [Ad+ On the other hand, Lemma 4 yields [hl-{H)l +...+ Ihd+l-{HII = I(h1 hd+l)=

...

< number of parallels of H 5 nd + x. Together we get nd + x 2 f(1) + d + n - (d-l)(c+l) + x. Since f(1) > nd-d(c+2)+2x, we conclude n 2 (2d-l)(c+l)-2x. Hence implies e 5 0. But in view of d 2 2 and h(L1,L2) condition ( 6 ) -> I { H I 1 = 1, this contradicts condition ( 4 ) . Therefore, there are exactly d maximal cliques. N o w Lemma 3 and 4 show that every parallel of H is contained in exactly one of the sets Mi. This proves (a); (b) is obvious.0

In the following corollary, we handle an important particular case. COROLLARY 1.2. Let S be a finite linear space of order n, and let H be a line with kH 5 n such that every point outside H has degree n+l. Let the integers d, x, z be defined in the following way: The number of lines of S is b = n2+n+l+z, kH = n+l-d, and H has exactly nd+x+z parallels in S . Suppose that there exists positive integers and 5 with the following properties: 1) n+l-d 5 kL 5 n+l-i for every parallel L of H. 2) 2n > (d+l)(da’ + d - 2dG + :2 - 2) - 2dx + d(d-l)z. 3 ) n > (2d-l)(d-l)(d-l) + a* - 1 + (2d-3)x + 2(d-l)z. Then assertion (a) of Theorem 1.1 is true. Furthermore, the sets h I; are parallel classes of S. x ’ = x+z. PROOF. Define a = dd-d-d+x+z, c = dd-d-a+x+z and e = d’ Using 2) and 3 ) we see t h t (5) and (6) of Theorem 1.1 are satisfied (note that x‘ replaces x and x is replaced by x+z). Obviously, the conditions (l), (21, (7) of 1.1 are fulfilled. Now, let L be a parallel of H with k = ntl-d’. Since L intersects kL(d-l) parallels of H and H kas nd+x+z para lel lines, we get

c

A . Beutelspacher and K.Metsch

44

h(L,H) = nd + x + z - kL(d-l) - 1 = n + (dd'-d-d') + x + z. So, condition ( 3 ) of Theorem 1.1 is true (note that 2 5 d' 5 d ) . Let L1 and be two different intersecting lines parallel to H, and put kLL2= n+l-di (i = 1 , 2 ) . Then 1

h(L1,L2) = dldZ + z 5 e l which shows that also condition (4) of Theorem 1.1 is fulfilled. Therefore, the corollary follows from Theorem 1.1.0 REMARK. If pl, ...'p are the points on H and if we denote the degree of p. b9'AFb-d. then we have x = d 1+. +dn+l-d in the corollary. IA particula;, x = 0, if every point of S has degree n+l.

..

2. CONSTRUCTION OF THE PROJECTIVE PLANE In this section, S = (p,L,I) denotes a finite linear space of order n with b = n2+n+1+z lines. First, we show the following theorem. THEOREM 2 . 1 . Suppose that S satisfies the following conditions: (a) b 2 n'. ( b ) For every line L of S there is an integer t(L) with the following property: If kL = n+l-d, then there are exactly d maximal sets M of mutually parallel lines with L € fi and ( M I 2 t(L) Furthermore, every line parallel to L appears in exactly one of these d sets M . Then S is embeddable in a projective plane of order n. For the proof of this theorem we shall use the following notation. A c Z i q u e is a maximal set fi of mutually parallel lines with ] M I t(L) for at least one line L E f i . A clique M is called norrnaZ, if I f i l = n. By we denote the set of all cliques of S . For p € p and 18 € j we define p M , if p 1 L for at least one line L of M . we put For h E

-

51(h)

= IF)'

Z,cm

=

I

fi' € j r mnh' = @ I ,

G1(M) u { M I , 4 p I ) = Ip For every normal clique M we define

I

p E p, p

1.

MI.

Now we can dFfine the incidence structure S ' = ( P d , L the following way: pI'L * P I G for all p € p r L € L ; p I' 5 ( f i ) p E 4(M) for all p 6 p , G ( f l E 2; MI'L L E A for all M E j, L E L ; M I' * fi E G ( M ' ) for all ill E /3, 5 ( f i ) E 2. As in section 1, we shall prepare the proof by several lemmas. From now on we suppose that S satisfies the hypotheses of Theorem 2.1.

-

Q

~(4')

LEMMA 1. (a) A line of degree n+l-d is contained in exactly d cliques. ( b ) If L and are parallel lines, then there is a unique E A. clique f i l with

Embedding Finite Linear Spaces

45

PROOF. Let L be a line of degree n+l-d. Then there are exactly d cliques Mi with L E Mi and I/tlil 2 t(L). Furthermore, every line parallel to L appears in exactly one of theses cliques. Assume that there is another clique fl with L E h. Then I f l l 5 t(L). By definition, h contains a line L' with Ihl 2 t(L'). In L' 4 L, and L' is parallel to L. Let j be the index with L' E M . . Then we have L,L' E f l . , f l , and ~ h . ~ , 1.~ t(L'). f l ~ Now condition of Theorem 2 . 1 gives M 3 = M . contraaicting IhI < t(L) 5 I f l j I . O

(a)

1

LEMMA 2. (a) Let L be a line, and denote by p a point off L. If kL = n+l-d and r = n+l-y, then there are exactly d-y cliques fl with L E h and p' y f l . (b) If p is a point of degree n+l, then p % /11 for every clique

m.

(c) We have 1/31 + v = n2+n+l. PROOF. (a) There are exactly r -kL = d-y parallels of L through p. Therefore, the assertion folaows from Lemma 1 (b). (b) Let L be a line with p 1 L. From (a) and Lemma l(a) we infer p % /rl for every clique M with L E A . (c) Let p be a point of degree n+l, and let L1f...,Ln+l be the lines through p . If the degree of Li is n+l-di, then we have n+l n+l v-1 = (kL.-l) = (n+l)n di. 1 i=l i=l In view of (b) and Lemma 1 we conclude

2

n+l

n+ 1 I{fi

=

€

13,

Li E M I 1

i=l Together, our assertion fol1ows.U

=

1

di.

i=l

LEMMA 3. Let fl be a normal clique, and denote by L a line with = n+l-d, and l e t t denote the number of points p L 4 h. Put on L with % h . Then there are exactly 1-t cliques which are disjoint to fl and contain L. In particular, we have 0 5 t 5 1. PROOF. L has kL-t points p with p M . Therefore, there are exactly m := I f l l - (kL-t) = d + t - 1 lines LA! L in h which are parallel to L. Let hi be the clique w ich coatains L and L.. Since L { f l , we have h f hi m . for i f j . Obviously, for all i. Therefore, by Lemma l(b), hi I I { M ' I h ' E /3, M n f l ' $. @, L E f l ' l l = m. Now, Lemma l(a) shows 1 { h ' I h ' E 13, h n h ' = @, L E h ' l l = d - m = 1 - t.0

kb

Q

...,

LEMMA 4 . Denote by fl a normal clique. Then (a) l G 2 ( M ) l 5 1. In other words: There is at most one point p with P 1. f i . (b) l G ( h ) I = n+l. PROOF. (a) Assume that there are two distinct points p and q with h. Let L be the line which passes through p and g , and dep,q fine t as in the preceding lemma. Then t 2 2 , contradicting our Lemma 3.

+

A . Beutelspacher and K. Metsch

46

M = iL1,

(b) If

...,Ln)

and

k L , = n+l-d. 1

I G 2 ( M ) I = 1 Ip

I p

€ p

M}I = v

Q

then n

- 1 i=l

(n+l-di).

+

n'+n+l.O

From Lemma 1 we get

and the assertion follows in view of

v

=

LEMMA 5. Let hl be a normal clique. (a) If L is a line, then L and G ( M ) intersect in S ' in a unique point; i.e. one of the following cases occurs: (i) There is a unique clique in G ; , ( A ) containing L. If G 2 ( h ) f 0, then L is not incident with the point of 4 ( M I . (ii) NO clique of G ( M ) contains L, I G , ( M ) I = 1 an& L is incident with the point 4 (&). (b) Any two cliques of Gl(3) are disjoint. PROOF. (a) We may suppose L E M . Using the notation of Lemma 3 we get t € I0,l) by Lemma 3 . Moreover, Lemma 3 implies that (1) (or (ii)) occurs if and only if t = 0 (or t = 1, respectively). (b) is a consequence of (a1.0

02

LEMMA 6. Let

M1

and

M2

4(A1) = 5 ( A 2 )

(a)

be two distinct normal cliques. Then A 1 n M2 =

0

0.

(b) 14(M1) 5 ( M 2 ) I = 1 0 Mln M 2 9 0. PROOF. (a) One direction is obvious. Suppose therefore M1n M 2 = 0. Then M € 51(fll); hence, by Lemma 5(b), cl(hl) c Gl(M2). SirnilarZy, we have G1(h2) c G l ( f l l ) , hence equality. In view of Lemma 4 we may assume without loss of generality that 5 ( 4 . ) = {p.) with points p1 and p Lemma 5(a) sa s that a line L2 it incidknt with p. if and only ig no clique of contains L. Since G1(Mlt = T1(fi2), this shows that a line is incident with p1 if and only if it is incident with p2. Since r 1. 2, we Pi have 5,(fll) = 5 , ( M 2 ) . (b) In view of (a), one direction is obvious, Let us suppose A 1 n M 2 f 0. Then fil and f12 intersect in a line L.Define T = r M I ki € ,E, M nMl = 0 fl n M 2 } . G ( M 2 ) , we have Since M1, M 2 [ 5(M1)

.

+

I5(A1) n 5 ( A 2 ) I = IM E

F

= rM E

i~ I

=

I

MnMl

=

MnM1 =

ia

= MnM2)1

$11

-

+ 152(M1)n52(M2)1

li-I + 142(fil)n52(M2)1

- li-l + 1 5 2 ( M 1 ) n 5 2 ( M 2 ) 1 .

IG1(fl1)l

Now we distinguish three cases. C a s e 1.

52(/111) = 0.

Then by Lemma 5(a) every line is contained in a unique clique of ( 4 ) Because no clique of G1(hl) contains two lines of M 2 , we have' 171 = IA2-ILfI = n-1, and so

T

.

l5(fl1)n5(fi2)l since

IG1(M1)I + 1

=

=

151(f11)l - ( T I = n - (n-1) = IG(M1)I = n+l.

IT1(M1)I

=

1,

41

Embedding Finite Linear Spaces

152(fi1)I

=

1

G2(M1)

and

= 52(fi2).

G 2 ( f i , ) = S 2 ( f i 2 ) , no line of fi2 is incident with the point fi 1. As in case 1, this implies I T ( = I f i -ILlI = n-1. Since I 1= 1 we now have I G , ( A , ) I = n-1, and t6is imp1 es

15(fi1)nG(f12)l

=

lGl(fil)l - I T 1

+ lG2(fi1)nG2(fi2)l

I G 2 ( f i 1 ) I = 1 and G2(fi1) c G 2 ( M 2 ) . Since 4 ( f i ) G 2 ( M Z ) , there is a unique line L' in incident2wi&h the point of G ( f i l l . In view of L € f i l L', and so 171 = ( M 2 -{L,L')y = n-2. This shows again

= 1.

Case 3 .

IG(PIl)n G(M2)l = I G 1 ( M 1 ) I

-

f i 2 which is we have L $.

I T 1 = 1.

Since IG2(fii)I 5 1, we have handled all cases. Thus. Lemma 6 is proved. 0 LEMMA 7. Any two distinct lines of S ' intersect in a unique point of S ' . PROOF. If one of the two lines is an element of l, we already proved the assertion in Lemma 6(a) and Lemma 7. If both lines are elements of L , the assertion follows from Lemma l(b1.O Now we are ready for the proof of theorem 2.1. * Let S* = L L u L , p u p , I ) be the dual incidence structure of S ' . By is a linear space with n2+n+l lines (Lemma 2(c)) and Lemma 7, S at least n2 points (hypothesis (a)).Furthermore, in view of Lemmas l(a) and 4(b), any poipt of S has degree n+l. Now, by the theorem of VANSTONE [ 7 ] , S is embeddable in a projective plane of order n. But then also S ' is embeddable in a projective plane P of order n. This completes the proof of Theorem 2.1.0 We remark that S' = P if b > n2. (Assume to the contrary that S' P. Since I p I + 1/71 = n2+n+1, there is a line L of P which is not a line of S ' ; so, L ( Lu 2. Because S is a linear space, at most one of the points P 1 t - - - r P n + l incident with L is a point of p . Every line of S is in P incident with exactly one of the points pi. Since b > n', this shows that there are at least two points among the p.'s which are incident with n lines of S . But one of these two po$nts, say pl, is an element of /3. Hence p1 ,is a normal clique of S , and G(p,) = tpl,...,pn+ll = L, contradicting L t Z.) The following theorem is probably the main result of this paper. THEOREM 2 . 2 . Suppose that the hypotheses of part (a) of Theorem 1.1 or its corollary are satisfied for every line of S which has not degree n+l. If b 2 n', then S is embeddable in a projective plane of order n. PROOF. We show that for every line L of S there exists an integer t(L) such that the hypothesis (b) of Theorem 2.1 is fulfilled. If kL = n+l, we put t(L) = 2 . If kL n+l, then Theorem 1.1 (or its corollary) show that such an t(L) exists. Therefore Theorem 2.2 follows from Theorem 2.1.0

48

A . Beutelspacher and K. Mehch

3 . LINEAR SPACES WITH CONSTANT POINT DEGREE

Let A be a finite set of nonnegative integers. We say that the linear space S is A-semiaffine, if r -k < A for every non-incident point-line pair (p,L) of S . TRe kinear space S is called A-affins, if it is A-semiaffine, but not A‘-semiaffine for every proper subset A ’ of A. Throughout this section, S will denote an A-affine linear space in which every point has degree n+l. Because the lO)-affine linear I 0 1 throughspaces are the projective planes, we will assume A out. Denote by a the maximal and by a the minimal element in A-fO}. The integer z is defined by b =-n’+n+l + z. The following two facts shall be used frequently. For any line L whose degree is not n+l we have n+l-a 5 k n+l-g. If L is a line which has at least one parallel line,LtEen kL n+l. LEMMA 3.1. (a)- We have z 2 - g z . (b) If n > aa(a-a), then z 5 (a-a-l)a. PROOF. (a) Since a € A and since every point has degree n+l, there is a line L- of degree n+l-a. Let L ’ be a line intersecting L at a point q . Then every point of L’ other than q is on precisely 5 lines parallel to L. Thus, L has at least (kL,-l)g 1. (n-a)a parallels. On the other hand, L intersects exactly k L m n = (n+l-a)n lines. Hence, n2+n+1+z = b 2 (n-a)g + (n+l-a)n + 1 = n’+n+l - 22, i-e. z -aa. ( b ) If S has a line of degree n+l, then b = n2+n+l, and so z=O. Therefore, we may assume n+l-a 6 k n+l-a for every line X . Let L be a line of degree n+l-a, and 8 e k t e by 4 the set of all lines parallel to L. Then (rp-kL) = (v-kL )a [ M I (n+l-a) 5 x k X = -X€A PkL Because every line has at most n+l-g points, we have v 5 kL + n(n-g). Together it follows (v-k )a n(n-g)g (Z-l)a(Z-a-l) IMI 2 L - ,= na- + ( 2 - g - 1 )+~ n+l-Z n+l-Z n+l-Z Our hypothesis yields n+l-Z > gZ(Z-5) + 1 - a 1. (a-1)3(Z-g-l), It follows therefore I M I 2 ng + (2-2-1)s. b = 1 + kL*n + I M l 5 n2+n+l + (a-g-l)~, i.e. z 5 (2-g-1)a.m Suppose b 5 n’+n+l and assume that following conditions: (1) n > i ( ~ 2 - 1 ) (~’+~-2g+2 + ) T1 z ( f i - 1 ) z I

THEOREM 3.2.

S

satisfies the

49

Embedding Finite Linear Spaces

+ 2(?-1)z, (3) b n2 or n 2 g; - 1. Then S is embeddable in a projective plane of order n. (b) If b > n2+n+l, then one of the following inequalities holds: n 5 q(52-1)(~2+~-2g+2) 1 + k2 ( ~ - l ) z , or ( 2 ) n > 2(Z-1)(Z2-Z+l)

n 5 2(Z-l)(Z2-Z+1) + 2(Z-l)z. (a) If n 2 a;-1, then b = n2+n+l + z 2 n2 by Lemma 3.1. Hence we have b 2 : ’ in any case. In view of Theorem 2.2 it suffices to show that for every line L with kL n+l the hypotheses of Corollary 1.2 are fulfilled. Consider therefore a line L- of degree n+l-a 5 n. Put d = a, = a, 4 = g and x = 0. Then d 2 d 2 4, and our hypothesis ( 1 ) shows PROOF.

a

-

> (2d-l)(d-l)(a-l) + 8’ - 1 + Z(d-l)z. Therefore, the hypotheses of part (a) of Corollary 1.2 are fulfilled. Hence the assertion follows in view of Theorem 2.2. (b) Assume that our statement is false. Then, as in part (a), we would be able to embed S in a projective plane of brder n, contradicting b > n’ +n+l .O

COROLLARY 3 . 3 . If b 5 n2+n+l and n > +(a2-l)(Z‘+Z-2g+2) is embeddable in a projective plane of order n.0 1 -

then

S

+ 3.4. If b > n’+n+l, then n 5 7(a2-l)(a2+z-2a+2 f-OROLLARX -_ Ta(a l)(a-g-l)g. PROOF. Since b > n2+n+l, there is no line of degree n+l. Assume first a = a. Then every line has degree n+l-a, we have v = 1 + (n+l)(n-g) and v(n+l) = b(n+l-a). We obtain b 5 n’+n+l, a contradiction. Hence we may suppose 1 5 g C a. Assume that our statement is false. Then 1. ~ ( ~ - 1 1 2 n > $ ( ~ ’ - 1( )~ ‘ + ~ - 2 a + 21.) L2 ( ~ 2 - 1 ) -> -aZ(Z-a), and from Lemma 3.1 we get z 5 (Z-a-l)g. In view of z > 0 we have a 23+2. Now we get n > $ ( ~ 2 - 1 ) ( ~ ’ + ~ - 2 g + 2+) $(i-l) (~-5-l)C > 2(Z-l)(2-Z+l) + 2(Z-l)z. This is a contradiction to Theorem 3.2(b).0

(%‘-a)

-

A . Beutelspacher and K. Metsch

50

In the remainder of this last section we shall study the case IAl = 2. Let a and c be non-negative integers with a c, and denote by S an ta,c)-affine linear space in which every point has degree n+l. Then every line has either n+l-a or n+l-c points. We call a line of degree n+l-a l o n g : the other lines are said to be s h o r t t . The number of long lines (or short lines) of S is denoted by ba (or b , respectively). Let t be the (constant) number of long lines Fhrough a point. Then we have v-1 = t(n-a) + (n+l - t)(n-c) = t(c-a) + (n+l)(n-c). The proof of the following assertion is straightforward and will be omitted here. LEMMA 3.5. (a) We have ba and (b) If

=

(n-c+a)t +

bc = n2+n+l - c

((c-a)(t-a)+l-a)t n+l-a

-

(n-c+a)t - (t(c-a)+l-c)(t-c) n+l-c

b = n2+n+l, then (c-a)’t2

-

(c-a)[(n+l)(c-a)+n]t

+

cn(n+l-a) = 0,

or JD t = t to 2(c-a)

‘

where n+l 2(c-a) and

D

=

[ ( ~ - a + 1 ) ~ - 4 c ]+n ~ 2(c2+a2-c-a)n + (c-a)’ . U

COROLLARY 3.6. Let c be a positive integer, and denote by S a finite tO,c)-affine linear space in which every point has degree n+l. If 1 n > ?(c2-1) (c2-c+2), then S is the complement of a maximal c-arc in a projective plane of order n. In particular, c divides n. PROOF. Since there is a line of degree n+l, we have b = n2+n+1. By corollary 3.3, S is embeddable in a projective plane P of order n. Hence there is a set c of points of P such that S = P-c. It follows that c is a set of class (0,cI of P, so c is a maximal c-arc of P . 0 REMARK. Corollary 3.6 is a slight generalisation of a theorem of THAS and DE CLERCK [6]. COROLLARY 3.1. Let c 1 be a positive integer, and denote by S a finite tl,cl-affine linear space in which every point has degree n+l. If 1 n > zc(c-1)(c2+3c-1), then S is embeddable in a projective plane P of order n. Moreover, one of the following cases occurs: ( a ) S is the complement of c concurrent lines of P. (b) There is a maximal (c-1)-arc c and a line L in P such that L does not contain a point of c . S is obtained by removing the line L and the points of c from P. (cl b = n2+n+l, c-1 divides n, and c(c-4)n’ + Zc(c-l)n + ( ~ - 1 ) ~

Embedding Finite Linear Spaces

51

is a perfect square. PROOF. First we show that S is embeddable in a projective plane of order n. From our hypothesis we get 1 + zc(c-l)(c-2). 1 n > 7(c2-l)(c2+c) (*) Therefore, in view of Corollary 3 . 3 , we may assume b > n’+n+l, i.e. z > 0. Hence, in view of Theorem 3 . 2 , it suffices to show that (2) n > 2(c-l)(c2-c+l) + 2(c-l)z. Let L be a long line. Then through every point outside L there is precisely one line which is parallel to L. Hence L together with its parallel lines forms a parallel class n of S. Since b = kL*n + I I I 1 , we have I I I l = n+l+z. Since t 5 n we have (n+l+z)(n+l-c= = I~~l(n+l-c) 5 v = 1 + t(c-1) + (n+l)(n-c) -< n2+l-c. Now we claim z 5 c-2. (Otherwise, we would have (n+c)(n+l-c) 5 n’+l-c,. so

n 5 c2-2c+l = (c-l)2I contradicting ( * ) . I NOW (1) and (2) follow immediately from ( * ) . Hence S is embeddable in a projective plane P of order n. In particular, b 5 n2+n+l. Let L and n be as above. We distinguish two cases. II. C a s e 1 . All long lines are contained in Then t 5 1 and s o t = 1. From Lemma 3 . 5 we get b = b + b = n’+n+l - c. a c L in P which are not Hence there are exactly c lines L1, lines of S. Since b = n2 + 1 III , we have Y I I l = n+l-c. Now it is easy to see that S is the complement of the c concurrent lines L1,. ,LC‘ C a s e 2 . There is a long line L ’ outside II. Since every point of L ’ is on a unique line of n , we have I n 1 2 n, and so b 2 n2+n. This means b € tn2+n,n2+n+11. Consider first the possibility b = n‘+n. Then there is exactly one line X in P which is not a line of S . Because all the points of S have degree n+l, none of the points of X is a point of S. Adding X to S we get a Il,c-ll-affine linear space s ‘ , in which every point has degree n+l. Corollary 3 . 6 shows that S ’ is the complement of a maximal (c-1)-arc. Suppose finally b = n2+n+l. Then I n 1 = n+l. Let s be the number of lomg lines in n . Then v = sn + (Inl-s)(n+l-c) = s(c-1) + (n+l)(n+l-c). On the other hand, we have v = 1 + t(c-1) + (n+l)(n-c). Together we get n = (t-s)(c-1). Therefore, c-1 divides n. From Lemma 3.5(b) we obtain furthermore that c(c-4)n2+2c(c-1)n+(c-1)’ is a perfect square. Thus, Corollary 3 . 7 is proved completely.0

...,

..

REMARKS. 1. Corollary 3 . 7 has already been proved by BEUTELSPACHER and KERSTEN [l] under the additional hypothesis b 5 n’+n+l.

52

A . Beutelspacher and K. Metsch

2. Case ( a ) of Corollary 3.7 can be obtained from the theorem of MULLIN and VANSTONE [ 5 ] . 3. If P is a projective plane of order (c-l)’, and c is a Baersubplane of P, then P-c meets the conditions of 3.7(c). c, COROLLARY 3 . 8 . Let a and c be positive integers with 2 5 a and let S be a finite Ia,cl-affine linear space in which every point has degree n+l. Suppose that S satisfies the following conditions: ~) - 2 n > T1( C Z - ~ ) ( C ‘ + C - ~ ~++ c3z’+4(c-a+1)2 (1) and 1 n > z(c2 -1)(c2+c-2a+2). (2) Then S is embeddable in a projective plane P of order n and one of the following cases occurs: (a) There is a positive integer x with a = x 2 + 1 and c = x’+x+l; S is the complement of a subplane of order x in a projective plane of order n. (b) n’+n+l-a 5 b 5 n2+n, c-a divides n and c-1. In particular, c 5 2a-1. (c) b ;n 2 + n + l , c-a divides n, and [ ( ~ - a + 1 ) ~ - 4 c ] n ~ + 2 ( c ~ + a ‘ - ~ - a ) n +(c-a) is a perfect square. PROOF. From Lemma 3.5 we get b = n’+n+l + z = n2+n+l - c + f(t), where f(t) = [(t-a)(c-a)+l-a]t - [t(c-a)+l-cl(t-c) (3) n+l-a n+l-c Obviously, f(t) is a polynomial of second degree with negative coefficient in t‘, which has its maximum in t = (n+l)(c-a) + n 2(c-a) From (2) we get (4) f(0) = - c(c-l) > -1, and n+l-c n a(a-1) > c-a-1. f(-) c-a = f(n+l) = c-a - n+l-a First we show that S is embeddable in a projective plane of order n. If z 5 0, this follows from ( 2 ) and Corollary 3.3. Therefore, we may assume z > 0. Then

.

(n+l)’ n+ 1 < n+l < < (n+l-a)(n+l-c) n+l-a-c n+l-2c -

53

Embedding Finite Linear Spaces

Z(C*-C3+3C2+C-4) 1 + 1 < 51 -(c‘-c3 +3c2+c-4)+1-2c 2

c3-c2+4

c3-c2

Thus I

and c3-c2+4 2 = f(t) c 5 f ( t ) - c < 4(c,-cd)(c-n+l)’ (5) In view of ( l ) , this implies 1 1 n > -(c2-1)(c2+c-2a+?) + Zc(c-1)z. 2 On the other hand, Theorem 3.2(b) yields

-

-

c.

n 5 2(c-l)(c2-c+l) + z(c-1)~. Together, we have a = 2, c = 3 and z 2 2 0 , which contradicts ( 5 ) . Therefore, b 5 n’+n+l, and S is enbeddable in a projective plane P of order n. Denote by x the set of points of P which are not points of S. Consider a point p of x . Since the lines of S through p constitute a parallel class of s , every line L of degree n+l-d is contained in exactly d parallel classes nl(L),...,IId(L). Furthermore, every line parallel to L lies in exactly one parallel class ni(L). It follows in particular b = 1 + (n+l-d)n + I nl(L) I +.. .+- I nd(L) 1 . Now we distinguish two cases. C a s e 1 . There is a parallel class n of S having exaczly n+l lines. If s denotes the number of long lines in TI, then Inl(n+l-c) + s(c-a) = v = 1 + (n+l-t)(n-c) + t(n-a). Hence s(c-a) = t(c-a) - n, so c-a divides n. Furthermore, t = s + -n 2 L . c-a c-a ‘ hence ( 4 ) implies b 2 n2+n+1-a. If b = n2+n+1, then Lemma 3.5(b) shows that we are in case (c) of Corollary 3.8. Therefore we may assume that n‘+n+l-a 5 b 5 n2+n. Let X be a line of P which is not a line of S . Since every point of S has degree n+l, each point p. of X lies in x (i E t 1,. ,n+l}). Let hi be the number o* lines of S through pi. Then h . 5 n and h + + hn+l = b. Since b n2+n+l-a, there with h . = n. Therefore the lines of S is a j l € {l,...,n+l{ through p . form a parallel ciass n ’ with exactly n elements. deAotes the number of long lines in n ‘ , then If s ’ n(n+l-c) + s’(c-a) = v = 1 + (n+l-t)(n-c) + t(n-a).

..

...

Hence s’(c-a) = t(c-a) - (c-1). Consequently, c-a c-1, and now we are in case ( b ) of Corollary 3.8. Case 2 .

Consider

is a divisor of

Every parallel class of S has at most n elements. a short line L of S . ThenC n’+n+l-c 5 b = 1 + k n + ( I I I ~ ( L , )5~ -n2+n+l-c. ~) i=l

C

A . Beutelspacher and K. Metsch

54 b = n’+n+l-c

Hence

and

...

Ini(L)I = n (i E [l, ,cl) for every short line L. In particular we have f(t) = 0. By (4) we obtain t < A. c-a Consider now o long line L of S . We get a n2+n+l-c = b = 1 + k n + ( Ini(~)I-l)5 n’+n+l-a. i=l Hence there exists a parallel class n which contains L and has fewer than n elements. Since n is a parallel class corresponding to a point of x, it follows in view of ( 6 ) t.hat every line of n is long. Hence, n+l-a divides v ( = l+(n+l)(n-c)+t(c-a)); so, (7) n+l-a I (t-a)(c-a) + 1 - a. Since t 5 n we have (t-a)(c-a)+l-a < n+l-a On the other ha.nd, (6)

.

-

=,

we get from (2) that (t-a)(c-a)+l-a 2 -a(c-a)+l-a > -c(c-1 +l-c = -c2+1> -(n+l-a), so, I(t-a)(c-a)+l-al < n+l-a. NOW, ( 7 ) implies t-a)(c-a)+l-a = 0, in particular -a v = (n+l-a)(n-c+a), and t = a(c-a)-1 c-a Toaether with [ (c-a)(n+l)+n] + c(c-1) (n+l-a) 0 = f(t) = t 2 (c-al2 - t.(c--a) (n+l-a)(n+l-c) we get ~i[a!c-a)~ +2!a-1) (c-a)-(c-I)*] = (a-1)[a(~-a)~+Z(a-l) (c-a)t(c-1)2 1. n f a-1, we obtain a(c-a + 2(a-l)(c-a) ( ~ - 1 = ) ~0 , and so c = a + Ja-1. Therefore, there is a positive integer x satisfying a = x2+l, c = x’+x+l, v = (n-x2)(n-x), b = n’+n+l (x2+x+l), ba = x’+x+l)(n-x), and bc = n2+n+l - (x2+x+1) ba. Moreover. if n is a parallel class corresponding to a point of P outside S , then one of the following possibilities occurs: Since

-

-

n contains x2 long and n-x2 short lines; V n+3.-a , and n consists of lomg lines only.

(I) 1x1 = n, and (11)

1 !1

=

Using these properties it is now easy to see that we are in case (a) of our corol1ary.n REMARKS. 1. Suppose a = 2 and c = 3 . If we are in case (a) of the above corollary and if n > 42, then S is the complement of a triangle in a projective plane of order n. (This result has been proved in case n > 7 by DE WITTE 18.1.) 2. The existence of strectures in case (b) and ( c ) satisfying (1) and (2) is not known to the authors. COROLLARY 3.9. If S is a finite (2,4l-semiaffine linear space in which every point has degree n+l, then n E I5,7,131. PROOF. Since a short line has n-3 points, we have n 2 5 . By 3.5 we get b2 = (n-2)t + (2t-5)t (81 n-l , b4 = n2+n-3 - (2t-3jft-4) n-3 and b = n2+n-3 + f(t) with f(t) = (2t-5)t - (2t-3)(t-4). n-1 n-3

Emhediiiug Finite Linear Spaces

55

Obviously, f is a polynomial of second degree with negative leading coefficient, which takes its maximum at t = (3n+2)/4. Since n 2 5, we have

therefore From n-1

f(t) 5 3. 1 (2t-5)t and f(t) 2 f(4)

=

n 2 5

t 1. 4; consequently,

we get

12 n -l > 0

or

f(t)

2 f(n+l)

=

2

-

1-l n

1.

with s E (1,2,3}. By (l), we obtain 3n+2r JD with D = (9-4s)n2 + (16s-36)n + 52-12s. t = 4 Now we distinguish three cases.

Hence

f(t)

=

s

C a s e 1 . s = 3. Then D = -3n’ + 12n + 16 C a s e 2 . s = 2. Then D = n’ - 4n

we get

+ 28

=

2

0, and therefore

(n-2)’

+

24. Since

n = 5. C

is a perfect square,

n = 7.

C a s e 3 . s = 1. Then D = 5(n2-4n+8) and b = n‘+n-2. Assume n > 135. Then, by Corollary 3.3, S is embeddable in a projective plane of order n. Because b < n’+n+l, there is a parallel class n of S with n elements. If s denotes the number of lonq lines in n , then v = lnl(n-3) + 2s = n2 - 3n + 2 s .

On the other hand, we have v = 1 + (n+l)(n-4)

+ 2t

=

n2

-

3n - 3

+ 2t.

Together we get 2(t-s) = 3, a contradiction. Consequently n 2 135. Since D is a perfect square, it follows n E 16,13,31,78}. In view of t = (3n+2tJD)/4 and t 5 n we get n 6,31,78. So, n = 13.0 REMARK. The authors do not know, whether the structures considered in 3.9 exist in the cases n = 7 or n = 13. For n = 5 we give the following example. Let A be an affine plane of order 4, and let S ‘ be the linear space which is obtained by removing one of the points of A. Then there are five lines of degree 3 in S ’ . Replacing each of these lines by three lines of degree 2, we get a 12,4l-affine linear space S of order 5 with 15 lines of degree 4 and 15 lines of degree 2. REFERENCES

[ l ] Beutelspacher, A. and Kersten, A., Finite semiaffine linear spaces, Arch. Math. 44 (1985), 557-568. [2] Bose, R.C.: Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math. 13 (1963), 389-419. [ 3 ] Bruck, R.H.: Finite nets 11. Uniqueness and imbedding, Pacific J. Math. 13 (19631, 421-457. [ 4 ] Hall, M.: Projective planes, Trans. Amer. Math. SOC. 54 (19431, 229-211.

A . Beutelspacher and K. Metsch

56

[5] Mullin, R.C. and Vanstone, S.A.: A generalization of a theorem of Totten, J. Austral. Math. SOC. A 2 2 (1976), 494-500. [ 6 ] Thas, J.A. and De Clerck, F.: Some applications of the fundamen-

tal characterization theorem of R.C. Bose on partial geometries, Rend. Sc. fis. mat. e nat. 59 (1975), 86-90. Lincei

-

[7] Vanstone, S . A . , The extendability of (r,l)-designs, in: Proc. third Manitoba conference on numerical math. 1973, 409-418. [ 8 ] De Witte, P., On the complement of a triangle in a projective

plane, to appear.

Annals of Discrete Mathematics 30 (1986) 57-68 0 Elsevier Science Publishers B.V. (North-Holland)

57

VERONESE QUADRUPLES Alessandro B i c h a r a D i p a r t i m e n t o d i Matematica I s t i t u t o " G . Castelnuovo" U n i v e r s i t a d i Roma "La Sapienza" 1-00185 - Rome, I t a l y

ABSTRACT. The c l a s s i c a l Veronese v a r i e t y r e p r e s e n t i n g t h e conics i n a p r o j e c t i v e plane i s generalized s t a r t i n g f r o m Buekenhout o v a l s . T h i s l e a d s t o t h e d e f i n i t i o n o f a Veronese quadruple which i s c o m p l e t e l y c h a r a c t e r i z e d as a p r o p e r i r r e d u c i b l e p a r t i a l l i n e a r space c o n t a i n i n g two d i s j o i n t f a m i l i e s o f suspaces s a t i s f y i n g s u i t a b l e axioms. 1. INTRODUCTION The p a i r -S = ( P , L / i s s a i d t o be a p r o p e r i r r e d u c i b l e p a r t i a l l i n e a r space

( P L S ) i f P i s a non-empty s e t , whose elements a r e c a l l e d p o i n t s , L i s a p r o p e r f a m i l y o f subsets o f P , l i n e s , and t h e f o l l o w i n g h o l d [3]: ( i ) Through any p o i n t of S t h e r e i s a t l e a s t one l i n e . ( i i ) Any two l i n e s have a t most one p o i n t i n common.

(iii) Any l i n e o f S i s on a t l e a s t t h r e e p o i n t s .

( i v ) There e x i s t two d i s t i n c t p o i n t s such t h a t no l i n e c o n t a i n s b o t h o f them. Through t h i s paper

,?

=

( P , L J denotes a p a r t i a l l i n e a r space.

Two d i s t i n c t p o i n t s p and q i n S a r e s a i d t o be c o l l i n e a r ,

if t h e y l i e on

a common l i n e ; i n t h i s case we w r i t e p s q .

A subset H o f 7' i s s a i d t o be a p r o p e r subspace o f

2,

H consist o f col-

if

l i n e a r p o i n t s , a t l e a s t t h r e e o f which a r e n o t on t h e same l i n e . Now we c o n s t r u c t an i r r e d u c i b l e p r o p e r PLS

5

c o n t a i n i n g p r o p e r subspaces.

L e t P = ( 7 , B ) be an i r r e d u c i b l e p r o j e c t i v e p l a n e o f o r d e r g r e a t e r t h a n t h r e e and denote by ? t h e s e t o f a l l unordered p a i r s [ l , s ] P,

of lines i n

P. F o r a l i n e 1 i n

we d e f i n e

al

= t

Such a

!I,s

nl

1

: s E i31.

i s n a t u r a l l y endowed w i t h t h e s t r u c t u r e o f a p r o j e c t i v e p l a n e

A . Biclzara

58

(three pairs [l,sl

s,,

s,,

s,

1,

[l,s,],

[ l , s , ] a r e s a i d t o be c o l l i n e a r i f t h e t h r e e l i n e s

a r e c o n c u r r e n t i n P ) . Such a p l a n e i s isomorphic t o t h e dual p l a n e

o f P. Def ine

PI =

Next, f o r p

a

P

E

In

1

: 1 E DI,

3 define

t[l,sl,

pE1,

PES,

1, S E 01

I f a Buekenhout o v a l [ 2 ] B ( p ) i s d e f i n e d on t h e p e n c i l F ( p 1 o f l i n e s i n P t h r o u g h p, t h e n t h e s t r u c t u r e o f l i n e a r space lows: t h e t h r e e p a i r s [ l , ,sl

1,

[lp,sz

as f o l P [ l 3 , s 3 1a r e c o l l i n e a r i f e i t h e r an i n -

1,

( B ( p ) ) can be g i v e n t o

a

v o l u t i o n o f B ( p ) i n t e r c h a n g e s li ans s ( i = 1,2,3), o r 1 , = l 2 = 1 ). Therefore, i t h e l i n e a r spaces a ( B ( p ) l i s isomorphic t o t h e dual space o f t h e one c o n t a i n i n g B(p). i t w i l l be assumed t h a t f o r any p o i n t p i n 3 a Buekenhout

I n what f o l l o w s ,

o v a l B ( p ) i s g i v e n . THen we d e f i n e P , = ( a ( B ( p ) ) : p € 9 1 . Denote by L t h e f a m i l y o f l i n e s b e l o n g i n g t o some elment i n P,u

P 2 . Hence

the p a i r S = ( P , L ) i s a proper i r r e d u c i b l e PLS c o n t a i n i n g t h e c o l l e c t i o n s P I and

P, o f p r o p e r subspaces. The quadruple ( P , L , PI, P I ) w i l l be s a i d t o be t h e Veronese space o f P ass o c i a t e d w i t h t h e f a m i l y i B ( p ) : p~ 71 o f Buekenhout o v a l s .

I n o r d e r t o c h a r a c t e r i z e t h e Veronese space o f a p l a n e a s s o c i a t e d w i t h a c o l l e c t i o n o f Buekenhout o v a l s , we d e f i n e a Veronese quadruple as a quadruple

( P I L, P , , P 2 ) s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n s , (i) The p a i r S = ( P , L ) i s a p r o p e r i r r e d u c i b l e PLS. ( i i)

P

and

P,

a r e two d i s j o i n t f a m i l i e s o f p r o p e r subspaces such t h r o u g h

any l i n e o f S t h e r e i s a t l e a s t one subspace o f $ , U p ,

(1.1)

F o r any n E P , , and any a E P , e i t h e r

(1.2)

Any two d i s t i n c t subspaces i n common.

P

i

ann =

(i= 1,2)

0

and

o r any l i n e i n a meets n . have p r e c i s e l y one p o i n t i n

Veronese Quadruples

59

Through any p o i n t t h e r e a r e a t l e a s t t h r e e elements o f P I u P ,

(1.3)

most two elements o f P I

.

I f a p o i n t p i s on two elements o f

(1.4)

and a t

P,, t h e n p i s on e x a c t l y one element

o f PI. Any t h r e e elements i n Y',

(1.5)

meeting t h e same element i n P , have a common

point. Any t h r e e elements o f P I meeting t h e same element i n P, meet e v e r y e l e -

(1.6)

ment i n 7 , i n c o l l i n e a r p o i n t s .

P , i n c o l l i n e a r p o i n t s then

I f t h r e e elements i n 7 , meet a subspace i n

(1.7)

t h e r e e x i s t s an element i n

P, h a v i n g a n o n - t r i v i a l i n t e r s e c t i o n w i t h

each o f them. An isomorphism between two Veronese quadruples ( P . L , P l $P,) and ( ? ' , L ' , ?',

,PI2) i s a bijection f : P

+

p ' such t h a t

( 1 .a)

Both f and f - ' map l i n e s o n t o l i n e s .

(1.9)

Both f and f - ' p r e s e r v . t h e two c o l l e c t i o n s o f p r o p e r subspaces. It i s easy t o check t h a t t h e Veronese space ( P , L , P ,

.P,

) o f a projective

( 7 , f l ) a s s o c i a t e d w i t h a f a m i l y i B ( p ) : p E S ) o f Buekenhout o v a l s i s

plane P

a Veronese quadruple. Furthermore, i f

P can be c o o r d i n a t i s e d by a (commutative)

f i e l d K and each B ( p ) i s a s s o c i a t e d w i t h a c o n i c i n P , t h e n ! p z L P I $, 1 i s i s o morphic t o t h a t p a r t o f t h e c u b i c s u r f a c e Mt i n PG(5,K),

representing the conics

i n P which s p l i t i n t o two l i n e s i n P [ l ] . I n t h i s paper t h e f o l l o w i n g r e s u l t s w i l l be proved.

I. - If Q = (P,L,P,,P,

i s a Veronese quadruple, t h e n t h e r e e x i s t s a p r o -

j e c t i v e p l a n e o f o r d e r g r e a t e r t h e n t h r e e such t h a t f o r each p o i n t p i n P a Buekenhout o v a l B ( p ) i s d e f i n e d on t h e p e n c i l o f l i n e s t h r o u g h p. Furthermore, Q i s i s o m o r p h i c t o t h e Veronese space o f P a s s o c i a t e d w i t h t h e f a m i l y ( B ( p ) : p ~ P l

When p i s f i n i t e , axiom ( 1 . 7 ) w i l l be shown t o be a consequence of t h e r e mai n i ng ones. 11. PLS and

?1

Let Q and

=

( ? , / . , P I ,?,)

be a quadruple i n which ( P , L ) i s an i r r e d u c b l e

f', a r e two f a m i l i e s o f p r o p e r subspaces such t h a t t h r o u g h any

S t h e r e i s a t l e a s t one subspace o f P I U ,y' line of -

f i l s axioms ( 1 . 1 )

,..., ( 1 . 6 ) .

Suppose moreover t h a t Q

If S i s f i n i t e , then also (1.7) holds.

ul-

A . Bichara

60 2. SOME PROPERTIES

OF VERONESE QUADRUPLES

L e t Q = (P,.L, PI,?',)

be a Veronese quadruple.

111. Denote by V t h e s e t o f a l l p o i n t s i n ment i n

P , passesand by A t h e s e t o f a l l p o i n t s

2 t h r o u g h which e x a c t l y one e l e i n 2 t h r o u g h which p r e c i s e l y two

elements pass o f P I . Then b o t h A and V a r e non-empty and P = A uV. __ Proof.

By (1.3),

2

t h r o u g h any p o i n t p i n

a t most two elements pass of

P I . I f p IE A,

i.e. t h r o u g h p a t most one element passes o f P I

exist i n 8,

c o n t a i n i n g it; t h e r e f o r e ,

two elements

t h r o u g h p p r e c i s e l y one subspace passes

o f P , ( s e e ( 1 . 4 ) ) and p e V . Consequently, P Next, A #

, then

A uV.

=

,

0

w i l l be proved. Take q E P and v , a subspace i n P , t h r o u g h q,

( b y t h e p r e v i o u s argument such a subspace e x i s t s i n P I ) . S i n c e ( P , L ) i s a prcjpe r e r PLS and

of P,

V,

i s a subspace a p o i n t q, e x i s t s i n

t h r o u g h q 2 . Obviously,

~ l ,

f

P\

vl. Let

and by (1.2) t h e p o i n t

IT,

n2

be an element belongs t o

n,na,

A; t h u s , A # 0.

ai

Finally, V #

0 will

P2, i

passes (see ( 1 . 3 ) ) .

E

Assume

= 1,2,

a, =

az

and l e t

be proved. Through b o t h q

CI

If a , #

be an element i n

a2,

and q 2 a t l e a s t one element then

C I , ~CI,

i s a p o i n t i n V.

P2 t h r o u g h a p o i n t q o f f

a , ; thus

a n a , E V and t h e statement i s proved.

The n e x t p r o p o s i t i o n i s a s t r i g h t f o r w a r d consequence o f axioms ( 1 . 3 ) and (1.4).

I V . Through any p o i n t i n V a t l e a s t two d i s t i n c t subspaces o f P ,

pass and

t h r o u g h any p o i n t i n A p r e c i s e l y two subspaces o f P I and one o f P , pass. V.

If n € P , ; then I n n V I ~ l .

P r o o f . Assume

TI

c o n t a i n s two d i s t i n c t p o i n t s i n V, say p , and p 2 . By prop.

I V , t h r o u g h p , two d i s t i n c t subspaces a , and a : o f

p o i n t p , (see (1.2));

hence, t h e y have a non-empty i n t e r s e c t i o n w i t h

ly, t w o subspaces a,and

subspaces

a., a ' . , 1

1

P2 pass which share j u s t t h e

a; e x i s t s i n P,meeting

i = 1,2,

o f P , share a p o i n t b y (1.5).

belongs t o a, na; so t h a t a , n a ~ > t p , , p , ] ;

TI.

Similar-

a t p, and n o t skew w i t h

IT. The

Therefore, p I =

a c o n t r a d i c t i o n , s i n c e p 1 # p 2 . The

statement f o l l o w s .

V I . If T E P , , a E P , , then e i t h e r P r o o f . Assume __

nna =

0 o r l n n a l 2 3.

a n n # 0; t h u s , a p o i n t p e x i s t s i n

a t h r o u g h p and q a p o i n t i n a n o t on

1 (since

a

anv.

L e t 1 be a l i n e i n

i s a p r o p e r subspace i t i s n o t

Veronese Quadruples coincident with

1;

61

q e x i s t s ) and q ' a p o i n t on 1 . By ( 1 , l )

hence,

-

S and - o b v i o u s l y t h r o u g h q and q ' meets n a t a p o i n t p ' E spaces

and

TI

If

EP,,

TI

then e i t h e r

a€$,,

P r o o f . By pr0p.s V and V I ,

if

. The

p 1 and p , o f A l i e on n n a

nna =

0 or

111

nna

By prop.

nanA.

I V , through p

ses o t h e r t h a n n. Moreover, meet

a

( a t l e a s t a t pi,

hence, p l ,

sub-

T,

resp.

EL.

l i n e 1 i n L t h r o u g h them i s c o n t a i n e d i n n n a ,

i'

i = 1,2,3,

nn = p , .

TI

1 , t a k e a p o i n t p, i n

n o =

p r e c i s e l y one subspace

The t h r e e subspaces

and b y axiom (1.6) meet

p , a r e c o l l i n e a r and

p,,

2 3.

# 0 t h e n a t l e a s t two d i s t i n c t p o i n t s

Tna

as two subspaces meet a t a subspace. To p r o v e t h a t TI

p ' # p. The

o f ( P , L ) meet a t a subspace which c o n t a i n s t h e l i n e t h r o u g h i t s

d i s t i n c t p o i n t s p and p ' . Since (F',L) i s i r r e d u c i b l e ,

VII.

the l i n e

nna

n A c l . If n

T

T., 1

TI

i

o f $,pas-

i = 1,2,3

i n c o l l i n e a r points;

n a =

T I ~ G ~ tAh e, n t h e

statement i s ?roved. On t h e o t h e r hand, i f t h e r e e x i s t s a p o i n t q o f V i n then,

by prop. V,

nna

nV

=

q, whence

nna =

i s an i r r e d u c i b l e l i n e a r space, t h e r e f o r e ,

all

nna,

1 U L q ) . The subspace n n a o f ( $ , L )

q €1 o t h e r w i s e any l i n e i n

nn

(I

join-

i n g q w i t h a p o i n t p on 1 would c o n s i s t s of j u s t p and q, a c o n t r a d i c t i o n . The statement f o l l o w s .

3. THE PLANE (P2,B)

Let TI

be an element i n P 1 . By prop. V I I ,

TI

e v e r y subspace a o f , ' Y

a t a p o i n t meets i t a t a l i n e i n L. Thus, t h e f o l l o w i n g subset o f

veeting

P 2 i s defin-

ed B(n) = { a € $ , : ( 1 n n ~ L 1 .

VIII.

If

elements i n Proof. -

Ti

E

P I , t h e n B ( T ) i s d i s t i n c t f r o m P , and c o n t a i n s a t l e a s t two

P,; hence, By prop.

P, passes which meets a p o i n t p, i n

TI\

B ( n ) i s a p r o p e r subset of

IV,

t h r o u g h any p o i n t p

n at a line 1 i n

1 t h e r e i s a subspace

O f course, ~ , E B ( T I ) and s i n c e a , € B ( n ) ,

,in

P2. TI

a t l e a s t one element

a2

other than

a1

P , meets

TI

Through

IB(TI)I,Z.

P 2 . Thus, any sub-

=

i n a l i n e i n L , so t h a t , by axiom (1.5);

i n V t h r o u g h which a l l t h e elements i n P , pass. Therefore, axiom ( 1 , 4 ) ,

B(n).

o f P, ( p , € a , and p,BaI).

To p r o v e t h a t B ( T T ) # P 2 , assume, on t h e c o n t r a r y , B ( T I ) space o f

(I,

L ( s e e prop. V I I ) ; t h u s ,

of

t h r o u g h q p r e c i s e l y one element

n' E

PI

a point q exists

s i n c e I B ( T I ) ~ by ~ ~ ,

passes.

TI'

contains t h e

A . Bichara

62

p o i n t q which i s on a l l elements i n P 2 ; hence,

i s met by e v e r y subspace o f

P , a t a l i n e i n L t h r o u g h q (see prop. V I I ) . Next, l e t q ,

and q z be two p o i n t s

o t h e r t h a n q and n o n - c o l l i n e a r w i t h i t . By prop.V,

q , and q , b e l o n g t o A

in

9 '

and t h r o u g h t h e n t h e subspaces and

TIi

#

IT,

. Through

TI,

ses, The subspace u meets

and n ,

TI,

the point q' =

meets

p r e c i s e l y one element a EP? pas-

n,nn,

~

l

'

(since

' 1 . Thus, by axiom (1.11, t h e l i n e 1

Consequently, q , i s on 1 , and s i n c e

IT,.

l a r argument, q, E elements i n ?,, TI';

TI

, TI, #

TI,

on

a t a p o i n t . Since 1, i s c o n t a i n e d i n n, i t passes t h r o u g h t h e p o i n t

11'

IT'^

q1 =

P 1 pass; moreover,

of

a t a l i n e 1 , i n L and i s n o t skew w i t h

TI,

no e l e m e i t i n 8 , i s d i s j o i n t f r o m a

, resp.

. The subspace

D

TI

D.

By a s i m i -

a E P , passes t h r o u g h q, t h e p o i n t on a l l t h e

t h u s i t c o n t a i n s q, q l ,

hence, I q,q , , q , ~ c u n

q , l i e s on

I,CU,

and q 2 . These t h r e e p o i n t s a r e a l s o on

a c o n t r a d i c t i o n s i n c e these p o i n t s a r e n o n - c o l -

I,

l i n e a r (see prop. V I I ) . The statement f o l l o w s . I f n , , n 2 ~ P I yIT, # TI^

IX.

, then

P r o o f . Through t h e p o i n t p

~

prop. I V ) . O b v i o u s l y , l i n e 1, E L and meets n 2 Since

IB(n,)nB(n21I

= n,nn,

a t a p o i n t pi E n 2 . Since p '

TI^ and TI^

a u n i q u e element

E

P,

passes (see

U E B ( T , 1 n B ( n , 1. Take D ' ~ B ( T I , ) ~ B ( T I , D' ) ; meets a t a l i n e 1 , L. ~ 1

IT,

1.

=

i s contained i n

El

, and

a'

and by axiom ( 1 . 1 )

1 ,C n,, p ' E n, whence p ' E TI^

meet j u s t a t p, p ' = p. Therefore,

at a

TI,

Tin,.

a ' passes t h r o u g h p and i s

c o i n c i d e n t w i t h u t h e unique element t h r o u g h p i n P 2

.

Thus,

D

= B(n,)nB(n,)

and t h e statement i s proved. By prop.

IX,

B = { B ( n ) : n c P , } o f p r o p e r subsets o f P 2 i s d e f i n -

the family

ed.

X. The p a i r (F'2yB) i s a p r o j e c t i v e plane. ~

P r o o f . By pr0p.s

V I I I and I X ,

0 i s a p r o p e r c o l l e c t i o n o f proper subsets

o f P , and any two d i s t i n c t elements i n 13 share p r e c i s e l y one element i n P 2 .

Next, we prove t h a t two d i s t i n c t elements a u n i q u e element o f

B. Set p = a , n a ,

P I passes (see prop. I V ) . Obviously, u , , XI.

TI

€ P I implies I n n V l

a, and

a2

i n P z are contained i n

; t h r o u g h p e x a c t l y one element a, E

TI

iri

B ( T I ) and t h e statement f o l l o w s .

1 . Furthermore, t h e l i n e s i n

TI

which a r e t h e i n -

t e r s e c t i o n s o f n by elements i n B ( T I ( G P ~ )a r e p r e c i s e l y t h e l i n e s i n t h e pencil (in

TI)

with centre a t the point v =

nnV.

Thus, p a i r w i s e d i s t i n c t p o i n t s i n

n \ { v l a r e c o l l i n e a r w i t h v i f t h e y belong t o t h e same element i n B(TI).

__ Proof. L e t

a,

and

a2

be two d i s t i n c t elements o f

B(n

1 ( t h e y do e x i s t by

Verotiese Quadruples prop. V I I I ) and s e t v = a l n sume v

a

lines 1

(see axiom ( 1 . 2 ) ) ; o f course, v belongs t o V . A s -

a2

n; t h r o u g h v a u n i q u e element

,

i = 1,2,

= a,

63

n n ' and 1, = a, n

and belong t o a, and

n'

P,

of

IT'

a r e d i s t i n c t , as l a , n a , resp. Since

az,

passes o t h e r t h a n

1

=

I T .Thus,

1

1 and

the

1.1 2 2 , 1

a , and a, belong t o B ( n ) ,

the

l i n e s 1, and 1, meet n a t t h e p o i n t s q, and q,,

r e s p e c t i v e l y , which a r e d i s t i n c t ,

o t h e r w i s e v would be c o i n c i d e n t w i t h q,

i m p o s s i b l e as v a n . Hence, ~ n n '

= q,,

c o n t a i n s t h e two p o i n t s q, and q 2 , a c o n t r a d i c t i o n ( s e e axiom ( 1 . 2 ) ) . v belongs t o

IT

and by p r o p . V

By axiom ( 1 . 5 )

meets n

n

~i

Therefore

nV = [ v ) .

any element i n B ( n ) c o n t a i n s t h e p o i n t v = a , n

; thus, i t

a2

a t a l i n e i n L t h r o u g h v (see prop. VII: S i n c e t h r o u g h e v e r y p o i n t i n

t h e r e pass a t l e a s t one element i n

are t h e sections o f

n

P,,

by t h e elements i n

hence i n B ( I I ) , t h e l i n e s i n n , which are p r e c i s e l y t h e l i n e s through

B(n),

v. The statement f o l l o w s . XII.

(P2,5) i s a n i r r e d u c i b l e p r o j e c t i v e plane.

Proof. Take n as

n

E

P , and v

=

i s a p r o p e r subspace of

i~

n V. A l i n e 1 does e x i s t i n n n o t t h r o u g h v,

( a , L ) . Since

(a , L )

i s irreducible,

l i n e s j o i n i n g v w i t h p o i n t s on 1 a r e d i s t i n c t and c o n t a i n e d i n least three l i n e s e x i s t i n

IT

t h r o u g h v. By prop. X I ,

111 3 3 . The n

.

Thus,

at

on any l i n e i n ( P z , B ) a t

l e a s t t h r e e p o i n t s l i e and t h e statement i s proved.

4. THE PROOF OF PROPOSITION I

I n t h i s s e c t i o n prop. I w i l l be proved. Take

C Z EP

, and La be t h e s e t o f a l l l i n e s i n L on

F ( a ) o f t h e l i n e s i n t h e p r o j e c t i v e plane ( p ,

,o)

a;

consider t h e p e n c i l

through t h e p o i n t a

E

P 2 ; ob-

v i ously , F(a) = iB(n) €5: a n

I f l c L a , a correspondence i

1

il( B ( v 1 )

(4.1)

\

= B(n')

1

= B(n)

~i

EL 1

: F(a) w

+

F(a) i s d e f i n e d as f o l l o w s

n n n '

n

VE

€1,

n

#

n',

I.

By axiom ( l . l )any , element i n P I m e e t i n g a i n a l i n e shares a p o i n t w i t h 1. Moreover,

t h e p o i n t s on 1 n o t i n V b e l o n g t o A and (see p r o p . 111) t h r o u g h

each of them e x a c t l y two elements o f P , pass. T h e r e f o r e , il i s a b i j e c t i o n and

64

A . Bichara

and i n v o l u t i o n o f F( a) whose f i x e d l i n e s a r e a l l t h e l i n e s B ( n ) i n F ( a ) such t h a t n nV € 1 ; F(a) i f

n n s '

f u r t h e r m o r e , i i n t e r c h a n g e s t h e l i n e s B ( n ) and B ( n ' ) , 1 ~ 1. Thus, t h e n e x t statement has been proved.

I f a e P 2 and l E 6 , t h e b i j e c t i o n i : F ( a ) 1 an i n v o l u t i o n .

XIII.

If

a E

P2 t h e n t h e f a m i l y

6

e(a) =

.

1'

+

* #

11')

of

F ( a ) d e f i n e d by (4.1) i s

1 E L a } o f i n v o l u t i o n s o f F ( a ) i s de-

fined. I f a ~ 7 ', ~t h e n IF( a l l

XIV.

24. Furthermore,

the pair (F(a),e(a)) i s a

Buekenhout o v a l . Proof. -

each b i j e c t i o n i : F ( a ) F ( a ( 1 EL,) i s an i n v o l u 1 I F ( a ) l '3. Next, i t w i l l be shown t h a t ( F ( a ) , e ( a ) ) i s a

By prop. X I I I ,

t i o n . Since 111 '3,

+

Buekenhout o v a l , i .e. t h a t (21 ( i l e v e r y element o f e ( o ) i s an i v o l u t i o n o f F ( a ) ( i i ) for any two p a i r s (B(n,

# n'

i , j = 1,2 j' o f l i n e s i n F ( a ) p r e c i s e l y one i n v o l u t i o n e x i s t s i n e ( a ) i n t e r c h a n g i n g B( n,)

and

B(n,)

and

From prop. X I 1 1 ( i )f o l l o w s . Thus, B(n;),

the points

nl

mi

w i t h B(n:).

B(n:)

and B ( n ; ) =

and (B(n:),B(n:)),

),B(n2))

then

( i i ) w i l l be proved.

= n 2 and n I l =

11,

.A

I f B(n, ) = B(n, 1

unique l i n e 1 e x i s t s i n

#

through

nV and n', n V, b o t h on a s i n c e B ( n , ) , B ( n : ; E F ( a ) . I f t h i s occurs

t h e n i i s t h e unique element i n e ( a ) f i x i n g b o t h B(n,) and B(n:). 1 On t h e o t h e r hand, i f B(n, 1 # B(n,) and B ( n , ' ) # B(n:), t h e n n 1 n;

a

nI2; t h e two p o i n t s

TT,

n n 2 and

and

i i s the 1 and B ( n , ' ) w i t h B(n;).

L a t h r o u g h b o t h of them. Again,

i n t e r c h a n g i n g B ( n , ) w i t h B(n,)

e(a)

n2

i n A belong t o a and a r e d i s t i n c t ;

n ' , nn',

hence, t h e r e i s a unique l i n e 1 i n unique element i n

#

A s i m i l a r argument proves ( i i ) i n t h e r e m a i n i n g cases. The statement f o l 1ows.

--

-.

.

Next, l e t ( F ' , L , P , , P collection {(F(a),

O(a)):

be t h e Veronese space o f a E p 2 1 0 f Buekenhout o v a l s .

Consider t h e mapping [B(n),

p2,8) a s s o c i a t e d w i t h t h e

@

: P

+

P

B( n')] o f d i s t i n c t l i n e s i n Li(i.e.

a s s o c i a t i n g w i t h e v e r y unordered p a i r n f n ' ) t h e p o i n t n n n ' E P and w i t h

the p a i r [B(n),B(n)] o f coincident lines i n D the point

XV

. The mapping

Proof. -

(4.2)

-

0 i s an isomorphism between

By pr0p.s 111, X I and axiom (1,2), @ i s one-to-one

and o n t o .

5

TI

nV in P

.

-

(rJ,P1,F2)

and ( P , L , P l , p 2 ) .

Vrronrse Quadruples

65

From t h e d e f i n i t i o n o f (F(a),e(n)) t h e n e x t statement f o l l o w s .

-

(4.3)

-

Any l i n e i n L on an element i n

L o n an element i n P , ;

L on an element o f

mapped by @ o n t o a l i n e i n

f u r h t e r m o r e , t h e i n v e r s e image under @ o f any l i n e i n

8, i s a l i n e i n

By axiom (1.61,

, is

L

on an element i n

d,.

t a k i n g i n t o account prop. X I ;

Any t h r e e c o l l i n e a r p o i n t s o f an element i n

(4.4)

P , a r e mapped

t h r e e c o l l i n e a r p o i n t s o f an element i n P 1 . From axiom ( 1 . 7 )

by 0 onto

and prop.

X I the

n e x t statement f o l l o w s . Three c o l l i n e a r p o i n t s on an element i n P , a r e mapped b y

(4.5)

t h r e e c o l l i n e a r p o i n t s on an element i n

F1.

@-' onto

The statement f o l l o w s f r o m ( 4 . 2 ) -

(4.5). From pr0p.s X, X I I , X I V ,

XV, prop. I f o l l o w s .

5. THE PROOF OF PROP. I 1

remark t h a t a l l p r e v i o u s r e s u l t s b u t ( 4 . 5 ) were proved w i t h o u t

Firstly,

t h e h e l p o f axiom ( 1 . 7 ) . Thus, w i t h t h e same n o t a t i o n as b e f o r e , t h e n e x t propos i t i o n can be s t a t e d . XVI.

Under t h e assumptions i n prop.

I 1 f o r t h e mapping

@

:

P

* P(4.21,

( 4 . 3 ) and ( 4 . 4 ) h o l d . Next, any space

-=

4

n

B(

fl

E

n =

E?,

determines b o t h t h e l i n e B(.

b e l o n g i n g t o B and t h e sub-

b,in ( P , L ) d e f i n e d b y ( s e e s e c t . 1 ) : n

B( n )

= ([B(n),B(n')]:

Consider t h e mapping 6 :13

V'E

PI].

o f t h e dual p l a n e o f ( 2 , ,/I) o n t o ? d e f i n e d

+

by (5.1 1

6 ( B ( n ' ) ) = [B(n),B(n')

1

.

C l e a r l y ( s e e s e c t . 11, (5.2)

$

i s an isomorphism betwecn t h e dual p l a n e o f ( P 2 , D ) and t h e sub-

space % o f (P , L ) . D e f i n e a mapping a ' : ii

(5.3)

@'([B(n),B(n')

1)

n by

= @([B(a),B(n')

1);

o b v i o u s l y , 0' i s one-to-one and o n t o . S i n c e @ and @ a r e b i j e c t i o n s ,

(5.4)

t h e mapping @'6: 13

+TI

i s a bijection.

Next, a s s m e 7) i s f i n i t e . Then P ,

i s f i n i t e and so i s t h e p r o j e c t i v e .

A . Bichara

66

p l a n e ( ' P , , B ) ; i f i t i s o f o r d e r q, t h e n 101 = q z t q t 1; hence (see ( 5 . 4 . ) ) ;

lfll

(5.5) By (5.2), (5.6) in

(5.31,

0'

J,

q

= q2 t

1.

t

and (5.4)

maps t h r e e c o n c u r r e n t l i n e s o f B

onto t h r e e c o l l i n e a r p o i n t s

TI.

Furthermore, t a k i n g i n t o account prop. X I , a l i n e i n point v

n n

=

passes

through

V if i t i s t h e image under $ ' 6 o f a p e n c i l o f l i n e s i n

the

h'zJ

t h r o u g h a p o i n t on B ( n ) . Consequently, (5.7)

There

are

precisely q

t

1 lines i n

through v =

n V and on

each o f them q t 1 p o i n t s l i e . Next, l e t 1 be a l i n e i n t o v b y a l i n e , by ( 5 . 7 ) 111 (5.8)

Any l i n e i n The q

t

q

t

5

1~

n o t t h r o u g h v. S i n c e e v e r y p o i n t on 1 i s j o i n e d

q + 1; hence

c o n s i s t s o f q t 1 p o i n t s a t most.

1 lines i n

(P z , B ) a l l have s i z e q

ped by O ' J I o n t o a l i n e i n n ( s e e (4.4) q

t

q

t

t 1 and each o f them i s map-

and (5.8));

t h e r e f o r e , on

1 l i n e s l i e each o f them having s i z e q t 1 . Since

*

11

at least

i s a subspace of

( B , L ) , by (5.51,

(5,9)

1~

i s a p r o j e c t i v e p l a n e o f o r d e r q and t h e mapping 0 'J, i s an i s o -

morphism between t h e d u a l p l a n e o f L e t pi,

i = 1,2,3, @-'(pi)

( P 2 , B )and

n.

be t h r e e p o i n t s i n n. C l e a r l y , = ( e ' l - ' ( p . 1)

Since b o t h JI and

$ I $

= (J,dJ-l(O')

')(Pi)

= J,(O'J,)-'(pi)

a r e isomorphisms t h e t k r e e p o i n t s

Q-l(p.1 are c o l 1

l i n e a r iff t h e p o i n t s p . a r e c o l l i n e a r whence ( 4 . 5 ) f o l l o w s . 1

By t h e p r e v i o u s argument and prop. X V I ,

P L,?,, $:

under t h e assumptions i n prop. 11,

+ ? s a t i s f i e s ( 4 . 2 ) t o ( 4 . 5 ) so t h a t i t i s an isomorphism between Q =

F2)

(d,

and Q = ( P , L ,P,, P J . Since f o r t h e Veronese space Q axiom ( 1 . 7 ) h o l d s ,

t h e same i s t r u e f o r Q and prop. I 1 i s proved.

REFERENCES

[ l 1 E. B e r t i n i ,

I n t r o d u z i o n e a l l a geometria p r o i e t t i v a d e g l i i p e r s p a z i , Pisa,

E. S p o e r r i (19071. [2]

F. Buekenhout, Etude i n t r i n s e q u e des o v a l e s , Rend. d i Mat. V (1966) 333-393

Veronese Quadruples [3]

G.

Tallini,

Spazi p a r z i a l i d i r e t t e ,

Sem. Gem. Comb. Univ. Roma 14 ( 1 9 7 9 ) .

61

s p a z i p o l a r i . Geometrie subimmerse,

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 69-84 0 Elsevier Science Publishers B.V. (North-Holland)

S-PARTITIONS OF

69

GROUPS

AN0 STEINER SYSTEMS

Mauro Biliotti Dipartimento di Matematica Universit2 di Lecce Lecce - ITALIA

In this paper we investigate a special class of S-partitions of finite groups. These 5-partitions are used for the construction of resolvable Steiner systems. Several classification theorems are also given.

The concept of S-partitions may be traced back to Lingenberg [ 131 although the actual introduction was made by Zappa [ 2 4 ] in 1964. Zappa developed some ideas of Lingenberg so as to provide a group-theoretical description o f linear spaces with a group of automorphisms such that the stabilizer of a line acts transitively on the points of that line. Afterward Zappa [ 2 6 ] and Scarselli [17] mainly investigated the following question: find conditions on a S-partition Z: o f a group G relating the existence o f C to that of a partition - in the usual group-theoretical sense - o f a subgroup o f G. In this case the linear space associated to C is simply the translation AndrC structure associated to that partition [ 3 ] . From a geometrical point of view, the work of Zappa [ 2 5 ] , Rosati [16] and Brenti [6] on the so-called Sylow S-partitions seems to be more interesting as Sylow S-partitions are useful in constructing some classes of Steiner systems. In this connection, another class of S-partitions is noteworthy. These S-partitions are those considered by Lingenberg [ 131 and later bv Zappa [ 2 4 ] , We shall call these S-partitions "Lingenberg S-partitions". Lingenberg S-partitions were inspired by a reconstruction method of the affine geometry A G ( n , K ) , K a field, by means of a special class of subgroups of SL(n,K). In this paper, we study Lingenberg 5-partitions o f finite groups. We mainly investigate "trivial intersection" 5-partitions which we call type I S-partitions (see section 2). For type I S-partitions, we give a "geometric" characterization and somewhat determine the corresponding group structure and action. Also we obtain a classification theorem for Lingenberg S-partitions o f doubly transitive permutation groups. We note that for some simple groups, Lingenberg S-partitions are useful in constructing resolvable Steiner systems. In these cases, the Steiner systems might be regarded as a natural affine geometry for the groups.

70

M . Biliotti

1. PRELIMINARIES

Groups and incidence s t r u c t u r e s considered here are always assumed t o be f i n i t e .

I n general, we s h a l l use standard n o t a t i o n . I f G i s a group and H 2 G, K 9 G, then O(G) i s the maximal normal subgroup o f odd order o f G, S (G) i s t h e s e t o f a l l P Sylow p-subgroups of G and HK/K i s denoted by A. If H l l K = <1> then K X H denotes the s e m i d i r e c t product of K by H. I f G i s a permutation group on a s e t R and r G f i then G

r

denotes t h e g l o b a l s t a b i l i z e r o f ?i i n G. A s e t R i s a G-set i f t h e r e i s a

homomorphism cp from G i n t o the symmetric group on G. Usually we s h a l l w r i t e GR instead o f v(G). L e t G be a group and S a subgroup of G with SzG. A s e t C o f n o n - t r i v i a l subgroups o f G such t h a t ICl22 i s s a i d t o be a (keguLatr) S-pwrLLtiion o f G i f the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : (i) (ii) (iii)

S H n s K = S f o r each H , K e Z with H#K;

f o r each g H

4

t

G t h e r e e x i s t s H t C such t h a t g f SH;

C i m p l i e s d-'Hs t C f o r each o E S.

The above d e f i n i t i o n i s due t o Zappa [ 2 4 ] . Here we are i n t e r e s t e d i n t h e f o l l o w i n g special class o f S-partitions: a S - p a r t i t i o n C o f a group G i s s a i d t o be a L i n g e n b a g S . - p a t , t i L i o ~ w l t h respect t o t h e subgroup T o f G i f t h e f o l l o w i n g hold:

(j) T 6 S < NG(T) < G; (jj) G , xtnGs x = < i > ; (jjj) C = { T * : x e G - N G ( T ) } u { N G ( T ) } . We p o i n t out t h a t C i s determined by the t r i p l e (G,S,T).

Now we g i v e some geome-

t r i c a l definitions.

II i s t h e s e t o f p o i n a and R i s a f a m i l y o f subsets o f Il whose elements are c a l l e d lines, such t h a t two d i s t i n c t

As usual a Lineah npace i s a p a i r ( T I , R ) , where

p o i n t s l i e i n e x a c t l y one l i n e . Here we assume a l s o t h a t 11 contains a t l e a s t t h r e e non-collinear points.

(n,R,//), where ( n , R ) i s a l i n e a r is an equivalence r e l a t i o n on R such t h a t each equivalence c l a s s

An AndkL btkuctuhe ( A - n h c t u h e ) i s a t r i p l e

space and

1'//1'

gives a s e t - t h e o r e t i c a l p a r t i t i o n o f Il. L e t 1' = ( I I , R , / / )

be an A-structure and l e t I I o = I I . Assume t h a t whenever P;Q,Rt

no

w i t h P#Q then the l i n e PQ and t h e p a r a l l e l l i n e through R t o PQ are wholly contained i n

no.

I f we s e t R o = { h : h b R , itcIIo}then

(or a l i n e ) and i t i s s a i d t o be a dubspace o f

(no,Ro,//) i s an A - s t r u c t u r e

Y . Isomorphisms and automorphisms

o f l i n e a r spaces and A-structures a r e d e f i n e d i n t h e u s u a l manner with t h e r e q u i r ement t h a t the p a r a l l e l i s m i s preserved f o r A-structures. A h u o t w a b t e SXeinet oybtem w i t h pakametm ( v , f i ) i s an A-structure with v p o i n t s such t h a t each l i n e contains e x a c t l y 12 p o i n t s .

71

Spartitions of Groups ond Steirrer Systems

Following Zappa [24] a q u w i - . t n a ~ h . t i o n A-4tJLuctuhe is an A-structure $=(N,K, / I ) together with a set ( o r pencil) 8 of lines and a family 8 = I T ( & ) : a c 8) of automorphism groups of 0 satisfying the following conditions: (Q,) for each P c IT there exists exactly one line JL t 0 such that P c h ; (GI2) for each n E R there exists exactly one line k E 0 such that c / / 4 ; ( Q 3 ) for each a E o the group T(K) fixes every point of k and every line parallel to t, leaves 0 invariant and acts transitively on the points of each line 4 such that 4 / / @ and b # t . In the following we shall say that (D is a quasi-translation A-structure with respect to the pencil 0 and the family 8. It is very easy to see that, starting from 0 , a new A-structure 5 may be obtained by adding to 0 a new point 0 incident with precisely the lines of 0. 5 will be called the compLetion of 0. Any automorphism a of @ extends to $ by setting a(0) = 0. Lingenberg S-partitions are essentially the geometrical counterpart of quasi-traslation A-structures. If Z is a Lingenberg S-partition of a group G with respect t o the subgroup T, define [G,S,T] as the triple given by - the set of right cosets o f S in G; - the set S ( C ) of the complexes of the form SXg with X E 1, y 6 G ; - the following relation "//" on S(Z): i f a = SN ( T ) q = NG(T)y, then a'//& if and only if either 4 ' = h or a' = STXz G with TX#T and Tg=Txz; if J L = STXy with TX+T, then ~ ' / / Jif L and only if either h ' = N G ( T ) z where TZ=TXY or J L ' = STWz with Tw#T and Txg=Twz. The definition of the parallelism seems to be very involved, but it will make clear in the following. h a t h e 6 a t t o w i n g pkopek.tiu: G o d t h e neR 0 6 t i g h t coht& 0 6 5 i n G i n t o a6 autornohphi~mo 06 [G,S,T] w h i c h i b u point-tJLaMnitiue gkaup

PROPOSITION 1.1 (Zappa [24],2.2). [ 11 t h e

mappingh

it4et6

60m

ib0tnOtphiC

( 2 ) [G,S,T]

: Sx

+

The t G p L e [G,S,T]

Sxy w i t h g E

Ro G ;

i n a q u u n i - t t a m t a t i o n A-n,i3uctutuhe

eLemen& a m t h e l i n e n NG(T)x w i t h x

E

w i X h henpect t o

t h e pen&

0 whobe

Y

G, and t o the 6 a m i L y 9 = I T x : x t G j .

Obviously in [G,S,T], we have that the point Sx belongs to the line SXg if and only if S x c S X y . Also the converse holds. Indeed, we have: PROPOSITION 1.2 (Zappa [24],2,1). Let 0 be

a quaoi-tkumLaLhn A-Athuctuhe w i t h

rtenpect t o .the penciL 0 und t o t h e t ; a m d g 9 = I T ( & ) 2, a cvmpte.te d a b

06

conjuga.te 4 u b g ~ ~ o u0p5 ~G

06

06

8 . t h o u g h P and S = Gp,

@,

p & t h e Pine

u{NG(T(p))}

0

2

IG,S,T(p)l.

u

.-

El}. Fuhthemohe unnume 3 < T ( k ) : J L C 8>. 16 P 0 a p o i n t : JL 6

.then C

=

{T()L) :

aE

8- { p } } LJ

Linyenbety S - p a h R i L o n o d G d R h ~ ~ e n p e otot T ( p ) and

M. Biliotti

I2

L e t 5 be a q u a s i - t r a n s l a t i o n A-structure w i t h respect t o the p e n c i l 0 and f o r each J L E O l e t H ( h ) denote t h e group o f a l l t h e automorphisms o f 5 f i x i n g every p o i n t o f

and every l i n e p a r a l l e l t o

4.

JI

and which leave 0 i n v a r i a n t . Then @ i s a quasi-

- t r a n s l a t i o n A-structure with respect t o Q and t o t h e f a m i l y 8 = { H ( d : k c O } . Moreover, 9 i s a complete c l a s s o f conjugate subgroups o f G = c H ( h ) : h e @ > so t h a t 5 may always be represented i n t h e form [G,Gp,H(p)l,

2. PROPERTIES AND EXAMPLES LEMMA 2.1.

each x,y

t

where P E ~ E O .

OF LINGENBERG S-PARTITIONS

d e t e m i n e u Lingenbetlg S - p a k L i t i o n , t h e n TXnTY= 6vtl

LeR (G,S,T)

G w i t h TX#TY. W

N

Paood. Consider t h e A-structure [G,S,T] and l e t ? t T X n T Y . Each p o i n t P of [G,S,T] i s on a p a r a l l e l l i n e t o NG(T)x and a l s o on a p a r a l l e l l i n e t o NG(T)y. Since these l i n e s a r e d i s t i n c t and b o t h are f i x e d by

?

then ?(P) = P and t h e r e f o r e

?

= I.

Lingenberg S - p a r t i t i o n s may be d i v i d e d i n t o two classes according t o t h e f o l l o w i n g definition:

a Lingenbetlg S-pwc.tLtion SnTX

ench

<1> d o t

=

bay t h a t

xt

it i~ a6 t y p e

C ad G w i t h kenpect t o t h e bubgtloup T LA 06 t y p e 1 4 6 ld t h e S - p a h t i L i o n 0 not 06 t y p e I,we 4haU

G with TX#T. 11.

For Lingenberg S - p a r t i t i o n s o f type I we have:

d e t e m i n e n a Lingenbekg S-pakLLtihion 06 .type 1 4 and o d y id t h e automotlpkiom gkoup 7 0 6 [G,S,T] act6 berniheguLahey on t h e Linen 0 6 0 d i b .tinct 6kom NG(T). Fuhthemohe, 4 (G,S,T) d e t e m i n e n a Lingenbekg S-pahtLtion 0 6 t y p e It h e n t h e duUowing hold: ( I ) 7 u c . i ~ t l e g U y on each .!he p a k a f i d t o NG(T) and dinLLnct @om .them, (G,S,T)

PROPOSITION 2.2.

121 N G ( T ) n T X = <1> 6ok each TK#T, and 131 [NG(T):S]

= ITI-1.

R o o d . Assume (G,S,T)

x

t

determines a Lingenberg S - p a r t i t i o n o f t y p e I and l e t

NG(T) n T y w i t h TY#T. Then

?

f i x e s both t h e l i n e NG(T) and each l i n e p a r a l l e l t o

N (T)q, so t h a t NG(T) i s pointwise f i x e d by G i t f o l l o w s t h a t x = 1 and ( 2 ) holds. Now l e t

w t G w i t h ?#T.

Then

and so z = 1 and

7

Z E

x. Therefore,

x c S and from SnTY=<1>,

z 6 T and NG(T)w,z = NG(T)w f o r some

NG(Tw) and hence z t NG(Tw)nT. But, by (21, N G ( T W ) n T = < l >

a c t s semiregularly on the l i n e s o f 0 d i s t i n c t from NG(T). The

argument may be reversed t o prove t h e converse. L e t k be a l i n e p a r a l l e l t o NG(T) and d i s t i n c t from them and assume ?(R)

= R f o r some

through R, which i s d i s t i n c t from NG(T), i s f i x e d by

on 0

-

{NG(T)},

z t. T,

z.

R 6 a. Then t h e l i n e o f 0

Since

?

acts semiregularly

i t f o l l o w s t h a t z = 1 and ( 1 ) i s proved. Now l e t lGl=g, ING(T)I=n,

I S ( = b and ( T I = t . I n G,

t h e r e e x i s t g/n - 1 d i s t i n c t complexes o f t h e form STX with

Tx#T and each complex contains e x a c t l y n t elements since S n T X = c1>. I f we take

73

S-Partitions of Groups and Steiner Systems SNG(T) = NG(T) c o n t a i n s n elements o f G and (G,S,T)

account o f t h e f a c t tha:

mines a S - p a r t i t i o n of G, then we must have ( n t - n ) ( g / n

- 1)

deter-

n = g so t h a t ( 3 )

t

now f o l l o w s .

16 (G,S,T) d e t e h m i n u a Lingenbefig S-pahtition 06 t y p e It h e n h a h u o l w a b d e SReinek nyotem w i t h paharneteu [ w , k l whehe w = [G:S1 + 1

COROLLARY 2.3. [G,S,TI

and k

=

ITI.

P J L U O ~ .This i s an immediate consequence o f (1) and ( 3 ) o f P r o p o s i t i o n 2.2.

Let 0 be a quabi-Xhun&!ativn A-bthuCtuhe w i t h henpeCt t o t h e pencil 0 and t o t h e damily 8 = { T ( k ) : h e 01. An auhomokpkinm a 0 6 @ c e n t ~ ~ a l i z eT(t) n

PROPOSITION 2.4. d o h each k e 06

0 id and o n d y

46

it 6 i x e 4 &Why &ne

06

0. A non-idenLicCLe automohpkcnm

0 6 i x i n g evehy f i n e 0 6 0 a c d 6 . p . 6 . on Rhe n e t ofi p o i n d 0 6 0.

Phoo6. I f u f i x e s every l i n e o f 0 then, by [24],3.1, a c e n t r a l i z e s T ( k ) f o r each

~ € 0 Conversely, . assume n c e n t r a l i z e s T(h) f o r each J L C O and t h e r e e x i s t s 6 E 0 such t h a t a ( n ) # n . T(o) f i x e s every p o i n t o f n and every l i n e p a r a l l e l t o 4 . L i k e wise, T(n) = a - ’ T ( n ) a f i x e s e v e r y p o i n t o f u ( n ) and every l i n e p a r a l l e l t o u ( n ) . Since n # a ( 4 ) , i t i s easy t o see t h a t t h i s y i e l d s T(n) = <1>, which i s impossible. Now l e t a. be an automorphism o f 0 f i x i n g every l i n e o f 0 and assume a(P) = P f o r

some p o i n t P. I f

JL

i s a l i n e through P and A / / & ,

6 E 0 then u ( b ) =

n and so a ( & ) / / &

which i m p l i e s ~ ( h =) fi. Therefore, a f i x e s every l i n e through P. L e t Q be a p o i n t d i s t i n c t from P and assume PQ=q{ 0. I f w denotes the l i n e o f 0 through Q, we have t h a t a ( Q ) = a ( q f \ w ) = a ( q ) n u ( w ) = q i ? w = Q. I f , on t h e c o n t r a r y , q t 0 then t h e r e l a t i o n a ( Q ) = Q can be obtained by u s i n g t h e same argument as above by s t a r t i n g from a p o i n t P ’ { 4 . The t h e s i s a = I now f o l l o w s . Now we s h a l l g i v e some examples o f Lingenberg S - p a r t i t i o n s . We assume the reader i s acquainted w i t h t h e s t r u c t u r e o f groups SL(2,q); PSU(3,qz), q = ph p a prime; S Z ( ~ ~ ~ ’ R ’ )( 3; 2 n f ’ ) , n > l , and a l s o w i t h the elementary

,

p r o p e r t i e s o f l i n e a r groups. General references are i n [ll] and [12]. I n p a r t i c u l a r , f o r Suzuki groups S Z ( ~ ‘ ~ ” ) , Ree groups R(3“”)

and PSU(3,q’)

see [20], [22] and

[23], [7]r e s p e c t i v e l y . EXAMPLE I. G

2

SL(2,q),

q = ph , q>2. L e t P c S (G) and assume T = S = P; then i t i s P

an easy e x e r c i s e t o show t h a t (G,S,T) I and t h a t [G,S,T],

determines a Lingenberg S - p a r t i t i o n o f type

the completion o f [G,S,T],

i s t h e a f f i n e plane over GF(q).

This i s t h e c l a s s i c a l example which i n s p i r e d t h e work o f Zappa [24]. I t a l s o exp l a i n s the d e f i n i t i o n o f t h e p a r a l l e l i s m i n [G,S,T] EXAMPLE 11. G

7

as given i n s e c t i o n 1.

S z ( q ) , q = Z Z n f ’ , ~ 2 1 .L e t P E S2(G) and l e t Z(P) be t h e c e n t r e o f P.

I f we assume T = Z(P) and S = P then (G,S,T)

determines a Lingenberg S - p a r t i t i o n

o f type I.Indeed, as i t i s w e l l known, NG(T) = N ( P ) and i f x { N G ( T ) then G NG(T)nTX = so t h a t NG(T) n S T X = S. Now l e t g E S T X n S T Y w i t h T # T X # T Y # T , then

M . Biliotti

74 g =

h,.t:

= h2RY w i t h b 1 , n 2 e S, R l , . t t 2 E T and hence

n;'n,

=

d1.t;1 1 x .

If. t , f l , t n f l

and G i s regarded as a c t i n g i n i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n o f degree

q z + l then the element .t$(.t;'JX, being t h e product o f two i n v o l u t i o n s w i t h o u t common f i x e d p o i n t s , f i x e s an even number o f p o i n t s . But h;'o1

l i e s i n a Sylow 2-subgroup

o f G and hence i t f i x e s e x a c t l y one p o i n t which i s a c o n t r a d i c t i o n . As we have p r e v i o u s l y shown, we cannot have R =1 f o r only one 4=1,2 and so R =1 for 4=1,2 and 4

~ E S This . y i e l d s S T x n S T Y = S. We =

S t i l l

have I T 1 = q , IS1 = q ' , :NG(T)I

q 2 ( q - I ) , I G / = ( q z + l ) q 2 ( q - I ) and hence, i f T X ' ,

...,TXQ2

=

are t h e q 2 subgroups o f

G which are conjugate t o T and d i s t i n c t f r o m them, i t i s e a s i l y seen t h a t t h e f o l -

lowing r e l a t i o n holds:

4'

1 ( I S T ' ~- ~ IS^)

This proves the a s s e r t i o n .

t

i- I

The completion o f the A-structure [G,S,T]

I N ~ ( T )=I I G ~ .

i s a r e s o l v a b l e S t e i n e r system w i t h

parameters (q(q2-q+I ) , q ) . EYAMPLE 111. G

2

PSU(3,qz), q = 2 h , h > l . L e t Pe S2(G). I t i s w e l l known t h a t NG(P) =

C, where C il c y c l i c o f order ( q 2 - I ) / d with d = ( 3 , q + l ) . Denote by Cl t h e subgroup o f C of order ( q t i ) / d and s e t T = Z(P), S = P X C,. Then (G,S,T) determines = P X

a Lingenberg S - p a r t i t i o n o f type I.Indeed, we have again NG(T) = NG(P) and, i f

x f NG(T), N G ( T ) n T X = <1>, so t h a t NG(T)nSTX = 5. Now l e t TX, Ty be such t h a t T#TX+TYgT. By w e l l known p r o p e r t i e s o f G, we have t h a t M = = SL(2,q) and M'ING(P) = <1>or Z ( P ) X C,, w i t h C, c y c l i c o f order q - I . Since q i s even, we have also t h a t ( q - I , q t ] / d ) = I and so, i f S n M # < l > then S n M = Z(P) = T. But, as we

I. (M,T,T) determines a Lingenberg S - p a r t i t i o n o f type I and

have seen i n Example

hence T X T Y n T = c l > . I t f o l l o w s t h a t T X T Y n S = <1>and t h e r e f o r e S T X n S T Y = S. The t h e s i s can now be achieved by a c a l c u l a t i o n s i m i l a r t o t h a t c a r r i e d out i n Examp l e 11. i s a r e s o l v a b l e S t e i n e r system w i t h

The completion o f the A-structure [G,S,T] parameters ( q ( q 3 - q 2 + I ) , q ) .

I n t h e case q=Zh, with h even, t h i s S t e i n e r system has

been already obtained by Schulz [la]. EXAMPLE I V . G

2

R(q), q=

32n+I

,

~ 2 1 .We s h a l l make use o f t h e r e p r e s e n t a t i o n o f G

i n P G ( 6 , q ) due t o T i t s [22],§5. L e t xI,x2,. ..,x7 be a coordinate system f o r P G ( 6 , q ) 3n+I Furthermore, l e t I be t h e hyperplane o f and l e t o c A u t ( G F ( q ) ) , a : x + x

.

P G ( 6 , q ) o f equation x , = O and denote by A the a f f i n e space obtained from P G ( 6 , q ) by assuming

I as the i d e a l hyperplane. Then x=x,/x7,

y=x2/x7,

z=x3/x,,

u=xs/x7,

u=xs/x7,

w=xb/x7

i s a non-homogeneous coordinate system for A . F i n a l l y , s e t and denote by

r -

( m ) : ( I ,O,O,O,O,O,O) { ( m ) } the s e t o f p o i n t s o f A whose coordinates s a t i s f y t h e

equations

(1)

- X Z t yo p y a - za f xy' xzo - x o + i y

u = xzy

p + 3

u

t

=

1o =

yz -

X'CJ'3 x2y2

-

- zz

X ~ a f 4

75

S-Partitions of Groups and Steiner Systems Then G

PGL(6,q)r

2

and G a c t s on

r

i n i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n

o f degree q 3 + l . L e t P be t h e unique Sylow 3-subgroup o f G l y i n g i n G(,+.

By u s i n g

t h e r e s u l t s o f T i t s [ 2 2 ] , § 5 , about t h e r e p r e s e n t a t i o n o f t h e elements o f P as w e l l as the f a c t t h a t I Z ( P ) I = q , i t i s n o t hard t o prove t h a t t h e p r o j e c t i v i t i e s l y i n g i n Z(P) are e x a c t l y those o f t h e form t c : (XI,xZ,x3,X4,x-,xb,X-)

+

+ ( X I, X L , X 3 + C X 7 , - C X I+ & + , c x ~ + X ~ - C ' X ~ ,C'X1-2CX3+X6-CZX7

c

,X7),

E

GF(r().

According t o T i t s [ 2 2 ] , § 5 , we have a l s o t h a t

: ,xZ,xJ,X4,x5,X6,x7) (X5,xb,X3,x2,xl ?-x7,-X6) i s an i n v o l u t o r i a l p r o j e c t i v i t y o f G which does n o t l i e i n G

fA

(a).

Therefore,

uZ(P)W i s the c e n t r e o f a Sylow 3-subgroup Q o f G which i s d i s t i n c t from P. Now

i t i s our aim t o prove t h a t i f c , d c GF(q), c + O , d#O, then

'""tCi does dd

n o t belong

t o any Sylow 3-subgroup o f G. Since NG(P)nNG(Q) = E, where E i s c y c l i c o f order

q-l

, and

Z(P)E i s a Frobenius group w i t h Frobenius k e r n e l Z(P) (see [23],111.4),

we can suppose, w i t h o u t

loss o f g e n e r a l i t y ,

d=l. We then have

dCdI : (X,,X2,X3,X,,Xj,X6,XI) ( X I + CX, + coxE,x 2 - cx j, ( J + ZC) x g X-'C - ( C + c P ) x 6 + x ,, - x + ( I - c)x, -c"xE,x 2 - 2cx3+( I -C+C~)X,+C'X 6-X7, X,-(2*2C)X3+CX,,'CuXSf

( It2CtC"c')X6-X7

2CX3'CoXg-c2Xgfx7)

A s t r a i g h t f o r w a r d c a l c u l a t i o n shows t h a t d c d Jpossesses the eigenvalue I whose

eigenspace i s generated by t h e v e c t o r ( 0 , I ,l/'Z,-cu~*,O,l/c,l).

Now suppose d c d J

l i e s i n a Sylow 3-subgroup o f G, then t h e f o l l o w i n g hold: ili.tc does dI n o t have any eigenvalue d i f f e r e n t from I , f o r we are i n charac-

-

t e r i s t i c 3;

-

LK

C

wx

I

must f i x a p o i n t o f

r

- {

(m)}.

From t h a t which we have proved p r e v i o u s l y , we can i n f e r t h a t t h e f i x e d p o i n t o f

d2dl

on

r -

{ (a)] must have non-homogeneous coordinates ( O , l , I /2, - c O - ~ ,0, I /c)

.

But these coordinates do n o t s a t i s f y (l), a c o n t r a d i c t i o n . Now we may argue as i n the previous examples t o show t h a t i f we s e t T = Z ( P ) and S = P then (G,S,T) determines a Lingenberg 5 - p a r t i t i o n o f type The completion o f t h e A-structure

[G,S,T]

I.

i s a r e s o l v a b l e S t e i n e r system w i t h

parameters ( q ( q 3 - q '+ I ) ,q 1. EXAMPLE V. G = SL(n,q),

q = ph , n>3. L e t K = GF(q), V = K" and U a 1-dimensional

subspace o f V. Denote by T ( g , p ) t h e t r a n s v e c t i o n ptHomK(V,K) w i t h p(2) = T(U) =

0,p#O.

y

-f

l-u(l)awhere g c

b of u(b)=gj.

For a f i x e d non-zero v e c t o r

{I, T(b,u) : O # U E HomK(V,K),

V and

U, s e t

Then T(U) i s a subgroup o f G (see [11],11, H i l f s s a t z 6.5). F i n a l l y , denote by S(U) t h e subgroup o f G f i x i n g U pointwise. Then (G,S(U),T(U))

determines a Lingenberg

S - p a r t i t i o n o f type 11. I t i s indeed enough t o observe t h a t t h e A - s t r u c t u r e which i s obtained from t h e a f f i n e space A associated t o V by removing t h e o r i g i n 0

i s a q u a s i - t r a n s l a t i o n A - s t r u c t u r e w i t h respect t o the p e n c i l 0 o f the l i n e s

M.Biliotti

16 through

0 (disregarding

the point

which a r e conjugated t o T(U).

0)and

t o t h e f a m i l y 3 o f t h e subgroups o f G

Furthermore, a t r a n s v e c t i o n o f T(U) with hyperplane

ff f i x e s a l l the l i n e s o f 0 l y i n g i n

H , so t h a t

T(U) i s n o t semiregular on 8 . The

a s s e r t i o n now f o l l o w s from P r o p o s i t i o n s 1.2 and 2.2.

3. FURTHER RESULTS ON LINGENBERG S-PARTITIONS OF TYPE I We w i l l r e q u i r e t h e f o l l o w i n g lemma. LEMMA 3.1.

Let G be one 06 t h e 6oCCowing ghoupn: h

SL(?,q), q = p , p p’Lime, 4 2 4 ; S z ( q ) , q = p2 n f 1 p=2, @ I ; S U O , ~ ~,) q - p

i,,

p phime, 4 > 2 , 3 1 ~ 7 ; h P S U ( ~ , ~ ’ ) , q = p , P phime, ~ 2 2 ; R(q), q=pZn*l, p=3, el; and l e i P be a SgCow p-nubgmup

conditioMn: (I1 /TI - I (21

TnZ(G)

0 4 G. 16

T

a nomat dubgtoup

06

NG(P) b ~ ~ q 4 u 2 g

[NG(P):Tl and =

,

then T = Z(P).

Pmod.

We s h a l l i n v e s t i g a t e t h e v a r i o u s cases separately.

L e t G = sL(z,q),

q s 4 . Assume q i s odd. Then I Z ( G ) ( = 2, ] P I = q , ING(P)I = q ( q - 1 )

= N (P)/Z(G) i s a Frobenius group w i t h Frobenius k e r n e l p. Since T Q N, then G by [ l l ] , V , Satz 8.16, we have t h a t e i t h e r 7 < or h P. I n t h e f i r s t case, i t

and

r

f o l l o w s t h a t T < PZ(G). But, T n Z ( G ) = <1>and hence T < P, which i m p l i e s T = since P i s a minimal normal subgroup o f NG(P). I n t h e l a t t e r case, c o n d i t i o n (1) y i e l d s T = P. Since P i s elementary abelian, t h e p r o o f i s achieved. The case q even i s s i m i l a r . L e t G = S z ( q ) . N (P) i s a Frobenius group w i t h Frobenius k e r n e l P, moreover G

/ N G ( P ) / = q 2 ( q - l ) , lP1 = q z ,

I Z ( P ) I = q . We have t h a t e i t h e r T 2 P or T c P. Con-

d i t i o n ( 1 ) i s u n s a t i s f i e d when T 2 P. I f T < P, then e i t h e r T 2 Z(P) or Tn Z(P) = = <1>s i n c e Z(P) i s a minimal normal subgroup o f NG(P). As T Q NG(P), we have t h a t

q-l

1

IT[-I.

So, i n t h e former

case, i t f o l l o w s T = Z(P) from c o n d i t i o n (1). I n

the l a t t e r case we have P = T X Z(P). However, P/Z(P) i s a b e l i a n and hence P must be a b e l i a n which

is a c o n t r a d i c t i o n .

L e t G = S U ( 3 , q 2 ) , 3 1 q + 7 . We have /Z(G)I = 3 and N (P) = P X C , where / P I = 9’ G

C i s c y c l i c o f order q2-I and c o n t a i n s Z(G).

-

Moreover, I Z ( P ) )

= q.

N = NG(P)/Z(P)Z(G) i s a Frobenius group with Frobenius k e r n e l

plements isomorphic t o clearly

p

c. Since io N, we must have e i t h e r

i s a minimal normal subgroup o f

and hence

7=

and

and Frobenius com<

p or

r 2 P. B u t ,

<1> i n t h e f i r s t case.

By c o n d i t i o n ( 2 ) and since ( 3 , q ) = I , we then have t h a t T 5 Z ( P ) and thence T = Z(P)

77

S-Partitions of Groups and Steiner Systems because Z(P) i s a minimal normal subgroup o f NG(P). I n t h e l a t t e r case we cannot have T 2 P by c o n d i t i o n

(l), w h i l e T n Z ( P ) = <1> forces P t o be ( T n P ) Z ( P ) , b u t

as we have seen b e f o r e then P must be a b e l i a n which cannot be t h e case. L e t G = PSU(3,q'). Let G = R(q).

The p r o o f i s s i m i l a r t o t h e previous one.

We have NG(P) = P X C, where ( P I = q 3 and C i s c y c l i c of order q - 1 . = NG(P)/P' i s a Froben-

Moreover, Z(P) < P' = @ ( P I , IZ(P)I = q , I P ' I = q 2 . Since i u s group w i t h Frobenius k e r n e l have e i t h e r

7

=

<1> or

7 2 p.

p

(see [23],111.11),

as i n t h e p r e v i o u s cases, we

I n t h e f i r s t case T 5 P I . I f T n Z ( P ) # , then

T 2 Z(P) since Z(P) i s a minimal normal subgroup o f NG(P), b u t by [ 2 3 ] , I I I . 2 , T > Z(P) i m p l i e s T = P ' and c o n d i t i o n (1) i s n o t s a t i s f i e d . So T = Z(P). We canX E PI n o t have TfiZ(P) = -1> s i n c e if

der 2 q z (see [23],111.2)

-

Z(P) then i t s c e n t r a l i z e r i n NG(P) has or-

and hence I T ( > q , c o n t r a r y t o T S P I . I n t h e l a t t e r ca-

se, we cannot have T n P ' = <1>, since for each t Z(P) < PI (see [23],

[23],111.2)

xE

P - P' we have o ( x ) = 9 and

Theorem). Nevertheless, T n P ' # <1> i m p l i e s I T 1 2 q'

x3 6 (see

and again c o n d i t i o n (1) i s n o t s a t i s f i e d . This completes the p r o o f .

The f o l l o w i n g theorem i s concerned w i t h Lingenberg S - p a r t i t i o n s o f type I i n t h e case o f T being o f even order. THEOREM 3.2.

L e L (G,S,T)

d e t w i n e a ling en be^ S-pcmLiAon

06

even o t d m t h e n one 0 6 t h e doLLowing h u t & : G = O(G)T und T iA a Fhobeniun cornpLement; (u) h ( 6 . 1 ) G 2 SL(2,q), 9.2 , h62; T = S = P w i t h P t S2(G); ( b . 2 ) G 2 SZ(~),q - 2 2ntJ a21; T = Z(P), S = P w i t h P6S2(G); i, ( 6 . 3 ) G 2 PSU(3,qz), q = 2 , h22; T = Z(P) w i t h . P G S2(G) and S = ( q + l l / d , &eke d=13,4+11.

type I.1 6 T ha^

= P X

CI with

IC1I =

Phood. I n t h e A-structure [G,S,T], the p e n c i l o f l i n e s 0 i s a t r a n s i t i v e k s e t w i t h 10l>l. I f 4 = N G ( T ) t 0 then E = Ne(?) and, by P r o p o s i t i o n 2.2, ? a c t s semiJl = r e g u l a r l y on 0 - { a } . Then by [ l o ] , Theorem 2, e i t h e r t h e case ( a ) occurs or 2 S L ( ~ , C ( ) , S Z ( C ( ) , PSU(3,qz), SU(3,q') w i t h q = Z h , h > l . I n the l a t t e r case by [ l o ] , Lemma 3, E a c t s on 0 i n i t s u s u a l doubly t r a n s i t i v e r e p r e s e n t a t i o n o f degree q + l , q 2 + 1 , q 3 t 1 , q 3 + 1 r e s p e c t i v e l y . Then, i t is w e l l known t h a t h = Nc(F) with P"eS2(G) and hence Nc(?) = N c ( B ) . By t a k i n g account o f P r o p o s i t i o n 2.2, we see t h a t 7 s a t i s f i e s c o n d i t i o n s (1) and ( 2 ) o f Lemma 3.1. Therefore ? = Z ( P ) . When E = SL(Z,q),

e

Sz(q)

E

or

PSU(3,qz) we o b t a i n (b.1) - (b.3) i n view o f P r o p o s i t i o n 2.2,(3).

= SU(3,q2) w i t h 31qtJ we have

171

= IZ(p)l =

If

q and hence I ~ l ( = q - l Since . 1Z(G)I=3,

t h i s i m p l i e s t h a t 3 Iq- I by P r o p o s i t i o n 2.4, a c o n t r a d i c t i o n . Therefore, t h e case

G"

= SU(3,q2), w i t h 31q+l, cannot occur.

We p o i n t o u t t h a t Examples

I, I 1 and I11 o f s e c t i o n

2 show t h a t t h e cases (b.11,

(b.2) and (b.3) a c t u a l l y occur. On the c o n t r a r y , i t seems very d i f f i c u l t t o achieve a complete c l a s s i f i c a t i o n o f Lingenberg S - p a r t i t i o n s o f type I i n t h e case ( a ) .

M. Biliotti

78

I n succession, we g i v e some r e s u l t s and examples concerning t h i s case.

LeA (G,S,T)

PROPOSITION 3.3.

d&tehmine u Lingenbehg S - p a d t i o n ud t y p e I und

M-

E induced

a F h o b e u p m u t a t i o n gmup on t h e pen& 0 i n %he then Rhe ~oUoiu4ngh o l d : h T, whehe M A a nonabe14un n p e c b l p-ghoup 0 4 m d u q 2 m t ’ w i t h q = p ,

dume [ T I 2 3.

16

[G,S,Tl,

A-n&ucRuhe

( I )G = M X m,hLJ;

( 2 1 IZ(G)I = lZ(M)I = 4,

I T 1 = q + I , S = T,

NG(T) = TZ(M).

P m a 6 . By P r o p o s i t i o n 2.4, Z(E) i s t h e k e r n e l o f t h e r e p r e s e n t a t i o n o f E on 0. Therefore, Go = E / Z ( c ) and a c t s on 0 as a Frobenius group by our assumptions.

c’

Denote by

G

the Frobenius k e r n e l o f

By P r o p o s i t i o n s 2.2 and 2.4,

IT], I i l ) = I

Moreover,

since

= G/Z(G) and l e t M

we have t h a t IZ(G)I

I /TI-1

i G such t h a t M/Z(G) =

M.

and hence ( l T l , l Z ( G ) l ) = l .

i s contained i n a Frobenius complement o f

c.

There-

<1> and MT = M X T. I f x t G then, c l e a r l y , T X C M T and hence G = MT and F = i@ . We have = G 2 NG(T) 2 T, so t h a t T = NG(T) and NG(T) = TZ(G). Set (TI = R , then ( Z ( G ) ( = [NG(T):T] 2 [NG(T):S] = t-I by P r o p o s i t i o n 2.2. Since, fore, T n M =

r

as we have p r e v i o u s l y seen, IZ(G)I

ii

Note t h a t since

group of M with P

I

R - 1 , i t f o l l o w s t h a t / Z ( G ) I d-I and S = T .

i s n i l p o t e n t so i s M (see [11], V.a.7).

$

Z(G) and l e t N = PT. Since (G,S,T)

L e t P be a Sylow p-sub-

determines a Lingenberg

S-partitionof type I, t h e f o l l o w i n g r e l a t i o n holds

It‘

(2)

- $1

!nlRc - I )

+

Rc

c

y1

,

= I N ( and c = ( P n Z ( G ) ( . From (2), i t f o l l o w s t h a t t - l j c since n>tc and

where

hence c=X-I and Z(G) < P. This y i e l d s M = P. Consider the commutators o f t h e form [x,g]

with X E T and g t M-Z(G).

We have [x,g]

= X-’(g-’xg)

and hence [ X , g ]

E

TT’.

Each complex TTg c o n t a i n s e x a c t l y t-J n o n - i d e n t i c a l d i s t i n c t commutators o f t h e form [ x , g ] .

Moreover, i f T 9 # T6 then T T g n T T 6 = T and hence, t h e R-7 commutators

l y i n g i n TTg are d i s t i n c t from those l y i n g i n TT‘. by s e t t i n g [M,T].

(GI

Since

= 6 there e x i s t a t l e a s t ( X - I ) ( i - 7 ) t I

Since [ M , T ]

IITX

: X E G}

2 M and I M I = m ( X - I ) , i t f o l l o w s t h a t [ M , T l

= M.

t h e r e e x i s t s a c h a r a c t e r i s t i c a b e l i a n subgroup A o f M such t h a t A group AT contains e x a c t l y a = [ A : Z ( G ) n A ]

I

=

121,

then

d i s t i n c t commutators l y i n g i n

6

Now suppose Z(G). Then the

d i s t i n c t conjugate elements of T. By

using the same argument as before, we have ( t - I ) a 2 / A \ 2 I[A,T]I

2 (t-I)(a-I)tJ

t i o n y i e l d s Z(G) < A , 13.4(b),

,

( t - I ) a . The l a t t e r r e l a b u t t h i s c o n t r a d i c t s a r e s u l t o f Zassenhaus [11],III, Sat2

where a>?. From t h i s i t f o l l o w s t h a t A = [ A , T ]

and ( A ( =

since IZ(G)) > I. Therefore, a c h a r a c t e r i s t i c a b e l i a n subgroup o f M i s

c e n t r a l i n G. I n conclusion we have proved t h a t :

(I) ( I M I , I T I ) = I , (11) [ M , T l = M, (111)

T c e n t r a l i z e s every c h a r a c t e r i s t i c a b e l i a n subgroup o f M.

By a r e s u l t of Thompson [ L l ] , I I I ,

Satz 13.6, we then have t h a t M i s a nonabelian

79

S-Purtitiorisof G r o i q s arid Steirier Systems

special p-group. Moreover, since Z(M) is a characteristic abelian subgroup of M, we have that Z(M) = Z(G). Let lZ(G)I = t - 1 = p h , h21, and let l M l phtn. Since a h Frobenius complement of G/Z(G) has order . t = p h + I , it follows that p +llp"-l. From so this, we have that ph+ I I pn+ph=ph (pn-h+1 ) and hence ph+ 1 I pn-h- Ph =Ph (Pn-2h- I ph+ 1 1 p n - 2 h - I . Let b E P such that bhSn< Ib+ lih. By iterating the above procedure, it . completes the proof. is easy to prove that b must be even and P ' - ~ ~ - I = O This

.

A Lingenberg S-partition of type I satisfying conditions ( 1 ) and (2) of Proposition 3.3 and its associated A-structure will be called 4peciu.e. An example is given below. EXAMPLE VI. Ue assume the reader is familiar with [14],V,§32. Let ~1 be the projective plane over GF(qz), q=ph , and let p be a hermitian polarity of TI. It is well known that the absolute points and non-absolute lines of p make a Stciner system u with parameters b=q+l and v = q 3 + 1 , which is usually called the d u b b i c d unitul. Moreover the group P(U) consisting o f the projectivities of TI leaving U invariant is isomorphic to PGU(3,qZ). According to Bose [ 5 ] , § 6 , for each absolute line p of [ I , we may define a parallelism among the lines of U as follows: a class of parallel lines consists of a non-absolute line fi through p ( p ) and the non-absolute lines through p ( h ) . Note that p ( p ) E h implies p ( 8 ) 6 p. Therefore, the group T ( h ) consisting o f all ( p ( h ) ,&)-homologies lying in P(U) preserves the parallelism just defined in U , because it fixes the line p . The group T(a) fixes each line i? parallel to h and acts regularly on the points o f L lying in U because T(h) has order 9 t J . Moreover, there exists a unique Sylow p-subgroup M of P ( U ) which fixes p ( p ) and so p itself and acts transitively on the Q' non-absolute lines through p ( p ) . It follows that U - {PI is a quasi-translation A-structure with respect to the pencil 0 of non-absolute lines through p ( p ) and to the family 8 = { T ( h ) : h E O}. It is easily seen that: - G =
= M X T(b),

- T(A) acts semiregularly on o

b E @

;

- [A} ;

- IZ(M)I= q and Z ( M ) consists of all (p(p),p)-elations lying in P(U) and therefore

it fixes every line of 0; scts regularly on 0. From this, it follows that Z ( M ) = Z(G) and G/Z(G) is a Frobenius group. Therefore, (G,T(h),T(h)) determines a special Lingenberg S-partition. - I M I = q 3 and M/Z(M)

now consider case (a) of Theorem 3.2. Assume, Z 0 u L i ~ g e ~ b e hSg- p ~ ~ h L i . t i 0o6~ G 0 6 .type I ~ i R hhebpecR 20 .the bubghoup T w i t h IT I t 3 and (44)G = CT, whefie C 0 bo.&ubi?e and C Q G.

We

141

be the pencil of lines of [G,S,T]. By Propositions 2.2 and 2.4, we have that ( / T I , J Z ( c ) ) ) = 1 and hence Z(c) < ?. Let c/Z(E) be a minimal normal subgroup of G/Z(G) contained in e/Z(c). Since $ Z ( E ) and E/Z(c) acts transitively on 0, then, Let 0

e

M . Biliotti

80

i s the o r b i t o f

i f 4 = NG(T)C 0 and

k(E)

since

h under

?

i s elementary a b e l i a n f o r

t a r y abelian. I t f o l l o w s now t h a t

r,

we have t h a t Is11 > 1 . Moreover,

i s solvable, i t f o l l o w s t h a t

a c t s r e g u l a r l y on

R i n v a r i a n t and a c t s semiregularly on s1 -

LR

R. Moreover, ?R =

[ k } . From t h i s , we i n f e r t h a t

i s elemen-

7

leaves

?ITn

is a

Frobenius group. Now s e t F = , 5, = S n F , No = NG(T)nF, I S o \ = h a , = n o , ( T I = t and I F ( =

IN,(

6.

Since C i s a Lingenberg 5 - p a r t i t i o n o f G o f type I,

the f o l l o w i n g r e l a t i o n holds: (3)

(,t~o

S

[N:S] = . t - 1 .

- I ) t no 2 6. t-1. On t h e o t h e r hand we have t h a t

- bo)(d/na

From t h i s , i t f o l l o w s t h a t n o / b o t

i t i s n o t d i f f i c u l t t o see t h a t ifwe s e t R = : x t F 1 } u {NFl(T1)

then C I = { T f

[N,:S,]

S

Therefore, n o / b o = t-f and ( 3 ) h o l d s as an e q u a l i t y . Using t h i s ,

respect t o the subgroup TI.

But

GFs;,

s l =SJR,

T ~ IRA, =

F,= F/R,

i s a Lingenberg S - p a r t i t i o n o f F 1 o f type I w i t h

-R-R

-R

F I = I- T

and hence, by P r o p o s i t i o n 3.3,

XI i s a

b p e c i d lingenbekg S - p a t L i L i u n . From a geometrical p o i n t o f view, t h e previous r e s u l t can be expressed as f o l l o w s .

PROPOSITION 3.4.

L e t C be. a Lingenbehg S-pa/ttLtitian a6 G

0 6 type I

w c t h k e ~ p e c tt o

-the hubghaup -i w i t h T 2 3. Adbume G = CT, whem C i h a hoLwabLe n u m d hubghoup

cuntaim a bubbpace w h i c h i~ a b p e c i d

ud G. Then t h e A-bRhuctwle [G,S,T]

A-btkuc-

tuke.

Pkoo6. L e t [FI,S1,T1]

be the s p e c i a l A-structure r e l a t e d t o the s p e c i a l S , - p a r t i -

t i o n & o f Fi described above. I f S I X i s a p o i n t o f [F,,Sl,T1] the map from t h e s e t o f p o i n t s o f [FL,SI,T1] defined as f o l l o w s

n

-

: SIX

-t

sx

and

x = Kx,

l e t q be

i n t o t h e s e t o f p o i n t s o f [G,S,T]

.

I t i s s t r a i g h t f o r w a r d t o show t h a t 11 i s w e l l d e f i n e d and g i v e s an embedding o f [FI,Sl,TIl

i n t o [G,S,Tl.

4 . LINGENBERG 5-PARTITIONS OF DOUBLY TRANSITIVE PERMUTATION GROUPS determines a Lingenberg S - p a r t i t i o n . Since i n Examples I - V vie have

Assume ( G , I , T ) that: (a)

t h r gnoup

cicib

Z-tcanoiLivek?y un t h e pen&

06 f i n e d

0

06

.the A - ~ i h u c . t u c e

[G,S,Tl. then the n a t u r a l question a r i s e s whether i t i s p o s s i b l e t o c l a s s i f y a l l the t I i p l e s

(G,S,T) which determine Lingenberg S - p a r t i t i o n s s a t i s f y i n g c o n d i t i o n ( a ) . I n the f o l l o w i i i g , we s n a l l prove t h a t a r a t h e r s a t i s f a c t o r y answer t o t h i s question may be g i v e n provided t h a t t h e c l a s s i f i c a t i o n o f doubly t r a n s i t i v e permutation groups i s assumed.

As i t i s w e l l knowri, such a c l a s s i f i c a t i o n f o l l o w s f r o m t h a t o f F i n i t e

simple groups. THEOREM 4.1.

&Lion

Annwrie

(G,S,l) deXehmAneb a Lingenbehg S-pamLCi.on C

( u l . 16 C 0 ad t y p e It h e n une

06

t h e ,joXCawing h d h :

b a t ~ A 6 y ~ ncung

81

S-Partitionsof Groups and Steiner Systems

q = p c l , ph22; T = S = P l ~ i h i hP E S (C); P 2n+ I ntl; T = Z(P), S = P wi2h P 6 S 2 ( G ) ; ( 3 1 G 2 PSU(3,qz), q=2', h t 2 ; T = Z(P) wiRh PE S2(G), S = P A C I , wh&he I C I = ( q + l ) / d ,d=(3,y+ll; ( 4 ) G R ( q ) , q = 3 2 n C 7 , n 2 i ; T = Z(P), S = P w a h PES3(G). 18 C 0 0 6 t y p e I1 lhen PSL(n,q) S G/Z(G) S PTL(n,q) iyhehe n23.

(I]

G

(21

G

SL(Z,q),

Sz(q), Q =

2

I. By P r o p o s i t i o n

Pk006. Assume (G,S,T) -,@

t l i e group G

2.7,

I=

a LinyenbErg 5 - p a r t i t i o n o f type - determines s a t i s f i e s the following condition:

* G/Z(E)

(h) f o r each h e @ , t h e s t a b i l i z e r o f s e m i r e g u l a r l y on

o-

5

6'

in

c o n t a i n s a normal subgroup which a c t s

{t}.

From t h e c l a s s i f i c a t i o n theorem o f f i n i t e doubly t r a n s i t i v e permutation groups, we

(see [ 4 ] ) i s a c t u a l l y a theorem (see

h a ? t h a t t h e so c a l l e d "Hering conjecture"

[19],p.302)

c0

asserting that i f

i s 2 - t r a n s i t i v e on 0 and s a t i s f i e s ( h ) then one

o f t h e f o l l o w i n g holds:

c0

(j) c o n t a i n s a r e g u l a r normal subgroup, (jj) Eo PSL(Z,(i), q L 4 , Sz(q), PSU(3,q"), q > 2 , or R ( y ) , 4'3, -1

aiid

Go

a c t s on 0 i n

i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n . We s h a l l i n v e s t i g a t e these cases separately.

( j ) . Let

C~c.14

hence

Go =

E0

be t h e r e g u l a r rlorrnal subgroup o f

cO. We have t h a t Go

" G Z ( e ) i s a Frobenius group. If( T I = 2 then

then by P r o p o s i t i o n 3.3,

-,o

we must have I N

I

=

EB =

and

=

= SL<2,2). I f ( T I 1 3

q Z m , where q i s a prime power and m 2 1

tdureoveL, ( T I = q + ~ Since . Go i s 2 - t r a n s i t i v e on

B, i t

iollows that q2m-l=q+l.

T h i s i m p l i e s q = 2 , m = l and i t i s very easy t o show t h a t G = SL(2,3). Cane ( j j ) . Note t h a t i f L 5 G and LZ(G) = G then L = G s i n c e ( \ T l , l Z ( G ) l ) = I and

ext e n s i o n o f $). Since, i n t h e case under c o n s i d e r a t i o n , G i s a simple group ( s e e

G i s generated by T and i t s conjugates. Therefore, G i s a c e n t r a l i r r e d u c i b l e

[11],[12]) then t h e r e e x i s t s a unique r e p r e s e n t a t i o n group H o f of Z(H). Moreover, Z(G) = Z(H)/Z,. I f M(E')

f o r some subgroup Z, multiplier o f 2.2,

EB then

IZ(H)I = IM(CO)I ( s e e [21],Ch.2,59).

we have t h a t ( Z ( G ) I

I

IT1

- I,

i t f o l l o w s t h a t IZ(G)I

I

? and

G = H/Z,

denotes t h e Schur

Since, by P r o p o s i t i o n ( l M ( ~ O ) ~ , ~ T l - JNow ).

assume :

q = ph , h 2 2 . I f p i s odd then we cannot have G PSL(2,q) s h c e i n t h i s case every normal subgroup o f NG(P), where P e S (C), c o n t a i n s P and hence r e P l a t i o n ( 3 ) o f P r o p o s i t i o n 2.2 cannot be s a t i s f i e d f o r \ P I = q and ING(P)I = % q [ q - I I . Therefore, when q:J,9 by [11],V,25.7, we must have G = H = SL(2,q) and t h e t n e s i s f o l l o w s from Lemma 3.1. If9 - 4 then T has even order and, by Theorem 3.2, G = = S L ( 2 , 4 ) . L e t q - 9 . Since NG(T)/Z(G) is a Frobenius group o f order 9.4 with Fro-

'"G

= PSL(p,(i),

benius k e r n e l o f order 9 , i t f o l l o w s t h a t I T 1 I 9 . On t h e o t h e r hand,

-

where

101= I0 and hence IT

fore,

/ Z ( G ) I = 2 and G

I

= 9 . By [8],Table 1, we have t h a t M(Co)

SL(2,9).

IT1

= Z6.

I

I01-J There-

M . Riliotti

82 Let

Go --

Sz(g). I n t h i s case T has even order and (2) f o l l o w s from Theorem 3.2.

Let

to =

PSU(3,q2), q > 2 .

= SU(3,q').

The case G

By [ 9 ] , Theorem 2, we have e i t h e r G

2

PSU(3,q')

or G

2

SU(3,q2), 3 1 q + l , may be excluded by arguing as i n the

p r o o f o f Theorem 3.2. Assume G

^1

h

PSU(3,qZ), y = p , p an odd prime. By Lemma 3.1, we

have t h a t T = Z(P) with P t S (G) and S = P X

P

C1,

where C I i s c y c l i c of order

g+J/d,

d=(3,q+J). Moreover C , < C, where C i s c y c l i c o f order q2-J/d and C < NG(P). L e t C2 be t h e subgroup o f C of order q - J . As i t i s w e l l known, t h e r e e x i s t s a subgroup M o f G such t h a t MnNG(P) = TC2 and SL(2,q)

= M = for a s u i t a b l e conjugate

Let b = = .ti, where I f - t e T and i is t h e unique i n v o l u t i o n i n C2. As was shown i n Example I,

subgroup Tx o f T, where TX may be chosen i n such a way t h a t Cz < NM(Tx). determines a Lingenberg S - p a r t i t i o n o f type

(M,TX,TX)

we have t h a t b

b

E

{ NW(TX). Therefore, t h e r e e x i s t s

TXTY and hence T x T y n S # < l > since b E S. Note t h a t

t h a t (G,S,T)

I.

Since NM(T)nNM(TX) = C 2 ,

Ty < M with T#TY#Tx such t h a t

i E Ci

f o r q odd. I t f o l l o w s

does n o t determine a Lingenberg S - p a r t i t i o n . The t h e s i s can now be

achieved by using Lemma 3.1. Let

Go

R ( g ) , q>3. By [l], Theorem 1, we have t h a t M(?)

= clz and ( 4 ) again f o l -

lows from Lemma 3.1. Now assume (G,S,T)

determines a Lingenberg S - p a r t i t i o n o f t y p e I I . I f NG(T)x i s a

--

l i n e o f 0 then the s t a b i l i z e r NE(T"X)' o f t h i s l i n e i n group Tx'

Eo

c o n t a i n s the normal sub-

s a t i s f y i n g the c o n d i t i o n

(1) yX0f3?@= f o r each

?@such

that

T"yo#T"xxo.

Indeed, assume ( 1 ) does n o t hold. Then TXZ(G)i7TYZ(G) > Z(G) f o r some Ty with TX# fTy.

I t f o l l o w s t h a t T X T Y n Z ( G ) # < l >since TXnTY= by Lemma 2.1. So t h e r e ex-1

sists z

E

Z(G),

~ $ 1 ,such t h a t z c T T Y X E STY'-'.

Since z

E

NG(T) and STY'-'

n NG(T)=

5, b u t t h i s i s a c o n t r a d i c t i o n because Sn Z(G) = <1>. Moreover, by P r o p o s i t i o n 2.2, we have: (2) ' 3 does n o t a c t semiregularly on Q - {NG(T)x]. By a w e l l known r e s u l t o f O'Nan [ 1 5 ] , Theorem A, c o n d i t i o n s (1) and ( 2 ) i m p l y 6 PrL(n,q) with n23. So t h e t h e s i s f o l l o w s from P r o p o s i t i o n 2.4. PSL(n,q) 5 = S then z

E

As a f i n a l remark, we note t h a t t h e c l a s s i f i c a t i o n theorem o f doubly t r a n s i t i v e permutation groups i s r e q u i r e d o n l y when (G,S,T)

determines a Lingenberg S - p a r t i t i o n

o f type I and T has odd order.

REFERENCES.

[l] A l p e r i n , J.L. and Gorenstein, D., The m u l t i p l i c a t o r s o f c e r t a i n simple groups, Proc. Am. Math. SOC. 17 (1966), 515-519. [ 2 ] Andre, J., Uber P a r a l l e l s t r u k t u r e n , T e i l I : Gundbegriffe, Math. Z. 76 (1961),

85-102.

S-Partitions of Groups and Steirier Systems

[ 31

Andr6, J.,

83

Uber P a r a l l e l s t r u k t u r e n , T e i l I 1 : T r a n s l a t i o n s s t r u k t u r e n , Math. Z.

76 (1961), 155-163. [41 Aschbacher, M., F-sets and permutation groups, J. Algebra 30 (1974), 400-416. [5] Bose, R.C., On t h e a p p l i c a t i o n o f f i n i t e p r o j e c t i v e geometry f o r d e r i v i n g a c e r t a i n s e r i e s o f balanced Kirkman arrangements, in:The Golden Jub. Comm., C a l c u t t a Math. SOC. (1958-591,341-354. [6] B r e n t i , F., S u l l e S - p a r t i z i o n i d i Sylow i n alcune c l a s s i d i g r u p p i f i n i t i , Boll. Un. Mat. I t . (6) 3-8 (1984), 665-685. [ 71 Burkhardt, R., Uber d i e Zerlegungszahlen der u n i t a r e n Gruppen PSU(3,2 2 f 1 , J. Algebra 61 (19791, 548-581. [ 8 ] Griess, R.L.,Jr., Schur M u l t i p l i e r s o f t h e known f i n i t e simple groups, B u l l . Am. Math. Soc. 78 (19721, 68-71. Schur M u l t i p l i e r s o f f i n i t e simple groups o f L i e type, [9] Griess, R.L.,Jr., Trans. Am. Math. SOC. 183 (1973), 355-421. [ 101 Hering, C., On subgroups w i t h t r i v i a l normalizer i n t e r s e c t i o n , J. Algebra 20

(19721%622-629. [ll] Huppert, B., Endliche Gruppen I(Springer-Verlag, Berlin-Heidelberg-New York, 1979). [ 121 Huppert, B. and Blackburn, N., F i n i t e Groups I11 (Springer-Verlag, B e r l i n -Heidelberg-New York, 1982). [ 131 Lingenberg, R. , Uber Gruppen p r o j e c t i v e r K o l l i n e a t i o n e n , whelche e i n e perspect i v e D u a l i t a t i n v a r i a n t lassen, Arch. Math. 13 (1962), 385-400. [ 141 Luneburg , H., T r a n s l a t i o n Planes (Springer-Verlag, Berlin-Heidelberg-New York, 1980). [ 151 O'Nan, M.E., Normal s t r u c t u r e o f t h e one-point s t a b i l i z e r o f a doubly-trans i t i v e permutation group. I, Trans. Am. Math. SOC. 214 (19751, 1-42. [16] Rosati, L.A., S u l l e S - p a r t i z i o n i n e i g r u p p i non a b e l i a n i d ' o r d i n e pq, Rend. Sem. Mat. Univ. Padova 38 (19671, 108-117. [17] S c a r s e l l i , A., S u l l e S - p a r t i z i o n i r e g o l a r i d i un gruppo f i n i t o , A t t i ACC. Naz. L i n c e i , Rend C1. S c i . F i s . Mat. Nat. ( 8 ; 62 (1977), 300-304. [18] Schulz, R.H., Zur Geometrie der PSU(3,q ) , i n : B e i t r a g e zur Geometr. Algebra, Proc. Symp. Duisburg, 1976 (Birkhauser, Basel, 1977), 293-298. [19] Shult, E.E., Permutation groups with few f i x e d p o i n t s , i n : Geometry - von S t a u d t ' s P o i n t o f View, Proc. NATO Adv. Study I n s t . Bad Windsheim, 1980 ( 0 . R e i d e l P.C., Oordrecht, 19811, 275-311. [20] Suzuki, M., On a c l a s s o f doubly t r a n s i t i v e groups, Ann. Math. 75 (19621, 104-145. [ 211 Suzuki, M., Group Theory I (Springer-Verlag, Berlin-Heidelberg-New York, 1982) [22] T i t s , J., Les groupes simples de Suzuki e t de Ree, i n : Sem. Bourbaki, 13e annCe, 210 (1960/61), 1-18. 1231 ~- Ward, H.N., On Reels s e r i e s o f simple groups, Trans. Am. Math. SOC. 121 (19661, 62-89. 1241 ZaDOa. G.. S u a l i sDazi a e n e r a l i quasi d i t r a s l a z i o n e , Le Matematiche (Catan i a ) i9 (i9647, 127-143: S u l k S - p a r t i z i o n i d i un gruppo f i n i t o , Ann. Mat. Pura Appl. (4) 1251 . . Zappa, G., 74 (19661, 1-14. [26] Zappa, G., P a r t i z i o n i g e n e r a l i z z a t e n e i gruppi, i n : C o l l . I n t . Teorie Comb. 1973 (Acc. Naz. L i n c e i , Roma 1976), 433-437. i

,

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Annals of Discrete Mathematics 30 (1986) 85-98 0 Elsevier Science Publishers B.V. (North-Holland)

COLLINEATION GROUPS

85

STRONGLY

IRREDUCIBLE ON AN OVAL

Mauro Biliotti

Gabor Korchmaros

Dipartimento di Matematica Universitl degli Studi di Lecce via Arnesano, 73100-LECCE Italy

Istituto di Matematica Universitl degli Studi della Basilicata via N. Sauro, 85, 85100-POTENZA Italy

In recent years, Hering has written several papers concerning the composition series of collineation groups of a finite projective plane. Prominent in his studies is the noIion of o&zongcy u u z e d u u b l e cu.Umea.t*on p o u p o n a pnoaectcve p4me. one

which does not leave invariant any point, line,triangle or proper subplane. There

is a well developed theory of strongly irreducible collineation groups containing perspectivities, which has significant applications (see [4],[5],[15]).

However,

it should be noticed that only isolated results are known for such groups in the general case. It should be interesting to investigate also "local" versions of the concept of irreducibility. In this connection, here we consider a finite projective plane

TI

of even order with a collineation group r and a r-invariant oval n such that r does not leave invariant any point, chord or suboval of n. Here a suboval o f n is a subset of points of is

R

which is an oval in a proper subplane of

4.t/Lony4y u z n e d u c ~ 6 4 eon f i e o v a l n.

since it fixes the knot K of

n.

We say that r

Clearly r is not strongly irreducible on

TI

0.

Our main result states that if r has even order then r contains some involutorial perspectivities, i.e. elations. The subgroup



generated by all involutorial e-

lations is essentially determined. If r has a fixed line then



is the semidi-

rect product of O() with a subgroup of order two generated by an elation. If r has no fixed line then r acts a s a "bewegend group" [6] on the dual affine plane of

~i

with respect to the line at infinity K. From Hering's result [6] on bewegend

groups containing involutorial elations, it then follows that



is isomorphic to

one of the simple groups: SL(Z,q), Sz(q), PSU(3,q2), where q is a power of 2 and 4'4

*

Clearly any collineation group of

TI

mapping n onto itself and acting transitively

on its points is strongly irreducible on n. As we shall prove in Section 5 , such a

M . Biliotti and G.Korclirnaros

86

Hence, t h e o n l y non-solvable c o l l i n e a t i o n groups o f

group cannot i n v o l v e PSU(3,q‘). H

or

a c t i n g t r a n s i t i v e l y on R a r e t h e groups 1 f o r which e i t h e r SL(2,q)c Z P r L ( 2 , q )

Sz(q)c z c A u t S z ( q ) . The groups a r e always 2 - t r a n s i t i v e on

t h e former case,

IT

i s a desarguesian p l a n e o f o r d e r q and

R . Furthermore, i n

i s a c o n i c . For t h e

l a t t e r , we may o n l y a s s e r t t h a t , a t t h e p r e s e n t s t a t e o f our knowledge, t h i s s i t u a t i o n occurs i n t h e d u a l Luneburg p l a n e of o r d e r q z (see ~ l l ~ , [ 1 3 ] , [ 1 4 ] ) .

2. NOTATION AND PRELIMINARY RESULTS F a i r l y s t a n d a r d n o t a t i o n i s used. A c e r t a i n f a m i l i a r i t y w i t h f i n i t e p r o j e c t i v e planes as w e l l as w i t h f i n i t e groups i s assumed. For t h e necessary background t h e reader i s r e f e r r e d t o [ 2 ] , [ 9 ] . Throughout t h i s paper,

11

denotes a p r o j e c t i v e p l a n e o f even o r d e r n c o n t a i n i n g an

o v a l R . Here an o v a l i s d e f i n e d as a s e t o f n + l p o i n t s no t h r e e o f which a r e c o l linear. The f o l l o w i n g elementary r e s u l t s a r e used i n t h e p r o o f s . Through each p o i n t o f R t h e r e e x i s t s e x a c t l y one t a n g e n t o f a. The t a n g e n t s a r e c o n c u r r e n t ; t h e i r common p o i n t K i s c a l l e d t h e k n o t o f 0. Each l i n e t h r o u g h K is a tangent o f n. A l i n e o f

IT

is

an

e x t e r n a l l i n e o r a secant l i n e o f R a c c o r d i n g

t o whether Ir flnl=O o r 2. There a r e e x a c t l y n ( n - 1 ) / 2 e x t e r n a l l i n e s and n ( n + 1 ) / 2 secants o f n i n

H.

A c h o r d o f R i s t h e p a i r o f p o i n t s which n has i n common w i t h a

secant. L e t G be a c o l l i n e a t i o n group o f

TI

mapping

62

o n t o i t s e l f . Then G f i x e s K. I f G has

no f i x e d p o i n t on R t h e n i t has no f u r t h e r f i x e d p o i n t i n n . The o n l y element o f G w i t h a t l e a s t Jn+2 f i x e d p o i n t s on B i s t h e i d e n t i t y c o l l i n e a t i o n o f

H.

The r e s t r i -

t i o n map o f G on R i s a f a i t h f u l r e p r e s e n t a t i o n . Any n o n - t r i v i a l e l a t i o n o f G i s i n v o l u t o r i a l .

I t s c e n t e r does n o t belong t o R U ( K 1 .

Two d i s t i n c t e l a t i o n s o f G do n o t have t h e same c e n t e r . The a x i s o f any e l a t i o n i s a t a n g e n t o f 8. Any i n v o l u t o r i a l c o l l i n e a t i o n o f

II

i s e i t h e r an e l a t i o n o r a Baer-

i n v o l u t i o n . The s e t o f a l l f i x e d p o i n t s and l i n e s o f a B a e r - i n v o l u t i o n f i s a subp l a n e o f o r d e r hi, c a l l e d t h e Baer-subplane o f f .

BAER INVOLUTIONS MAPPING R ONTO ITSELF

87

Collineation Groups F d e n o t e t h e Buen-4ubpLune

u{

f. Thm

Since n i s even, n has an odd number o f p o i n t s . So, f has some f i x e d p o i n t

?noo[.

on 0. Given any f i x e d p o i n t P on R , t h e s e t o f f i x e d p o i n t s o f f on and a l l p o i n t s

Q

consists of P

Q f o r which t h e l i n e PQ b e l o n g s t o F. Other t h a n t h e t a n g e n t o f

a t P, t h e r e a r e e x a c t l y J E l i n e s t h r o u g h P b e l o n g i n g t o F. T h e r e f o r e ,

R

I R F I=,'ii+l.

T h i s proves ( 1 ) . There i s a unique l i n e r t h r o u g h R b e l o n g i n g t o F. F' Moreover, f does n o t f i x R and so r i s n o t a t a n g e n t o f R . L e t ( R , S } = r n a . Then a l L e t R be any p o i n t o f R- R

so SEn

-

T h e r e f o r e , r i s an e x t e r n a l l i n e o f RF i n t h e subplane F. Since F' 1Q . R I=n-Jii, we o b t a i n i n t h i s way each e x t e r n a l l i n e o f nF i n t h e subplane. T h i s F proves ( 2 ) . R

-

$.zoo{.

By way o f c o n t r a d i c t i o n , assume F=G. Choose a l i n e r b e l o n g i n g t o F which i s

an e x t e r n a l l i n e o f R and f(P)+P,

F'

By ( 2 ) o f Prop. 1, I r n n l = 2 . L e t P E r n n . Since f ( r ) = g ( r ) Hence f g ( P ) = P w it h P B F. T h i s i m -

g(P)+P, i t f o l l o w s t h a t f ( P ) = g ( P ) .

plies that f g i s the identity collineation o f

?mw,L L e t F ( r e s p .

TI

which is a c o n t r a d i c t i o n .

GI be t h e Baer-subplane o f f ( r e s p . 9). Since f g = g f , t h e n f l e a -

ves G i n v a r i a n t . L e t f ' denote t h e i n v o l u t o r i a l c o l l i n e a t i o n induced by f on G. S i m i l a r l y , l e t g ' denote t h e i n v o l u t o r i a l c o l l i n e a t i o n induced by g on F. A c c o r d i n g t o [ 2 ] 4.1.11,

we have e i t h e r

I c l f ' wid g ' u.te both Buen-uzvo4uicon4 d b o t h 4itbplunea F und G , ua

F

nG

c4 u 4 u b p l m e

OL

onden 4fi uz

M.Biliotti and G. Korchmaros

88

We prove that the former possibility cannot occur. Suppose that H = F n G is a subplane of order ' / A .

H such that It n It

n

n 0 is an oval of H by (1) of Prop. 1 , Choose a line t of H I=O. Applying ( 2 ) of Prop.1 to G, f' and nG, we can infer that

So 0 =H

H

n 1=2. Similarly. It G

n a F 1=2. This yields I t

=4. A contradiction, since n

So we may assume that

nlilt

n

Q

F I+lt n nG I+lt n nH I=

is an oval.

(ii) holds. In this case nF n aG=tr n a ] . The lines through C

which are secants of either n or F

G

belong to F fl G. Since R is an oval, such li-

nes are pairwise distinct. Thus

Suppose there is a point P E Q

-

( n fl r) fixed by fg. Then P # n F U

QG.

Set Q=f(P)=

g(P). Again, Q # OF U n The line t joining P and Q meets R - ( a U R 1 in two poG' F G ints. In particular, tfr. Both f and g leave t invariant. Thus, t belongs to H. By ( 4 1 , It n ( n U a ) I = 2 . It follows that t has four common points with a . Since n F

G

is an oval, this is impossible. Therefore, we have that f g has a unique fixed point on n. By ( 1 ) of Prof. 1 , this implies that fg is an elation.

denote the involutorial collineation induced by g on G. By way of

P m m F . Let g '

contradiction, assume that g ' is either an elation or the identity. Choose an external line r of n in the subplane G such that r is fixed by g'. Applying ( 2 ) of Prop.

G

1 to G , g2 and nG, it follows that r meets n

- nG in two points P and 8. As g lea-

ves r invariant, then g2 fixes P and Q. On the other hand g 2 fixes G pointwise. Since P , Q $ G , tions.

it follows that g* is the identity collineation, contrary to our assump-

Collineation Groups

89

Paou,f. By Prop. 3, S c o n t a i n s a unique i n v o l u t i o n . Then by [ 9 ] , 112. Satz 8 . 2 , S i s e i t h e r a c y c l i c o r a g e n e r a l i z e d q u a t e r n i o n group. We s h a l l prove t h a t t h e l a t t e r p o s s i b i l i t y cannot o c c u r . Denote by F t h e Baer-subplane o f t h e unique i n v o l u t i o n f o f S. Assume t h a t S is a g e n e r a l i z e d q u a t e r n i o n group. Then t h e c o l l i n e a t i o n group an elementary A b e l i a n subgroup tions i n

f

7

5

induced by S on F a d m i t s

o f o r d e r 4 . By Prop. 4 , each o f t h e t h r e e i n v o l u -

i s a B a e r - i n v o l u t i o n i n F. By Prop. 2 , t h e i r subplanes a r e p a i r w i s e d i -

s t i n c t . B u t such a s i t u a t i o n is excluded by a p p l y i n g Prop. 3 t o F, n involutions of

7.

and any two F F i n a l l y , t h e statement c o n c e r n i n g t h e o r d e r o f S f o l l o w s f r o m [ 2 ]

4. '1.10.

Piroof. L e t

P be t h e s e t o f f i x e d p o i n t s o f

o r c h o r d of a we have t h a t e i t h e r

~ = o0r

Y on a . As r l e a v e s i n v a r i a n t no p o i n t

1 ~ 1 2 3 .I f 1~123,Ly f i x e s a quadrangle

s i n c e t h e k n o t K o f 0 is a l s o f i x e d by Y . Thus, t h e f i x e d elements o f Y i n n f o r m a subplane

TI'

r

P

leaves

and p = n 11 ill i s a suboval o f 0. Since

r

i n v a r i a n t . As

Y

is a normal subgroup o f r , t h e n

i s s t r o n g l y i r r e d u c i b l e on n, t h i s i s i m p o s s i b l e . Thus P

i s empty. As Y is an elementary A b e l i a n p-group,

t h i s i m p l i e s t h a t p d i v i d e s 10.1.

Hence p I n + l . Now we s h a l l p r o v e t h a t

r

f i x e s e x a c t l y one l i n e i n t h e s e t E o f a l l e x t e r n a l l i n e s

of R . Since I E i = n ( n - 1 ) / 2 and ( n + l , n ( n - l ) / Z ) = I ,

t h e n Y f i x e s a t l e a s t one l i n e o f E .

The common p o i n t o f any two l i n e s o f E is d i s t i n c t f r o m t h e k n o t K o f 0. As P is empty, i t f o l l o w s t h a t Y cannot have f u r t h e r f i x e d l i n e s i n E. L e t r be t h e unique f i x e d l i n e o f Y i n xes r . But t h e n , by ( 2 ) o f Prop. 1 ,

r

E.

As 'Y is a normal subgroup o f

r,

then

r fi-

has no Baer i n v o l u t i o n .

Since a f i n i t e group w i t h c y c l i c Sylow 2-subgroups i s s o l v a b l e (see [ 9 ] , I V . Satz

2 . 8 ) , t h e n P r o p o s i t i o n s 5 and 6 y i e l d t h e f o l l o w i n g r e s u l t :

M . Biliotti and G. Korchmaros

90

4. COLLINEATION GROUPS STRONGLY IRREDUCIBLE ON AN OVAL

P ~ c J u ~We. d i L t i n g u i s h two cases a c c o r d i n g t o whether

r

f i x e s e x a c t l y one l i n e o r i t

has no f i x e d l i n e . Assume 1,

r

f i x e s a l i n e r o f n. C l e a r l y , r i s an e x t e r n a l l i n e o f a . By ( 2 ) o f Prop.

r c o n t a i n s no B a e r - i n v o l u t i o n . Hence, any i n v o l u t i o n of r i s an e l a t i o n whose

c e n t e r belongs t o r . B u t t h e n two d i s t i n c t i n v o l u t i o n s o f

r

cannot commute s i n c e i t

i s e a s i l y seen t h a t t h e i r c e n t e r s as w e l l as t h e i r axes must be d i s t i n c t . So any two d i s t i n c t involutions i n

r generate a d i h e d r a l group w i t h c y c l i c stem o f odd o r d e r .

By [ 3 ] , C o r o l l a r y 3, i t f o l l o w s t h a t < A > i s t h e s e m i d i r e c t p r o d u c t o f O() by a group o f o r d e r two generated by an i n v o l u t o r i a l e l a t i o n . Assume t h a t

r

has no f i x e d l i n e . By Theorem A , A i s non-empty.

So we can a p p l y He-

r i n g ' s main theorem on bewegend groups [6]. As t h e k n o t K of n cannot be t h e c e n t e r o f any e l a t i o n i n

r , our s i t u a t i o n corresponds, up t h e d u a l i t y , w i t h t h a t conside-

r e d i n Theorem 1 o f [ 6 ] . l t remains t o exclude the p o s s i b i l i t y t h a t =SU(3,qz), where q i s a power o f 2 and q24. I n such a s i t u a t i o n , Z() has o r d e r 3 and f i x e s t h e a x i s o f each e l a t i o n i n A . Thus, set of all fixed points of

Z() on

Z() R.

has some f i x e d p o i n t s on P. L e t p be t h e

r leaves

p i n v a r i a n t . As

r

leaves i n v a r i -

a n t no p o i n t o r chord o f n t h e n 1~123.But as we have shown i n t h e p r o o f o f Prop, 6

1 ~ 1 i~m3p l i e s t h a t P is a suboval o f P . Since

r

i s s t r o n g l y i r r e d u c i b l e on R , t h i s

i s impossible. A s i m i l a r argument shows t h a t t h e

r 5

Aut .

centralizer

o f i n

r i s t r i v i a l . Therefore,

Collineation Groups

91

In the following, we shall be concerned with some geometrical properties of the set D o f all points which are centers of involutorial elations of A. Also the set D U S U U I K 1 will be considered. Here S denotes the subset of n consisting of those points

which are fixed by some involutorial elation of

we have seen in the proofs

A . As

o f Prop. 6 and Theorem 6 , the following statement holds:

Now we shall prove

B 4 4 . In our situation,

tions in





has exactly one class of involutions. So all involu-

are elations.

Given any point P t S , let

a

denote an involutorial elation of

be the center of the unique Sylow 2-subgroup L of



A

fixing P. Let z(Z)

containing u. The involutions

o f < A > commuting with o are exactly those belonging to Z ( L ) . Each o f them fixes P

and has axis PK. Conversely, any two involutions of



with the same fixed point P

on S commute because they have the same axis PK. Thus, < A > acts on S as the corresponding simple group acts on the set o f its Sylow 2-subgroups. This completes our proof. Assume ~SL(2,q), q=2' U iK1

and q24. By a result of Hering [7], Theoren 2.8.c, D U S U

is a desarguesian subplane

r is strongly irreducible on

n'

of order q o f n and

n, then we must have S=n

.

S is

a

Hence,

suboval of n. Since n'=n

.

Moreover, n

is a conic. Therefore, we have

Assume either c a > - S z ( q ) , q=Za q>4, or zPSU(3,q2), q=2' q24. The present state o f

M.Biliotti and G.Korchmuros

92

our knowledge does n o t a l l o w u s t o determine t h e u n d e r l y i n g p l a n e rical

For a geomet-

TI.

approach t o t h i s e s s e n t i a l q u e s t i o n , i t may be o f i n t e r e s t t o know t h e c l a s s

o f D as w e l l as t h a t o f has c l a s s [XI,

...,x

D U S U I K } . Here, a s e t U o f p o i n t s o f a p r o j e c t i v e p l a n e

] when I r I l U l belongs t o t h e i n t e g e r s e t I x l ,

...,x

K

1 f o r any

l i n e r i n t h e plane. R e s u l t s about s e t s w i t h p r e s c r i b e d c l a s s a r e g i v e n i n [ E l , ~ 1 7 1 , [IE

P/LUO[. L e t

u 1 and a 2 be any two d i s t i n c t i n v o l u t o r i a l e l a t i o n s b e l o n g i n g t o A. We

s h a l l denote t h e i r c e n t e r s by R . ( i = 1 , 2 ) and t h e l i n e through them by r . We want t o 1

determine

1 r n D I.

We a l r e a d y remarked t h a t each i n v o l u t i o n o f < A > i s an e l a t i o n . Moreover, as i t was shown i n t h e p r o o f o f Theorem B, < A > does n o t l e a v e r i n v a r i a n t . Hence D g r n D. Assume f i r s t a 1 a 2 = a 2 a , . An argument s i m i l a r t o t h a t used i n t h e p r o o f o f Prop. 8 shows t h a t r i s a t a n g e n t o f 0 such t h a t rfl R E S , and r f l D c o n s i s t s o f t h e q-1 cent e r s o f t h e i n v o l u t i o n s i n Z ( E ) , where Z i s t h e Sylow 2-siibgroup o f < A > c o n t a i n i n g 011a2.

Assume now a l a 2 f a 2 a 1 . Then r i s n o t a t a n g e n t o f R. By [ 1 4 ] , Lemma 5.1, r i s t h e unique f i x e d l i n e o f u 1 a 2 which does n o t pass t h r o u g h K. L e t A be any d i h e d r a l subgroup o f < A > which c o n t a i n s u 1 a 2 . Then A a l s o Thus, t h e c e n t e r s o f i n v o l u t i o n s o f with a E A ,

A

l e a v e s r i n v a r i a n t and f i x e s K .

b e l o n g t o r , a l s o . If A i s such t h a t no



u B A , leaves r i n v a r i a n t then r n 0 consists o f the centers of the invo-

l u t i o n s i n A and

Irn Dl=lAl/Z.

We p o i n t o u t t h a t t h i s s i t u a t i o n occurs when A i s a

maximal subgroup o f . We s h a l l prove t h a t such a d i h e d r a l group A e x i s t s i n b o t h o f cases under c o n s i d e r a t i o n . Assume zsZ(q), q=Za and q>4. Then < A > admits e x a c t l y t h r e e c o n i u g a t e c l a s s e s o f d i h e d r a l subgroups which a r e n o t p r o p e r l y c o n t a i n e d i n any o t h e r d i h e d r a l subgroup. They have o r d e r s 2(q-1) of .

o r 2 ( q V G + l ) . Those o f o r d e r 2(q-1)

a r e maximal subgroups

Moreover, each o f those o f o r d e r 2 ( q + / q + l ) i s c o n t a i n e d i n e x a c t l y one ma-

93

Collineation Groups x i m a l subgroup o f o r d e r 4 ( q + J q + l ) ,

which has c y c l i c Sylow 2-subgroups (see [161,

Theorem 9 ) . Thus, e i t h e r I r ; l D I = q - l

or l r n D l = q + d z + l .

Assume =PSU(3,q2), q=2a and 924. Then f o r any two d i s t i n c t Sylow 2-subgroups 6,

we have < Z ( 6 , ) , Z ( z z ) > = S L ( 2 , q )

( s e e [I.?],

Satz 4 . 3 . v i i ) .

El,

T h e r e f o r e , each d i h e d r a l

subgroup o f i s c o n t a i n e d i n some subgroups o f < A > i s o m o r p h i c t o SL(2,q).

By [ 9 1

11.8), t h e r e a r e e x a c t l y two c o n i u g a t e d c l a s s e s o f d i h e d r a l subgroups i n SL(2.q) which a r e n o t p r o p e r l y c o n t a i n e d i n any o t h e r d i h e d r a l subgroup o f SL(2,q). groups have o r d e r s 2(q-1)

These

o r 2 ( q + l ) a c c o r d i n g t o whether I r n S 1 = 2 o r 0. I n PSU(3,q')

t h e subgroups i s o m o r p h i c t o SL(2,q) a r e c o n i u g a t e . Thus, t h e above a s s e r t i o n h o l d f o r PSU(3,q2), also. From t h e s e f a c t s we can i n f e r t h a t < a l , a 2 > i s c o n t a i n e d i n a d i h e d r a l subgroup A o f o r d e r 2 ( q - I )

o r 2 ( q + l ) a c c o r d i n g t o whether l r n S I = 2 o r 0.

Assume q=8. L e t / r n S I = 2 . L e t us c o n s i d e r t h e subgroup Y o f < A > which l e a v e s

rnS

i n v a r i a n t . By [ 9 ] , 11.8, I i s t h e d i r e c t p r o d u c t o f A w i t h a c y c l i c group o f o r d e r 3. T h e r e f o r e t h e i n v o l u t i o n s o f Y a r e e x a c t l y t h o s e o f A . We may assume t h a t r and

S a r e d i s j o i n t . L e t M=SL(2,8) be t h e subgroup o f c o n t a i n i n g A , Denote by

the

0

subgroup o f o r d e r 3 o f < A > which c e n t r a l i z e s M. I f 2 i s t h e subgroup o f o r d e r 9 o f A

t h e n 0 x 5 i s a Sylow 3-subgroup o f < A > .

Since t h e c e n t r a l i z e r o f 3 i s c o n t a i n e d

i n O X M t h e n 0x5 i s t h e u n i q u e Sylow 3-subgroup o f < A > which c o n t a i n s f. L e t R de-

n o t e t h e s e t o f a l l i n v o l u t i o n s i n A whose c e n t e r s l i e i n r . For any P , , P ~ E R w i t h p,6p2,

we have t h a t cp,,p,>is

c o n t a i n e d i n a d i h e d r a l subgroup o f o r d e r 2.3'.

Thus

R i s a f u l l c l a s s o f c o n j u g a t e i n v o l u t i o n s i n < A > and Glauberman's theorem (see [ 9 ] B B and I/=3 Moreover, O x A has

.

Cor. 3) may be a p p l i e d . I t f o l l o w s t h a t 1
o r d e r 2-33 and c o n t a i n s e x a c t l y 9 i n v o l u t i o n s , which a r e e x a c t l y t h o s e o f A . Since 4

i s c o n t a i n e d i n , we have e i t h e r =E o r =@xf. Thus, < R k A o r =@xA

h o l d s . B u t the l a t t e r p o s s i b i l i t y c.annot ,occur because a l l i n v o l u t i o n s o f

OxA

are

i n A.

Assume qf8.

L e t < a 1 , a 2 > be any d i h e d r a l subgroup o f < A > o f o r d e r 2(q-1) o r 2 ( q + l ) .

I n o r d e r t o p r o v e t h a t each i n v o l u t i o n O E A , w i t h c e n t e r on r, belongs t o A , we

s h a l l show t h a t i f we deny t h i s t h e n t h e r e e x i s t two commuting i n v o l u t i o n s i n b o t h l e a v i n g r i n v a r i a n t , which is a c o n t r a d i c t i o n . I n f a c t , such e l a t i o n s must have d i s t i n c t c e n t e r s , s i n c e t h e y map ?i o n t o i t s e l f . T h e r e f o r e , t h e y must have t h e same a x i s t . But then, t h e y b o t h cannot l e a v e r i n v a r i a n t s i n c e t , b e i n g a t a n g e n t o f a , i s d i s t i n c t f r o m r. L e t I:, and

z

denote t h e Sylow 2-subgroups c o n t a i n i n g a , and

0,

r e s p e c t i v e l y . I f I:,=

M. Biliotti and G. Korchmaros

94 =Z , t h e r e

I f u g h t h e n =SL(2,q),

Z(Z)>.

(see [9], 11.8.27).

(2,q) now

s n o t h i n g t o prove. Otherwise, assume f i r s t t h a t < Z ( Z , ) , Z ( Z , ) > = < Z ( Z , ) ,

as A i s a maximal subgroup o f < z ( Z , ) , z ( Z ) > = s L

Hence, < o , A >

#.

c o n t a i n s two commuting i n v o l u t i o n s . Assume

Then N<*>(Z,) c o n t a i n s two d i s t i n c t c y c l i c subgroups

0 and o 2 o f o r d e r ( q + l ) / d w i t h d=(3,q+l)

which c e n t r a l i z e < Z ( Z , ) , Z ( Z ) >

Z ( z 2 ) > , r e s p e c t i v e l y (see [ 1 2 ] , S a t z 4 . 3 . v i i ) . i n v a r i a n t , since the centers o f a,

0,

and

a,

These c y c l i c subgroups b o t h l e a v e r l i e on r . We p r o v e t h a t

two commuting i n v o l u t i o n s by showing t h a t In V ~ ~ , @ ~ < Y = Zand ,O k,o/Z(z,)

0

over,

and

0,

=2

,

Z(Z,)1>2. By [121, Satz 4 . 3 . v -

z,.

More-

a r e F r o b e n i u s complements o f Y . Since 1Z11=q2, IBI=l@,l=iq+ll/d,

<e,o,>=!.

where ~ 2 1 s i n c e each element o f

Z(Z,).

admits

i s a Frobenius group w i t h F r o b e n i u s k e r n e l

i t i s n o t d i f f i c u l t t o show t h a t Y

and < Z ( Z , ) ,

I f <EI),@~>=Z,O

z,-Z(Z,)

Since t h e q-1 i n v o l u t i o n s i n Z ( Z , )

w i t h z,
/I,, f l z ( E , ) l

has o r d e r 4 and i t s square l i e s i n

a r e c o n j u g a t e d under N
o f them i s t h e square o f e x a c t l y q ( q + l ) elements of Z , - Z ( Z , ) . ( q 2 - 1 ) elements o f C,-Z(C,).

qf8,

t h e n each

But, C, c o n t a i n s 2''

Hence l E , f l Z ( Z , ) 1 > 2 .

From t h a t we have seen so f a r , i t f o l l o w s t h a t I r n D l = q - l o r I r n D l = q + l . Now c o n s i d e r D U S U I K I . A l i n e r through t h e k n o t K c o n t a i n s q-1 p o i n t s o f D and one p o i n t o f 5. Hence,

Ir fl ( 0 U S U I K 1 ) I=q+l.

Assume r fl S=(A,B),

w i t h A&.

Since

a c t s on S i n i t s u s u a l doubly t r a n s i t i v e r e p r e s e n t a t i o n , we have t h a t < A >

is a IA,Bl d i h e d r a l group w i t h c y c l i c stem o f o r d e r q-1 o r ( q * - l ) / d a c c o r d i n g t o whether ? Sz(q) o r PSU(3,q2) (see [ 1 6 ] and [ 1 2 ] , Satz 4 . 3 . v i ) .

I n t h e former case, I r n D I = q - I

I n t h e l a t t e r , we have a g a i n Ir n DI=q-I s i n c e

contains exactly q-I involuiA,B) t i o n s , namely those l y i n g i n i t s d i h e d r a l subgroup o f o r d e r 2(q-1). Conversely, i f I r f l O l = q - l t h e n r i s f i x e d by a d i h e d r a l group H o f o r d e r 2(q-1) which, o f course,does n o t f i x any o t h e r l i n e . B u t , H i n t e r c h a n g e s two p o i n t s o f S and hence i t f i x e s t h e l i n e through them. I t f o l l o w s t h a t such a l i n e must be r and so

lr flSl=2.

T h i s completes t h e p r o o f o f P r o p o s i t i o n 10.

5 . COLLINEATION GROUPS

OF EVEN ORDER WHlCH ARE TRANSITIVE ON AN OVAL

95

ColIimeation Groups

P/iou[.

Clearly r is strongly irreducib e on 0. So we can apply Theorem B. Since r

acts transitively on

R,

every point o f R is fixed by an involution of A. But then,

by Gleason’s lemma (see [2], 4.3.15).



also acts transitively on il. So, with

the notation o f Section 4 , we have S=R. If r leaves a line r invariant then every point o f r is the center of an involution o f Aand Propositions 6 and 7 yield (i). If r does not leave any line invariant then it turns out that either (ii) or (iii)

holds. In fact, actually r cannot involve PSU(3,q’). To see this, assume, by way of contradiction, that zPsu(3,q2) holds. Since S=R, we have n=q3 with q=2a, a 2 2 . With the notation of Section 4 , let P be any point P o f DUSUIK). By Prop. 10, for any line r through P, Irn ( D U S UtK):))_2 implies Irn [ D u s U{Kl)I=q+l. Since I D U S U{K}I=(q +l)(q+l).

U

it follows that no line in the plane meets D U S U t K l in a

unique point. Hence, D U S U i K 1 is actually of class [U,q+l], i.e. it is a maximal ((q’+l)q+l,q+l)-arc. By [ 8 ] 1 2 . 2 . 1 ,

this implies (q+1)l(q3+l)q+l, a contradiction.

Notice that the possibility +‘SU(~,C,~) can also be excluded by applying [I]. Research partially supported by G.N.S.A.G.A o f C.N.R. and by M.P.I.

REFERENCES

(1; Biliotti, M. and Korchmaros, G . , On the action of PSU(3.q’)

on an affine plane

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P., Finite Geometries (Springer Verlag, Berlin-Heidelberg-New York,

M.Biliotti and G.Korchmaros

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C e n t r a l elements i n c o r e - f r e e groups, J . Algebra 4 (1966) 403-

[ 3 ] Glauberman, G., 420. [ 4 ] H e r i n g , C.,

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On B e w e g l i c h k e i t i n a f f i n e planes, i n : F i n i t e geometries, Proc. Conf. Ostrom, Wash. S t a t . Univ. 1981, L e c t . Notes Pure Appl. Math. 82 (1983)

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E n d l i c h e Gruppen 1 ( S p r i n g e r V e r l a g , Berlin-Heidelberg-New York,

1967). [ l o ] Huppert, B. and Blackburn, N.,

F i n i t e Groups 111 ( S p r i n g e r Verlag, B e r l i n - H e i -

delberg-New York, 1982). S y m p l e c t i c groups, symmetric d e s i g n s and l i n e o v a l s , J . Algebra

[ l l ] K a n t o r , W.M.,

33 (1975) 43-58. [ 1 2 ] Klemm, M.,

f 2f C h a r a k t e r i s i e r u n g d e r Gruppen PSL(2,p ) and PSU(3,p ) durch i h r e

C h a r a c t e r t a f e l , J . Algebra 24 (1973) 127-153. Le o v a l i d i l i n e a d e l p i a n o d i Luneburg d ' o r d i n e 2

[ 1 3 ] Korchmaros, G.,

2r

che pos-

sono v e n i r mutate i n se da un gruppo d i c o l l i n e a z i o n i i s o m o r f o a 1 gruppo semp l i c e S Z ( ~ ~ A) t, t i Accad. Naz. L i n c e i , Memorie, C 1 . S c i . F i s . Mat. Nat.,

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( 1 982) 360-381 [161 Suzuki, M.,

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1 s t . Matem. Univ. N a p o l i (1973).

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This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 99-104 0 Elsevier Science Publishers B.V. (North-Holland)

ori

SETS

99

OF PLUCKER CLASS TWO IJ

PG(~,Q)

-2)

P a o l a R i o n d i and r J i c o l a Melone Dipartimento d i Matematica e A p p l i c a z i o n i "R. Cacc i o p p o l i" Universitd d i Napoli ITALY

I n t h i s paper t h e concept of t h e Plucker c l a s s o f a k-set i n PC(n,q) i s i n t r o d u c e d and c e r t a i n k - s e t s of P l u c k e r c l a s s t w o i n PG(3,q! a r e c h a r a c t e r i z e d .

IPJTt?ODlJCTIOfJ The c o n b i n a t o r i a l c h a r a c t e r i z a t i o n o f g e o m e t r i c o b j e c t s embedded i n P G ( n , q ) , t h e n - d i m e n s i o n a l p r o j e c t i v e space o v e r t h e G a l o i s f i e l c l G F ( q ) , i s one o f t h e m o s t the i n t e r e s t i n g p r o b l e m s in c o m b i n a t o r i a l geometries.The t h e o r y o f k - s e t s , i . e . i n v e s t i g a t i o n o f s u b s e t s o f s i r e k i n PG(n,q) w i t h r e s p e c t t o t h e i r p o s s i b l e i n t e r s e c t i o n s w i t h a l l s u b s p a c e s o f a g i v e n d i m e n s i o n ( s e e f o r i n s t a n c e b5], DG], [?Oil t u r n s o u t t o be q u i t e a p o w e r f u l a n d u s e f u l t o o l i n s u c h c h a r a c t e r i z a t i o n s . A k-set

in

I<

d-suhspnce

s

non-negative

PC(n,q)

we tiave rl

inteqer,

n.-secant d-space.If -J K i s o f type (m,,m2, class.

% for

i s s a i d t o be o f

lKns

1

E{m,,m2

class

,...,ms]

...,ms

j=l,Z, d

..., s

.The k - s e t

1,

2,...,ns

...

.If I K n S d l = m . ,

=(qd"-l)l(q-1) a l l

[ml,m

,(0<ml<m2<

some

m.-secant

J

d ;:

J d-space

i f f o r any ms<%

,

m, J

a

i s c a l l e d an exists,then

theory c l a s s i f i e s a l l sets i n a given

The f i r s t r e s u l t s f r o m t h i s p o i n t of v i e w w e r e o b t a i n e d by 9 . S e g r e LO], El], whose c h a r a c t e r i z a t i o n o f c o n i c s i n P C ( 2 , q ) ((1 o d d ) , a s k - s e t s o f c l a s s [0,1,2], i s w e l l known.

k21,

T h i s r e s u l t was g e n e r a l i z e d t o q u a d r i c s b y G . T a l l I . n i E3], b 4 ] a n d l a t e r on by F.Buekenhout ,as a r e s u l t o f t h e i n v e s t i g a t i o n o f s p e c i a l s e t s o f c l a s s b , 1 , 2 , i n PG(n,q) .Furthermore,a thorough i n v e s t i g a t i o n o f the c l a s s E,n,q+d 1 q+lI1 & J.A.Thas s e t s was c a r r i e d o u t b y M . T a l l i n i S c a f a t i E7] a n d J . W . P . H i r s c h f e l d

I?]

Whereas t h e l i t e r a t u r e i s f a i r l y a b u n d a n t f o r d = l , t h e r e a r e n o t so many r e s u l t s when d:l .For instance,sets o f t y p e (rn,n) were i n v e s t i g a t e d by k],G.Tallini A . D i c h a r a [l] , '.4.J.de R e s m i n i arld hl.de F i n i s [ 4 ] , M . f e F i n i s h 4 . T a l l i n i S c a f a t i [lU], Lg], J.A.Thas 1211. F i n a l l y , r e s u l t s o n more t h a n t w o c h a r a c t e r s e t s , a q a i n when d > l ,were o b t a i n e d e.g. by A . B i c h a r a [ 2 ] , 0 . F e r r i and r.l.hlelone [ g ] :

161

A c o m p l e t e c h a r a c t e r i z a t i o n o f k - s e t s w i t h more t h a n t w o c h a r a c t e r s w i t h r e s p e c t t o d-spaces, d > l , seems t o b e e x t r e m e l y d i f f i c u 1 t . t h e r e f o r e we s h a l l a d d some

n a t u r a l a s s u m p t i o n s o n t h e s e t s i n o r d e r t o c l a s s i f y them.Such o f e i t h e r a r i t h m e t i c o r s t r u c t u r a l nature.

___-----------(")

kiork s u p p o r t e d by I t a l i a n M.P.I.

a s s u m p t i o n s may be

P. Riondi and N. Melone

100

From t h i s p o i n t o f v i e w we d e f i n e t h e P l u c k e r c l a s s o f a s e t i n PG(n,q) a s f o l l o w s . D e n o t e b y K a k - s e t i n PG(n,q).A h y p e r p l a n e H i s s a i d t o be t a n g e n t t o K a t i t s p o i n t p i f any l i n e i n H t h r o u g h p i s e i t h e r a 1 - s e c a n t o r S i n PG(n,ql we d e f i n e m(K,S ) ( q + I ) - s e c a n t o f K . F o r a n y (n-21-space n-2 a s t h e number o f t a n g e n t h y p e r p l a n e s t o K ":$rough S . I f , f o r any 'n-2 K has i n PG(n,q) m(K,Sn-2)E [ O , l , m,q+l} w i t h m
,

...,

.

of and I n t h i s p a p e r we d e a l w i t h k - s e t s i n PG(3,q) P l u c k e r c l a s s m=2 . T a k i n g i n t o a c c o u n t t h e r e s u l t s i n types f o r such a s e t a r e (1,q+1,2q+1)2 (l,q+l) and ( +1,2q+1I2 . S i n c e t h e a r e t h e l i n e s and t h e ( q 2 ? 1 ) - c a p s remains t o sets o f type (l,q+l) and ( q + 1 , 2 q + 1 ) 2 and o f P l u c k e r c l a s s consider t h e s e t s o f Zypes (l,q+1,2q+1)2 two.The n e x t s t a t e m e n t sums u p t h e o b t a i n e d r e s u l t s .

,

hl1,it

o f c l a s s E,q+l,Zq+1J2 and P l u c k e r c l a s s Theorem.Let K be a k - s e t i n P G ( 3 , q ) t w o c o n t a i n i n g p 1 i n e s . D e n o t e by t t h e number o f t a n g e n t p l a n e s t o K.Then one o f the f o l l o w i n g occurs. I f K i s o f type (l,q+1,2q+1)2 w i t h P 2 2 ,then p 2 3 and K i s a cone ( i ) p r o j e c t i n g a ( q + l ) - a r c o f a p l a n e f r o m a p o i n t . F u r t h e r m o r e P = 3 i f f q=2. (ii)If K i s o f t y p e (1,q+1I2 ,then K i s either a line or a (q2+1)-cap ( i . e . an o v o i d ) . (iii) I f K i s o f type (q+1,2q+1I2 w i t h t > O and pa5 , t h e n K consists o f t h e p o i n t s on q + l p a i r w i s e skew 1 i n e s . w h i c h e i t h e r h a v e one o r t w o t r a n s v e r s a l s o r form a hyperbolic quadric. 1. k-SETS OF TYPE

(l,q+1,2q+1)2

I n t h i s s e c t i o n K i s assumed t o be a k - s e t i n PG(3,q) o f Plucker c l a s s two w i t h P 2 2 ,i.e. i n which t h e r e a r e a t l e a s t two l i n e s . and t y p e (l,q+l,2q+1)2 Consider a 1-secant p l a n e H and l e t p = K n H . S i n c e any l i n e i n K i n t e r s e c t s H i n a p o i n t , i t must pass ehrough p . f i o r e o v g r , s i n c e a n y p l a n e has a t m o s t 2:+1 p o i n t s i n common w i t h K,no t h r e g l i n e s c o n t a i n e d i n K c a n be c o p l a n a r . C o n s i d e r f i r s t t h e c a s e P 2 4 .Take f o u r l i n e s 11,12,13,14 l y i n g on K .Then t h e three d i s t i n c t planes 11,1 > <1,,13> , (1 ,1 7 a r e t a n g e n t t o K . S i n c e K i s !s f a n g e n t t o K . H e n c e , f o r a n y p o i n t o f P l u c k e r c l a s s two, any psane t h r o u g h 1 x on K \ 1 , t h e p l a n e j o i n i n g l1 and x1 i s t a n g e n t t o K a n d m e e t s K i n t w o lines,name!y l1 and x , T h e r e f o r e , K i s t h e u n i o n o f and t h e l i n e t h r o u g h p l i n e s through p .Let t i be a p l a n e :ot through p .Then K n H consists o f t h e p o i n t s i n which'the P l i n e s on K t h r o u g h po mee? H . S i n c e no t h r e e o f t h e s e l i n e s a r e c o p l a n a r , K n H i s a p - a r c . 0 n t h e o t h e r hand, H i s e i t h e r a (q+l)-secant o r a (2q+l)-secant plane o f K;thus,p=q+l and K i s t h e cone p r o j e c t i n g f r o m po t h e ( q + l ) - a r c KnH

<

,

P

.

Next,we c o n s i d e r t h e c a s e p=2 . L e t l,m be t h e t w o l i n e s c o n t a i n e d i n K,and no ( 2 q + l ) - s e c a n t p l a n e d e n o t e b y H1 t h e p l a n e t h r o u g h 1 and m . S i n c e P=2 d i f f e r e n t f r o m H i s a t a n g e n t plane.Thus,any o t h e r p o s s i b l e t a n g e n t p l a n e 1 . t h r o u g h 1 o r m is a ( q + l ) - s e c a n t plane.\rJe c l a i m t h a t n e i t h e r 1 n o r m i s on t h r e e t a n g e n t p l a n e s ; o t h e r w i s e , a n y p l a n e t h r o u g h 1 ( o r m , r e s p e c t i v e l y ) d i s t i n c t from t i i s a ( q + l ) - s e c a n t p l a n e w h i c h i s t a n g e n t t o K .Thus, K c o n s i s t s o n l y o f 1 t h e p o l n t s on 1 and m a c o n t r a d i c t i o n since K i s o f type (l,q+1,2q+l) Assume t h a t t h e r e i s a u n i q u e t a n g e n t p l a n e L f H1 t h r o u g h 1 .Then a l l o t g e r ( 2 q + l ) - s e c a n t planes.Hence, planes through 1 a r e

,

.

,

2

(1.1) Consequently,by equals

k = q +q+l

b5]

eq.

.

( 1 8 ) , t h e number

t2q+1

of

(2q+l)-secant

planes o f

K

Sets of Plucker Class Two in PG(3,q)

(1.2)

.

= q(q+1)/2

tzq+1

101

Next, take a 1-secant l i n e r o f K and denote by u,s,b t h e numbers l-secant,(q+l)-secant,(2q+l)-secant planes o f K through r o f respectively.Counting the points o f K on t h e p l a n e s t h r o u g h r , we get i n view o f (1.1)

,

(1.3)

b = u .

,

r' i s an n-secant o f K Analogously,if (q+l)-secant and denote t h e numbers o f through r' ,respectively, then (1.4)

s' and b' planes o f K

.

= n-1

b'

,

n 2 and (Zq+l)-secant

Each fZq+l)-secant plane other than H, meets H1 i n a l i n e that, by (1.4) ,passes through t h e P o i n t Po=l m c o n s i d e r now a 1 - s e c a n t r" through po i n , Then,by ( 1 . 3 1 , t h e r e i s a t l e a s t one l i n e 1-secant plane through r " !'On t h e o t h e r hand, i f t h e r e were two r" , then any plane through r" were a 1-secant planes through t a n g e n t p l a n e a n d we w o u l d g e t a c o n t r a d i c t i o n a s a b o v e . C o n s e q u e n t l y , H o t h e r than 1 and m t h e r e is through any l i n e through p i n = 2q+l , Comparing a unique 1-secant p l a n e . tignce,by 71.3) , t2 t h i s equality with ( 1 . 2 ) , w e have q=2 ; m o r e a v e r , K consists o f seven p o i n t s i n PG(3,2) . F i v e one o f t h e s e p o i n t s a r e on t h e l i n e s 1 and in, t h e remaining two a r e o f f t h e plane H1 and n o n - c o l l i n e a r w i t h p .Since t h i s c o n f i g u r a t i o n i s n o t t y p e (l,q+1.2q+1)2 ,we h a v e a contrzdiction.

.

+,

Idow we a s s u m e t h a t each p l a n e t h r o u g h

i s the only tangrnt plane through l1 i s a (2q+l)-secant plane and then

H

(1.5)

=

k

Now,equation ( 1 8 ) contradiction.

i n

15

implies

(q+l)

2

that

1

.

Thus,

.

there

i s

no

1-secant

plane,

a

=3. L e t l,m and r be t h e three F i n a l l y , w e d e a l w i t h t h e case l i n e s contained i n K ; denote by H 1 , H 2 , H3 t h e p l a n e s j o i n i n g 1 and m , 1 and r and m and r ,respectively.Assurne t h a t through (q+l)-secant plane. Since one o f t h e s e l i n e s , say 1 , t h e r e i s a K any plane through 1 d i s t i n c t from such a plane i s tangent t o H and 1i2 i s a tangent ( q + l ) - s e c a n t p l a n e . Thus, K i s the union t h r e e c o n c u r r e n t n o n - c o p l a n a r l i n e s , a n d s o q = 2 . On t h e o t h e r h a n d if a n y $ l a n e t h r o u g h 1, m and r i s a (2q+l)-secant plane, then 1-secant k=(q+l) and a c o u n t i n g a r g u m e l t shows t h a t t h e r e a r e no planes, a c o n t r a d i c t i o n (compare 15 eq.s ( 1 8 ) 1.

,

07.

T h u s we

have

proved the

following

statement

T h e o r e m I. L e t K be a k-set o f class (l,q+1,2q+l) and P l u c k e r c l a s s two in PG(3,q) such t h a t 2 .Then 3 a n d 'K i s t h e cone =3 i f f q=2. projecting from a p o i n t a (q+l)-arc.Furthermore,

2.

SETS OF TYPE

(q+1,2q+1I2

.

I n t h i s section, K denotes a k-set o f 0 c l a s s two i n PG(3,q) s a t i s f y i n g

.

P r o p o s i t i o n 2 . 1 .We is a ( 2q + l ) - s e c a n t

have k=(q+1)2. plane.

type

(qt1,2q+1)2

Furthermore,each

and

tangent

Plucker

plane

P.Biondi and N . Melone

102

P r o o f . S i n c e t h e t a n g e n t p l a n e s o f K meet s u f f i c i e n t t o prove that t h e planes through planes.In

K i n e i t h e r one o r t w o l i n e s , i t a l i n e o f K are (2q+l)-secant

our situation,the

is

equation ( 2 2 ) o f [15] r e a d s 2 2 q k -((m+n) 8 - q )ktmn = 0 2 2 For rn=qtl n=2q+l t h i s e q u a t i o n has t h e u n i q u e i n t e g r a l s o l u t i o n k = ( q + l ) C o u n t i n g t h e p o i n t s o f K on t h e p l a n e s t h r o u g h a l i n e i n K we g e t t h a t a l l these planes are (2q+l)-secant ,

.

,

,

P r o p o s i t i o n 2.2 . T h r o u g h a n y s e c a n t p l a n e s ( o l n l q + l ).

n-secant

line

1 there are precisely

n

.

(2q+l)-

Proof.Let b be t h e number o f (2q+l)-secant planes through 1 .Counting t h e p o i n t s o f K on t h e p l a n e s t h r o u g h 1 ,we g e t k = q 2 + ( b + 2 - n ) q + l .It f o l l o w s b=n, i n v i e w o f p r o p o s i t i o n 2.1

.

P r o p o s i t i o n 2.3

K

.Through any p o i n t on

t h e r e a r e a t most t w o l i n e s c o n t a i n e d i n

K. Proof.Assume t h a t t h e r e a r e t h r e e c o n c u r r e n t l i n e s l , m , r be t h e i r common p o i n t and d e n o t e b y H t h e p l a n e t h r o u g h

contained i n K .Let p 1 a n d m .Any l i n e r'f1,m t h r o u g h p i n H i s a 1 - s e c a n t .Hence,by p r o p o s i t i o n 2.2 , H i s t h e through r and r ' i s u n i q u e ( Z q + l ) - s e c a n t p l a n e t h r o u g h i t .Thus,the p l a n e H ' a ( q + l ) - s e c a n t p l a n e . C o n s i d e r i n g t h e p l a n e s t h r o u g h r ,we g e t t h e r e f o r e k = 3 q + l , w h i c h c o n t r a d i c t s p r o p o s i t i o n 2.1

.

Plow w e a r e r e a d y t o p r o v e t h e f o l l o w i . n g Theorem I 1 . L e t K be a k - s e t i n P G ( 3 , q ) t w o w i t h t >O.Then k = ( q t l ) 2 and a l l t h e planes.Furthermore,if P 2 5 then e i t h e r K K i w i t h e i t h e r one o r t w o t r a n s v e r s a l s , o r Proof.Denote by D F i x a tangent plane

of t y p e (q+1,2q+l) and P l u c k e r c l a s s tangent planes a r e 2 ( 2 q t l )-secant c o n s i s t s o f q + l m u t u a l l y skew l i n e s s a hyperbolic quadric.

t h e s e t o f p o i n t s on K w h i c h a r e on t w o l i n e s c o n t a i n e d i n K. Ho w h i c h meets K a t t h e l i n e s 1, and mo and l e t po = lonmo . S e t Do = D n H o and do = I D o . S i n c e p0€ Do , d o > l . S i n c e p > 5 , t h e r e c o n t a i n e d i n K. a r e a t l e a s t t h r e e l i n e s 11,12,13 d i f f e r e n t f r o m l o and mo These l i n e s meet 1, o r mo ; m o r e o v e r , b y p r o p o s i t i o n 2.3 the points o f i n t e r s e c t i o n a r e d i s t i n c t and d i s t i n c t f r o m po . C o n s e q u e n t l y , d o 2 4 . S o , t h e r e a r e a t l e a s t two p o i n t s a,b i n Do , w h i c h a r e c o l l i n e a r w i t h po ,say a,b€lo.Since K is o f P l i j c k e r c l a s s two each p l a n e t h r o u g h 1, i s tangent.Hence,by p r o p o s i t i o n 2.3 , l oC Do w h i c h i m p l i e s d o 2 q + l .If d = q + l , t h e n , b y p r o p o s i t i o n 2.1 , K c o n s i s t s o f q + l m u t u a l l y skew l i n e s w i t h a u n i q u e t r a n s v e r s a 1 , n a m e l y lo. If d o = q+2 , t h e n Do c o n s i s t s o f t h e p o i n t s on 1, and a u n i q u e p o i n t p 1 o t h e r t h a n po on (11 .Denote by m t h e l i n e t h r o u g h p 1 l y i n g on K and o t h e r t h a n m0 ; 1, and mo a r e skew and t h e q + l l i n e s of K on t h e p l a n e s t h r o u g h 1, meet K consists o f q+l p a i r w i s e skew l i n e s h a v i n g b o t h lo and m . C o n s e q u e n t l y , 1, and m . F i n a l l y , s u p p o s e d 2 q + 3 .Then,also moC_Do j u s t two t r a n s v e r s a l s , n a m e l y lMoreover,any l i n e 1 d 1 , i n K t h r o u g h a p o i n t o f lo m g e t s any l i n e mfmo i n K t h r o u g h a p o i n t o f mo .Hence, K i s a h y p e r b o l i c q u a d r i c

I

,

,

.

REFERENCES

]

[l

Bichara,A. ,Sui k - i n s i e m i d i S L i n c e i , ( 8 ) , 6 2 ( 1 9 7 7 ) 480-485 . 3 ' q

[2]

Bichara,A. ,Sui k-insiemi 13(1980)

[3]

Quekenhout,F. ,Ensembles (1969)306-318.

.

d i PG( r , q

d i tip0 (

d i classe

( n - l ) q + l , n q + l I2,Rend.Acc.Naz.

k,

1,2, n

J *,Rend.di

q u a d r a t i q u e s des espaces projectifs,Math.Z.

Mat. ,Roma,

,110

103

Sets of Plucker Class Two in PG(3.y)

[ 4 ]

de Finis,M. and de Resmini,M.J. ,On a characterization o f subgeometries PG(r, q ) i n PG(r,q),q a s q u a r e l E u r o p . J r n l . C o r n b . , 3 ( 1 9 8 2 ) 319-328.

[5]

de Finis,M.,On k-sets in PG(3,q) of type ( m , n ) with respect to planes,to appear.

[6]

Ferri.0. ,Su di una caratterizzazione grafica della superficie d i Veronese di un S5,q ,Rend.Acc.l\laz.Lincei,VIII,vol.LXI,6~1976~603-610.

[7]

Hirschfeld,J.W.P. and Thas,J.A. ,The characterization of projection of quadrics over finite fields of even order,Jrnl.London Math.Soc.,22(1980) 226-238.

[8]

Hirschfeld,J.W.P. and Thas,J.A.,Sets of type (l,n,q+l) in PG(d,q),Proc.London Ilath.Soc.,41(1980)254-278.

[!I]Melone,N.,The linear line geometry in PG(3,q) from a synthetic point of view, Pubbl 1st .Mat. "R. Caccioppol i" Un iv Nap01 i ,30( 1983) 7-1 5.

.

01 Ed rl

1 14 -

Se g re,'2.

,

,0Ja1s

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.

in a finite projec t iv e plane ,Canad. J rn1.Math ,7( 1955) 414-416.

Segre,B. ,Curve razionali normali e k-archi negli spazi finiti,Ann.Mat.Pura e Appl.,(4),39(1955)357-379.

Segre,R. ,Le geometrie di Galois,Ann.Mat., (4),48(1959)1-97.

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Tallini,G. ,Sulle k-calotte di uno spazio lineare finito,Ann.Mat. (41,42(1956) 119-1 64.

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Tallini,G. ,Caratterizzazione grafica delle quadriche ellittiche negli spazl finiti,Rend.Mat.,Rorna,l6(1957)328-351.

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.

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PG(r,q) e in

AG(r,q),Sem.Geom.Cornb.

Tallini Scafati , P l . ,Caratterizzazione grafica delle forme hermitiane di un S

kq

.

r,q

,Rend.Mat.,Roma,26(1976)273-303.

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S r,q '

b9]

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. .

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This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 105-106 0 Elsevier Science PublishersB.V. (North-Holland)

105

A FREE EXTENSION PROCESS YIELDING A PROJECTIVE GEOMETRY Flavio Bonetti Dipartimento di Matematica - Via Machiavelli, 35 - 44100 Ferrara Nino Civolani FacoltA di Scienze - UniversitA della Basilicata - 85100 Potenza

Summary. Presented is a free extension process, mainly based on the configuration of Veblen and yielding a projective geometry.

The basic definitions can be found e.g. in the following sources: partial plane, Desargues' condition (resp. configuration), projective plane, in [ I ] ; (reducible) projective geometry, dimension, in [ 21; free extension (resp. completion) process, free projective plane, in [3]. Let B = ( P , f , I) be a partial plane. The free projective plane T( B ) can be associated to B by the well-known free completion process 1. We will describe another free extension process, 3 ", yielding a projective geometry T pv(iJ),possibly reducible with dimension 2 3, hence mostly)) different from the free projective plane1 (C). To this end we utilize the notion of Veblen configuration { c, a, a', b, b', A, B, C, C'}(see Fig. l), i.e. five distinct points c, a, a', b, b' E P and four distinct lines A, B, C, C' E € such that: c, a, a' I A; c, b, b' I B; a, b, I C; a', b' I C'. The two lines C, C' are the enrering ones of the Veblen configuration, which is closed if they meet in a point.

Fig. 1 Now we define the free extension process' ' 1

B y ic:y

=,)':C(

o:

= B I:yI), where (P:yl, € p : y l = 0,'" u { ( C ,C' 1 c, C' f f ;v

I =

are the two entering lines of a nonclosed Veblen confeuration of

f :y

1 =

€ :v

u {(P,P'}I P, P' E 0 y

are distinct points not joined by any line of L

;

a!"};

106

F. Bonetti and N . Civolani consists of the pairs of 1:"

:1;

of

P t;y

i: Xy

1,

and also of those originated by the new elements

namely ((C, C'}, C), ({C, C'\, C'), (p, { P. P'}), (P',{P,P'}~

Clearly 3 "(6) is a projective geometry, and 3 '"( 6) = 6 if and only if B is a projective geometry. It is also easily seen that, if D is a non-closed Veblen configuration, then at each stage D Z n - l (n there appears at least one new point.

> 1)

The next results about this extension process are summarized by the following THEOREM. Let

'

D beapartialplane, with Fpv(D) # & . Then:

-

i) 3 has infinitely many distinct stages 5 ii) dim T p v ( 3 ) 2 3 TPV(D) is reducible; iii) dim 3 ' " ( D ) = 2 = + T P V ( D )=F(&).

x"

;

Proofs are straightforward.

REFERENCES [ 11 Dembowski,P.,Finite Geometries(Springer, Berlin-Heidelber-NewYork;1968).

[2] DubreilJacotin,M.L., Lesieur, L. and Croisot.R., Lqons sur la Thborie des Treillis, des Structures algbbriques ordonndes et des Treillis g6omCtriques(Gauthier-Viars,Paris, 1953). [3] Siebenmann, L.C., A Characterizationof free projective Planes, Pacific J . Math. 15 (1965) 293 - 298.

Annals of Discrete Mathematics 30 (1986) 107-1 14 0 Elsevier Science Publishers B.V. (North-Holland)

107

SYMMETRIC FUNCTIONS AND SYMMETRIC SPECIES

FLAVIO BONETTI GIAN-CARLQ ROTA (*) DOMENICO SENATO ANTONIETTA M. VENEZIA

Universitrl di Ferrara M.I.T. Boston Universitrl di Napoli Univenitrl di Roma 1

INTRODUCTION The idea of proving identities for symmetric functions by bijective arguments is quite old; it goes back to Lucas (Theorie des nombres, 1891) and probably earlier. To the best of our knowledge, the first glimmerings of a systematization of such bijective arguments goes back t o one of the present author (cf. 191); the idea was further developed by R.P. Stanley, wo gave a bijective proof of Waring’s formula by Mobius inversion on the lattice of partition of a set, and later by Doubilet, who gaves bijective proofs of several identities in the theory of symmetric functions. Joyal’s theory of species led us t o develop a systematic setting for such bijective proofs. We introduce here the notion of synirnetric species, which can be viewed as a set-theoretic (a category-theoretic) counterpart of the notion of a symmetric function. To each of the classical classes of symmetric functions we associate a symmetric species. Operations o n species, as introduced by Joyal, are generalized to symmetric species, and simple categorical operations yielded bijective proofs of all identities among elementary symmetric functions. By way of example, we give a bijective proof of Waring’s forniula, which we believe t o be new, and dispenses altogether with Mobius inversion, as well as bijective proofs of several related identities. This note is part of a communication presented in Bari at ctCombinatorics 84)).

I . DEFINITIONS AND PRELIMINARIES We denote by 9 the category of finite sets and bijections, and we denote by: I* :9-+a the contruiwiunt identify ficnctor. mapping every finite set E t o itself, and such that, i f u :E+F

is a bijection, then I*[u]

= u-’

Recall that a species (Joyal) is a functor

M :@ +@. We shall follow Joyal’s terminology for species. Let X be an infinite set, which will remain fixed throughout, whose elements will be called vuriubles.

To agree with current usage, we may occasionaly list the variables in linear order xl,x 2 , . . . through this listing is strictly speaking irrelevant. Recall, that, if N* is a contravariant functor of @ to 9 , the functor N from 9 t o the category of sets whose objects are -

(*) Research supported by N S F contract n. MCS 8104855.

F. Bonetti et al.

108

N[E] = Hom (N*[E], X) is covariant.

Indeed, if u E Horn (E, F) and if N*[u] : N*[F] +. N*[E] then N[u] : NLE] +. N[F] is defined by N[u](f) = = f o N*[u]. Let M be a species, and let N* be a contravariant functor of CP to functor defined as:

a . We denote by Pol (M, N*) the

Pol (M, N*) [El = M[E] x Horn (N*[E], X) and whose morphisms are, for u : E +. F, (M[u], N[u]) : M[E] x Horn (N*[E], X) +. M[F] x Horn (N*[F], X). A polynomial spades P is a subfunctor of Pol (M, N*), that is, for every object in Pol (M, N*) it is a subset P[E] C M[E] x Horn (N*[E], X) such that if u E Hom (E, F) and if (s, f) E P[E] then M u 1 (9, f 0 N*Lul) E P [Fl. In other words, the subset PIE] is functorially assigned. When N* = I*, we say that the polynomial species is ordinary. Let 9 : X -+ X be an isomorphism and let P be a polynomial species. We denote by Pv the polynomial species defined by: ( s , f ) E P v [ E ] C M[E]xHom(N*[E],q(X))-(s,

cp-’

of)EP[E].

Clearly, P +. Pv is a natural trasformation of functors. A polynomial species is a symmetric species when P = Pq for every isomorphism 9 : X +. X. EXAMPLE 1 . The elementary symmetric species

E.

E[E] = the set of all monomorphisms from E to X C Pol (I, I*) where I is the identity functor. EXAMPLE 2 . The p o w e r m m species

s

S[E] = the set of all functions from E to X of constant value C Pol (I, I*). EXAMPLE 3 . The disposition species

H.

Let S[E] be the set of all permutations on E. Take M = Exp (S) and let r of a partition II and a permutation on each block of n).

E Exp

( S ) [El (i.e. r consists

We let H[E] be the subset of Exp ( S ) [El x Horn (E, X) of all pairs (r, f) such that 1~ is the kernel of the function f : E +. X,that is, such that the blocks B E n are the sets f’(x)whenever f’(x) is non-empty, as x ranges on X. This defines the species of dispositions. EXAMPLE 4 .

K,

= the

monomial elementary species of class A.

Symmetric Functions and Symmetric Species

109

Let IT = { B, , . . . , Bk} be a partition of E, I E 1 = n, and let ri (1 Q i Q n j be the number of blocks of n with i elements. The class of IT is the partition of the integer n defined by: c l ( n ) = ( l r l , 2 2 , . . . , n r n j = (h,,X, where Xi

,

. , ,hkj

= I Bi 1.

Let h be a class (i.e. a partition of n), and let M,[E] be the set of all partitions of class X on set E. We define K,[E] to be the symmetric species of all pairs (n,f) where n is the kernel of the function f. We call this the monomial elementary species of class A. EXAMPLE 5 . The cyclic species C.

Let C[E] be the set of all cyclic permutations on E. We set C[E] to be the cyclic polynomial species on Pol (C, I*) of all pairs ( p , f) where p E C[E] and f is constant. EXAMPLE 6. The species

Hx[El ={(o,

f)

Hx

I u E S[E] and f(e) = x for ail e E E}C S[E] x Hom(F[E], X).

2. THE POLYNOMIAL OF POLYNOMIAL SPECIES

Let Z[X] be the ring of all polynomials in the variables x E X. We again denote by x the canonical image of x in the ring Z[X]. to denote the polynomial obtained from Let C be a cofinite subset of X. We write, if p E Z[X], p/,-, p by setting to zero all x E C. If p E Z[X; and C is a cofinite subaet of X, we let A(P; C) be the set of all q E Z[X] such that P/c=o

=

q/c=o.

This defines a topology on the ring Z[X]. The completion of Z[X] in this topology is denoted by Z[[X]]. Thus, an element r E Z[[X]] is an infinite sum of polynomials such that for every cofinite set C, dC= is an ordinary polynomial in Z[X]. Let { pc be a set of polynomials in Z[X]. We write lim pc = r C

when the set { pc } converges to r in Z[[X]] along the filter of all cofinite sets. A sufficient condition that ensures that ( p c ) converge is that, for C ’ C C

PC’/C=O = Pc. The element of Z[[ X]] defined as gen (0 =

11 ke) e€E

=

n

xlf”(x1

X€X

is called the generafing polynomial of the function f (cf. [9]). If P is a polynomial species on Pol (M, N*) we write:

110

F. Bonetti et al. gen(P[EI) =

sen(f). (s,nePI E 1

Noting that gen (PLE]) depends only on the cardinality n of E, we write gen (P, n) = gen (PIEI) for any set E of cardinality n. We call this the n-th polynomial coefficient of the polynomial species P.

Thegeneratingfiction of a polynomial species P is the element

of the algebra of formal power series in the variable z over Z[[X]]. PROPOSITION 1 . Let P be a symmetric species. Then the polynomial gen (P, n) is Symmetric, in other

words gen (P,n) is invariant for any bijection of X. EXAMPLE 1 . (cont. d.), The polynomial of the elementary symmetric species E is

gen(E,n) = n!

y7 L

i, <

...< i n

xi L xi’ . . . xi, = n! e n ( x l , . , ,)

that is, except for the factor n!, it is the n-th elementary symmetric function. rn EXAMPLE 2 . (cont. d.). The polynomial of the power sum species S is:

gen (s, n) =

x

x:

i

that is the powersum symmetric function s, EXAMPLE 3 . (cont. d,). The polynomial of the disposition species

H is:

gen (H,n) = n! h,(xl,. . J with h,(xl,.

. .) =

x

i, G ...G i n

xi,. . . . xi,

that is h,, is the elementary homogeneous function of degree n EXAMPLE 4 . (cont. d.). The polynomial of the monomial elementary species Kh of class X is:

gen(K, , n ) = p(n, A) r l ! r2! . . . k, where p(n, A) is the number of the partitions of class A on E, k, is equal to Zn:I ranges over all distinct monomials.

. . . q?,

Except for a coefficient, gen ( K h , n) is a monomial elementary symmetric function w EXAMPLE 5 . (cont. d.). The polynomial of the cyclic species Cis

and the generating function of the polynomial species C is:

and the sum

Symmetric Functions and Symmetric Species

111

EXAMPLE 6.(cont. d.) The polynomial of the species Hx is: gen(H,,n)=n!x" Thus 1

Gen(Hx z) = 1 - xz

Sum, product and exponential of polynomial species are defined as in Joyal. We recall the definition of product and exponential for ordinary species. Let Pi be a species on Pol (Mi, I*) (i = 1 , 2 ) . We let the product P, x P, be the polynomial species on Pol (M, x M,, I*) such that:

s = (sl, s2) and f

where E,

lEl

=

f , , f IE2 = f,

I

I + E2 = E ranges the set of all 2-scomposition of E.

PROPOSITION 2. Gen (P, x P2,z) = Gen (P, ,z) . Gen (P2.2). Let P be a polynomial species on Pol (M, I*) without constant term (i.e. P[@] = 4) and let i~ =(B,, . . . , B, } be a partition of E in k blocks. An assembly on E of order k of species P is defined as the set of all pairs (s, f) such that, if si represents the structure induced by s on Bi, we have (si, f/Bi) E P [Bi], for i = 1, . . . ,k.

The species of assemblies of species Pof order k is the polynomial species Expk(P) on Pol (Exp,(M), I*) defined as follows: Expk(P) [El= the set of all polynomial assemblies of species P on E of order k,where the partition ranges over the set of all partitions of E into k blocks. The species of fhe assemblies of species P is the polynomial species on POI (Exp(M), I*) defmed by:

PROPOSITION 3. (Theorem of the assemblies) Gen (Exp (P), z)

=

For present purposes, a family { Px ! x E conditions are satisfied:

of polynomial species will be called multipliable if the following

(i)

Px C Pol (M, I*);

(ii)

P,[@J]= { ( s , f :@-X)J,sEM[$];

(iii)

(s, f) E Px[E] iff f(E) = { x}.

F. Bonetti et al.

112

If Comp (C) denotes the complement of the cofinite set C and Pc = r i p W

we define the product

xEComp(C1

o f t h e family { P x } x e x as the lim Pc. C

PROPOSlTlON 4 .

Gen (lim P,, z) = lim (Gen (P, C

2))

C

EXAMPLE. There exists a natural transformation from the species H of the disposition to the product species of the family { H, I,, x .

As a consequence we obtain the classical identity:

11

W€X

1 -=

1 --z

h,z". n,O

3. A PROOF OF WARING'S FORMULA PROPOSlTION 5 . There is a natural transformation between the species

H of dispositions and the species

Exp (C)of assemblies of cyclic species, i.e. H = Exp (C)

Proof. The following algorithm gives a canonical bijection between dispositions over a finite set E and the cyclic species assemblies on the same finite set. STEP 1 .

Let ((ui}, f) E H[E] be a disposition on E. We shall write pi for to denote the cycles of ui and c(( the supports of these cycles. The assembly of associated. cyclic species is obtained in the following way: the partition of E is that in which the blocks are the Fj, on each block the cyclic permutation p/ and the function f/-. with constant value xiare defined.

4

STEP 2.

Given an assembly of cyclic species (s, f) relative to a partition II = { T I , . . , , F,,,. More explicitly, on each block i7. a cyclic permutation vj and a constant function are defined. The disposition associated with (s, i i thc fair ({(I%}, f) with ux a permutation whose cycles are those vj on the supported of which the fiiriction f assumes the constant value.

n

As a special case of propositions 3 and 5 we have the following: THEOREM (Waring's formula)

REFERENCES [ 1] Aigner. M.. Coinbinatorialtheory. Springer-Verlag.New York (1979). [ 2 ] B0urbaki.N.. Element de Mathimatique: A l p d m Commutative,Hermann, Paris (1965).

1-71

Coiiitrt,L., Advaiccd Combinatorics. Reidel. Dordrecht.Holland, Boston (1974).

Symmetric Functions and Symmetric Species

I I3

P.,On the foundations of Combinatorid theory. VII: Symmetric functions through the theory of Distribution and Occupancy, Studies in App. Math. Vol. 51 (1972).

[4] Doubilet,

[5] Cratzer,G., Universal Algebra,D. Van Nostrand,Princeton,NJ. (1968). [6] Joyal,A., Une th6orie Combinatoire des &ries formelles, Adv. in Math. v. 42 (1981). [7] MacDonaldJG., Symmetric functions and Hall polynomials, Clarendon Press, Oxford 1979. [ 8 ] Metropolis,N., Rota,G.C., Witt vectors and the algebra of necklace, Adv. in Math. vol. 50 (1983). [9] Rota,G.C., Baxter algebras and combinatorial identities I e 11. Bull. Amer. Math. Soc. (1969).

[lo] Rota,G.C., Finite operator Calculus, Academic Press, Inc. (1975).

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Annals of Discrete Mathematics 30 (1986) 115-124 @ EIsevier Science Publishers B.V.(North-Holland)

ON 'THICK (Q+2)-SETS

Rita Capodaglio Di Cocco Universita' di Bologna

Summary: Un k-insieme K di un piano proiettivo finito viene detto denso se da ogni punto del piano esce almeno una s-secante d i K , con s > l . Qui si studiano alcune proprieta' dei (q+2)-insiemi densi di un piano proiettivo d'ordine dispari.

INTRODUCTION About forty years ago, Bose [ 5 \ and Qvist 1201 introduced some subsets of a finite projective plane, called " non collinear systems of points". B. Segre and his school studied the same subsets, renamed k-arcs, and found very important results about them. In particular, if C is a k-arc of the plane PG(2,q) with q odd, we recall (see [28] p. 270-298) : 1 ) kO, and is "large enough relative to c " , then every k-arc i s contained in a conic, 5) if C is complete, then its secants fill up the plane, i.e. each point of the plane is contained in at least one secant of C. There arises the problem: for which values of k and q do complete As an answer, L. Lombard0 Radice k-arcs in PG(2,q) exist? constructed complete (q+5)/2 -arcs, k-arcs] with constructed complete and Pellegrino 1191 showed that if q Among the k-arcs then complete k-arcs with k <(q+1)/2 exist. which are not necessarly complete but are not contained in a conic, we recall the k-arcs contained in a cubic (see Di Comite [ 1 3 ] , 1141, 1151 and Zirilli [37]) . I t easy to see that the order of the known complete k-arcs is very "large" with regard to the theoretical valuation k > ( 3 + 1 / 1 t U q ) / 2 (see 1 2 8 1 ) and we are still a good way from finding a solution of the following problem: I f q is fixed, what is the smallest number k, such that a complete k-arc of PG(2,q) exists?

[

-

R. Capodaglio di Cocco

116

Recently, in order to obtain new resul.ts, the definition of has been generalized as follows (see [3]).

k-arc

9

(not DEFINITION 1: A k-set K o f , a finite projective plane necessary desarguesian) is called thick if V P E 9 there exists a s-secant of K passing through P with s>l. DEFINITION 2: A thick k-set K is called minimal subset of K is not thick,

if

every

proper

Obviously every complete k-arc is a minimal thick k-set, but the next examples show that the converse is wrong: 1) Let r and s be distinct lines of a projective plane of order q > 5 . If P = r n s , let AlrA2,...,Ac., be points of r and F , , B 2 be points of s , with Ai:bPkBi (i=1,~,...,q-1;j=1,2~. Then the set K = {A11A21...IAq-l,B,,B2} is a minimal thick (q+l)-set. 2) Let r,s,t be three non-concurrent lines of Pc(2,q). If we embed PG(2,q) in PG(2,q2), the set of the points of r,s,t is a minimal 3q-set of PG(2,q2). (E. Ughi's example). In the sequel, we will be interested in the minimal thick k-sets for which k takes the maximal value, i.e. qt2 (see [3]) . We point out that minimal thick (qt2)-sets exist: a (qt2,qtl)-arc is a trivial example of such sets. Moreover we will show that a minimal thick (q+2)-set can be represented by a permutation polynomial, and so the study of these sets is connected to a subject to which many important papers have been devoted (see [ 6 1 r [8I [ 9 1 I101 i r [12I). DEFINITION 3: A point N of a k-set K is called a nucleus of every line through N is a s-secant of K with s < 3 .

K

if

A (qt2)-set with a nucleus is obviously thick, but not necessarly minimal: for example let q=ph, with p*2,3 odd and qrl (mod. 3). In PG(2,q) assume r is an irreducible cubic with an isolated I t is easy to see that is a non-minimal double point N. (qt2)-set with nucleus N. Moreover if q = 5 , and F is a point of inflection of ,then {F} is a minimal thick 6-set. Remark: Irreducible cubics with an isolated double point are used in [IS] to construct (qt9)/2-arcs. So it seems that the following problems are the most important in the theory of minimal thick (qt21-sets: I: Has every minimal thick (q+2)-set one and only one nucleus? 11: For which number n is a minimal thick (qt21-set K a (qt2,n)-arc? 111: When is a (qt2)-set K with a nucleus minimal? In the following we suppose that the order of the plane is odd and we give partial answers to these problems. In particular in: problem I: we show that a minimal thick (qt2)-set has at most one nucleus, Moreover, if the plane is PG(2,q), we find conditions for the nucleus to exist.

r

r

r-

On Thick (Q+2)-Sets

I17

problem 11: we show that if K has a nucleus, then either n=q+l or n
1

Let 9 be any finite projective plane of odd order q. DEFINITION 4 : A point A of a thick k-set K is called essential if the set K - { A ] is not thick. Obviously, a thick k-set K is minimal if and only if every point o f K is essential. Suppose now that K is a minimal thick (q+2)-set, then for each X E K there esists at least a point Px such that every line through Px is an 1-secant for K - { X } . If X is not a nucleus o f K , then P,FX, whereas if K has a nucleus N and X # N , then any point Px is on the line NX. In this section we are interested in problem I, first we shall show that a minimal thick (q+2)-set has at most one nucleus. THEOREM 1: If a (q+Z)-set K has two distinct nuclei, then i t is not minimal. Proof: The plane Y contains no (q++arc, s o K has at least a s-secant r with s > 2 . Then every point A of K n r is not essential. Corollary; If the minimalqhick (q+2)-set K has a nucleus N and XEK, X Z N , then there exists at least one s-secant r3X of K with s>2. REMARK:

We point out that some (q+2)-sets with two nuclei exist. In fact, if N1 and N are any two distinct points and r is the 2 line N N , let ,ti (i=1,2) be the set o f the lines through N. and 1 1 f: + E 2 be any bijection. It differen$ from k, moreover let is easy to prove that K= { snf(s); s E g l } u { N1, N2 ] is a thick (q+Z)-set and Nl, N2 are nuclei of K . So the number of the is equal to the number of the (q+2)-sets with nuclei N1,N2 bijections from to , i.e. q!. On the other hand , if .iP = =PG(Z,q) , in[4] it is siown that a (q+2)-set can have more than two nuclei only if q is even. Now we find conditions for a minimal thick (q+2)-set to have a nucleus.

El

El

THEOREM 2 : Let K be any minimal thick (q+2)-set o f PG(Z,q), with q odd; then K contains two points C and D, such that, if the frame is conveniently chosen, it is possible to represent the set W = K - { C , D )

R. Capodaglio di COCCO

118

by an equation y=f(x), where f(x) is a permutation polynomial with 1) the polynomial f(x)-x has no root in G F ( q ) . Moreover if K has a nucleus, we have 2 ) v m € G F ( q ) , m-1-1,the polynomial f(x)-mx has only one root in GF(q). Proof : In the first place we suppose that K has no nucleus and we define an application r:K--->PG(2,q) in the following way: we choose a point X l € K and we pose r ( X 1 ) = P where Pq is one of the above-stated points. The line X , C ( X l f l 'intersects K at XI and at an other point, say X 2 . We pose z ( X 2 ) = Z ( X l ) . Then we choose a point X 3 f X 1 , X 2 and. we call z ( X 3 ) one of the points Px3' The line X 3 r ( X 3 ) intersects K at X 3 and at an Other point, say X4. If either X 4 = X 1 or X 4 = X 2 , we have nothing to define; if X 4 k X 1 1 X 2 we pose r i X 4 ) = r ( X 3 ) and so on. Since qt2 is odd, there must exist at least two points A and B of K such that the distinct lines A r ( A ) and B r ; B ) intersect K at the same point C, because otherwise the set K would have a partition in disjoint pairs. Obviously z ( A ) , r ( B ) and C are not collinear, so we can choose r ( A ) as the improper point of the axis x, r ( B ) a s the improper Let D be the only point of the axis y and C as the point ( 0 , O ) . point of K on the improper line, then, using the terminology of 1311, the set W=K-(C,D} is a diagram relative both to z ( A ) and to z(B) and so it can be regresented by an equation y=f(x), where f(x) is a permutation polynomial. If we choose the point (1,l) on the line CD, we obtain the cond. 1). Now let K have a nucleus N and A,B be distinct points of K, with AcN-fB. We choose a point PA (resp. a point PB) as the improper point of the axis x (resp. of the axis y) and we call D the only improper point of K. If we pose C=N, w e can repeat the above proof. The cond, 2 ) is satisfied, because C is the nucleus of K. THEOREM 3 : In P G ( Z , q ) , with q odd,let W be the set represented by the equation y=f(x), where f(x) is a permutation polynomial with 1) the polynomial f(x)-x has no root in GF(q) 2 ) v m E G F ( q ) , mtl, the polynomial f(x)-mx has only one root in GF(q). If C is the point ( 0 , O ) and D is the improper point of the line y=x, then the set K=Wu{C,D} is a (qt2)-set with nucleus C. Proof: Self evident.

I1

Now we shall deal with problem 11. In the plane 9 let K be a (qt2)-set w i t h a nucleus N. In conformity with the terminology of ( 3 6 1 , K is a (q+2,n)-arc for a convenient number n, and it has at least three characters, because i t has s-secants for s=1,2,n. Let t be a line which intersects K exactly in the n points B 1 , B2t

. . .1 0 , .

On Thick lQ+2)-Sets

119

THEOREM 4: Suppose the minimal (qt2)-set K has a nucleus N , then either n=q+l or n<(q-l) and so q > 4 . Proof: If n=qtl, we have nothing to prove. Suppose n=q; if C ~t is point different from Bl,B2,. . , B n r we have N,B1 ,B2,...,BnrA} , where A is a convenient point of the line but this means that A is another nucleus for K, in contrast with th. 1. Suppose n=q-1; if C1 and C2 a r the points of t different from B l r B 2 , . . , B n , we have K = N,B1 ,B?, . . . ,Bn,A1 , A # where A , is a convenient point of NC, (i=l,2), Since A1 cannot be a nucleus, the line A,A2 must pass through one of the points B ~ : but this is impossible because this point would be not essential. So we have n
.

.

Ifp=PG(2,q), the above result is improved by the next theorem and its corollary. set of PG(2,q) with a nucleus N . Then any line A1 A2, Ai E K, A, 3:N , A1,/zA2,is at least a s-secant df K with s > 3 . Proof: Let A1kN:bA2 be two disinct points of K. Let PA, be the point of the line NA, definied in the section I (i=l,2). For the instead of P ~ ~ . A f t ear B. Segre's sake of brevity we write P, scheme of proof (see 1281 ) , we chose P2 ,PI ,N as the fundamental triangle of a homogeneus coordinate system in PG(2,q). The line PIP2 contains only one point of K, say A3. Let U=P2A1 n N A 3 ; we choose U as (l,l,l), so we have A l = ~ O , l , l ~A3=(l,1,0) , and A2=(l,Ora) with a.kO. For each point CEK, C#.N,AlrA2,A3,the lines NC, P I C , Pf are represented respectively by the equations x1 =m2x0, x ~ = ~ x2=mOx1 ~ x ~ with , m,-=mom 2 If we consider all the points C of K, C=FN,Al,A2,A3, we obtain that m2 and mo take all the values of GF(q) different from 0 and 1, while m 1 takes all the values of GF(q) different from 0 and a. Since the product of all non-zero elements of GF(q) is equal to -1, we have a=-1. This means that the points A1,A2,A3 are collinear . Starting from the points A1 and A3 and putting P=P by the above arguments we have that the only point of K 3 $3 ' , vhich is on the line PI P3 is a so on the 1 ne A1A3 . If this point is distinct from A2, then the line A1A2 intersects K in at least four points; otherwise the point A4, the intersection of K with the line P2P3, is certainly distinct from A 1 and lies on the line AIA2. THEOREM 5 : Suppose K is a minimal thick

Corollary: Suppose the minimal thick set K of PG(2,q) has a nucleus N, then K is a (qt2,n)-arc with either n=qtl or 4
120

R. Capodaglio di Cocco I11

We need a new definition. Let q be an odd prime power. It is known [12] that a permutation polynomial for GF(q) is in reduced form if its degree is q , For the sake of brevity we put U(f(x))=vf. DEFINITION 5: A permutation polynomial is called pseudolinear if there exists a group d , subgroup of AGL(l,q), transitive on the elements of GF(q), except for at the most one element, and such that 1 V d e A the permutation polynomial a-'( vf 6 v ; ) is of the first degree. All the permutation polynomials xk , with GCD(k,q-l)=l, are pseudolinear because for them we can choose A = { x ---> ax; a E G F ( q ) , a-kO}. On the contrary,the polynomial corresponding to the permutation x--->x if'x-l.0~1; 0--->1:1--->0 is not pseudolinear. THEOREM 6: In PG(2,q) let H be the set represented by the equation y=f(x), where f(x) is a pseudolinear permutation polynomial and is the group of the affinities which map H onto itself. assume Then 17 is transitive on H , except for at the most one point. Proof : It is sufficient to point out that I1 cointains all the affinities x= a - $ 6 ) (xl ) ly= a- ( vf 6 v;' )(y') where ~ E A .

n

Now we treat the problem 111. Suppose the (q+2)-set K of PG(2,q) has a nucleus N. We choose a coordinate system different from the one we used in the proof of th. 2. Let A k N be a point of K , then we choose A N as the improper line and N as the improper point of the axis y . Since the set H=X-{N,A} is a diagram relative to N, so i t can be represented by an equation ( 1 ) y=f(x) where f(x) is a polynomial. THEOREM 7 : Suppose K is minimal, then it is possible to choose the improper point of the axis x such that f(x) is a permutation polynomial, Proof: self-evident.

On Thick (Q+2)-Sets

121

If in (1) f(x) is a pseudolinear permutation polynomial, then, by th. 6 , the group of the affinities that map H onto itself is not trivial. Now we suppose K is a (q+2,n)-arc with ncq+l and we prove that if some conditions are satisfied, then K is not minimal.

n

THEOREM 8: Let f(x) be a pseudolinear permutation polynomial. Assume 17 contains a subgroup 17, of the group of the is transitive on H, then A and N are the translations and only essential points of K . Proof: Let B be any point of H and t E 0 be a s-secant of K with s>l; assume T is the improper point of t. Since n < q + l , then there exists Q such that o(t)=t'+t. Obviously t' passes through T and it is likewise a s-secant of K . On the other hand, since any affine line through T is either external or s-secant of K , then no 1-secant of K passes through T. Let now Q be any point of the line BN: since the line QT is not a 1-secant of K - { B ) - , then the point B is not essential.

nq

THEOREM 9 : Let f(x) be a pseudolinear permutation polynomial. Assume any element of n fixes the improper points of the axes and a proper point 0 (obviously 0 is on H ) ; moreover assume 17 is transitive on the points of H different from 0, then 0 is not essential. Proof: Let X be the improper point of the axis x; since the line XO is an 1-secant of K , then there exists at least a line 2 3 0 that is a s-secant of K with s>2. The line r is a m-secant of H, where m=s-1 if A is on r , m=s otherwise. Since is transitive on H - { O } , then the q points of H are shared among d lines which pass through 0 and are m-secant H , with obviously q-l=d(m-l). It is easy to see that is transitive on the lines passing through X and different from XN and XO, and therefore also on the points of the line ON different from 0 and N. Suppose the line ON contains a point Q:bO,N such that any line u3Q is t-secant H with t<2, then the same is true for every point R of ON (RI0,N). But evidently this is absurd, So the point 0 is not essential. EXAMPLES :In PG(Z,q), with q=phodd, let N and A be respectively the improper point of the axis y and the improper point of the line y=x. Moreover let he the set represented by the equation xpt. . tch- xp%-I y=f (x)'COX where c i EGF(P) and GCD(cotclx +. . .+ck1xh-l,1-xh )=l. It is known [12] that f(x) pseudolinear because we can K=Hu{A,N}. Since the satisfied, no point of H is essential for K . 2 ) Let A , N , K have the same meaning d s in l), while H is the set represented by the equation

.

y=xk

As we have already proved, xk is a where GCD(k,q-l)=l. pseudolinear permutation polynomial. In this case the conditions

122

R. Capodaglio di Cocco

required by th. 9 are satisfied, so the point ( 0 , O ) essential for K.

of H

is

not

LITERATURE [l] A.Barlotti:Sui (k,n)-archi di un piano lineare finito. Un.Mat.Ital.11 (1956) 553-556.

Boll.

[2] A.Barlotti: Un'osservaziome intorno ad un teorema di B. Segre sui q-archi. Matematiche (Catania) 21 (1966) 287-395. [3] U.Bartocci: k-insiemi densi in piani di Ga1ois;in c o r s o di pubbl.su1 Boll.Un.Mat.It. [4] A.Bichara,G.Korchmaros: Note on (qt2)-sets in a Galois plane of order q . Ann. of discr.Math.14 (1982) 117-122 [5] R.C.Bose: Mathematical theory of the symmetrical factorial design. Sankhya 8 (1947) 323-338.

161 A.Bruen: Permutation functions on a finite field. Canad. Math.Bull.15 (1972) [ 7 ] :A.Bruen:The number of Lines Determined by n2 Points.Journ.of Comb.Theory (A)15 (1973) 225-241.

[8] L.Carlitz:Permutations in a finite field. Soc.4 (1953) 538.

Proc.Amer.Math.

[9] L.Carlitz: A Theorem on permutations in a finite field.Proc. Amer.Math.Soc.11 (1960) 456-459.

[lo] L.Carlitz: A note on permutation functions over finite field. Duke Math.J.29 (1962) 325-332.

[Ill L.Carlitz: Some theorems on permutation polynomials. Amer.Math. 68 (1962) 120-122. [lZ].L.Carlitz: Permutations in finite field. ged. 24 (1963) 196-203.

Bull,

Acta Sci Math-Sze-

[13] C. Di Comite: Sui k-archi deducibili da cubiche p ane. Acc.Naz.Lincei ( 8 ) 33 (1962) 429-435.

Rend.

On Thick (Q+Zl-Sets

123

[14] C. Di Comite: Sui k-archi contenuti in cubiche piane. Acc.Naz.Lincei ( 8 ) 35 (1963) 274-278.

Rend,

[15] C. Di Comite: Intorno a certi (q+9)/2-archi d i S(2,q). Acc.Naz.Lincei ( 8 ) 36 (1964) 819-824.

Rend.

[16] F. Karteszi: Introduzione alle geometrie finite. Feltrinelli 1978 (traduzione italiana)

[17] G. Korchmaros: New examples of complete k-arcs in PG(2,q). Eur.J. of Comb. [18] L.Lombardo Radice: Sul p r o b l e m dei k-archi completi di S(2,q).Boll.Un.Mat.It.l1 (1956) 178-181. [19] G.Pellegrino: Sur les k-arcs complets des plans des Galois d'ordre impair.Ann. Discr. Math. 18 (1983) 667-694. [20] B. Qvist: Some remarks concernin g curves of the second degree in a finite plane. Ann.Acad.Sci.Fenn.1 134 (1952) [21] L. Redei: Uber eindeutig umkerbare Polinome in endliche Korpern. Acta Sci.Math.Szegd. 11 (1946-1948) 85-92 [22] M. Sce, L. Lunelli: Sulla ricerca dei k-archi completi mediante calcolatrice elettronica. Convegno reticoli e geometrie proiettive.(Palermo 1957) Roma Cremonese 81-86 (1958) [23] B.Segre: Sulle ovali nei piani lineari finiti. Naz.Lincei (8) 17 (1954) 141-142.

Rend.Acc.

[24] B.Segre: Curve razionali normali e k-archi negli spazi finiti. Ann. Mat.Pura Appl. 39 (1955) 357-379. [25] B.Segre: Ovals in a finite projective plane. Canad. J. Math 7 (1955) 414-416. [26] B.Segre: Sui k-archi nei piani finiti di caratteristica 2.Rev. de Math.Pure et Appl. 2 (1957) 289-300. [27] B.Segre: (1959) 1-96

Le geometrie di Galois.

Ann.Mat.Pura Appl.

[ L 8 ] B.Segre:

Lectures on modern Geometry.

48

Cremonese,Roma 1961

[29] B.Segre: Ovali e curve nei piani di Galois stica due. Rend.Acc.Naz.Lince1 32 (1962) 785-790 [30] B.Segre: Introduction to Galois geometries. Lincei, 8 (1967) 135-236.

di

caratteri-

Mem.Acc.Naz.

R. Capodaglio di Cocco

124

[31] B.Segre,U.Bartocci: Ovali ed altre curve nei piani di Galois di caratteristica due. Acta Arithm. 18 (1971) 423-449 [32] B.Segre,G.Korchmaros: Una proprietd degli insiemi di punti di un piano d Galois caratterizzante quelli formati dalle singole rette esterne ad una conica,Rend.Acc.Naz.Lincei 62 (1977) 613-618.

[33] G.Tallin teristica p=2

Sui q-archi di un piano lineare finito di caratRend.Acc.Naz.Lince1 23 (1957) 242-245.

:

[34] G.Tallini: Le geometrie di Galois e le loro applicazioni alla statistica e alla teria del'informazione. Rend.Mat.e Appl. 19 (1960) 379-400. [35] M.Tallini Scafati: Sui k,n -archi di un piano grafico finito. Atti Acc. Naz. Lincei.Rend. 40 (1966) 373-378. [36] M.Tallini Scafati: k,n -archi di un piano grafico finito, con particolare riguardo a quelli con due caratteri (Note I e 1 1 ) Atti Acc.Naz. Lincei Rend. 4 0 (1966) 812-818; 1020-1025. [37] F.Zirilli: Su una classe di k-archi di un piano di Galois. Rend. Acc.Naz.Lincei 54 (1973) 393-397.

Annals of Discrete Mathematics 30(1986) 125-136 0 Elsevier Science Publishers B.V. (NorthHolland)

125

ON A GENERALIZATION OF INJECTION GEOMETRIES P i e r V i t t o r i o C e c c h e r i n i and N a t a l i n a Venanzangeli D i p a r t i m e n t o d i Matematica "G. Castelnuovo" U n i v e r s i t a d i Roma "La Sapienza" C i t t a U n i v e r s i t a r i a , 00100 Roma, I t a l y

I n j e c t i o n geometries have been i n t r o d u c e d i n [ 3 ] as a gene r a l i z a t i o n o f p e r m u t a t i o n geometries s t u d i e d in [ 1 1 . We p r e s e n t a g e n e r a l i z a t i o n o f i n j e c t i o n geometries, namely 7-geometries, i m p r o v i n g i n some cases p r o p e r t i e s o f i n j e c t i o n geometries s t a t e d i n [ 3 ] . 7-geometries have been i n t r o d u c e d i n [ 9 1, and a l s o preannounced i n [ 3 1 , i n a "Concluding remark" unknown t o t h e authors , F g e o m e t r i e s have been r e c e n t l y c o n s i d e r e d a l s o i n [ 4 1, under t h e name o f "squashed geometries" and i n [ 6 1 , [ 8 ] .

1. INTRODUCTION I n what f o l l o w s a l l s e t s w i l l be f i n i t e . L e t N be a non empty s e t and

2

N =

=NxN. I(x,y)

2

EN

f a E N , we g e t t h e g e n e r a t o r s g ( a ) = { ( x , Y ) E N2 : x = a) and g ( a ) = 1 2 : y = a]; t h e f i r s t (resp. second) system o f g e n e r a t o r s i s

= g 2 ( a ) : a € N 1. A ( p a r t i a l ) correspondence o f N 2 i s any subset o f N ; a p a r t i a l a p p l i c a t i o n ( r e s p . c o a p p l i c a t i o n ) o f N i s any sub2 s e t o f N which i s 0- o r 1-secant each g E G ( r e s p . g E G ) ; a subpermutation F 1 1 2 2 o f N i s any subset o f N2 which i s 0- o r 1-secant each g e n e r a t o r ; i f dom F = N,

G1

=

Igl(a)

aEN1 (resp. G

z

t h e n F i s a p e r m u t a t i o n o f N.

I n [ 7 1 s e t s and groups o f p e r m u t a t i o n s o f N a r e

studied ( w i t h special a t t e n t i o n t o t r a n s i t i v i t y p r o p e r t i e s ) from t h i s geometrical p o i n t o f wiew.

I n [ 2 ] c e r t a i n semigroups o f subpermutations a r e c h a r a c t e r i z e d i n

t h e c l a s s o f a l l semigroups. p e r m u t a t i o n geometries,

In [ 1 1 s p e c i a l s e t s o f subpermutations o f N (namely

c f . no. 2 ) a r e i n t r o d u c e d by means o f axioms s i m i l a r t o

m a t r o i d axioms; more g e n e r a l r e s u l t s a r e o b t a i n e d i n a s i m i l a r way i n ( 5 1 , where s e t s o f p a r t i a l a p p l i c a t i o n s ( o r c o a p p l i c a t i o n s ) i n s t e a d o f subpermutations a r e considered. Other g e n e r a l i z a t i o n s o f p e r m u t a t i o n geometries a r e i n j e c t i o n geomed t r i e s , c f . [31. I f d a l i s an i n t e g e r , i n t h e s e t N we have d systems o f genera-

tors :

P.V . Ceccherini and N. Venanzangeli

126

G . = t g . ( a ) : a e N ) , where g . ( a ) = I ( x l 1

1

1

,...,x d ) e Nd

: x . = a1 1

(i=ly...yd);

d . a subset o f N i s c a l l e d i n j e c t i v e i f i t i s 0- o r 1-secant each g e n e r a t o r ; an d i n j e c t i o n geometry i s a s e t o f i n j e c t i v e s e t s o f N s a t i s f y i n g s u i t a b l e axioms, which a r e s i m i l a r t o m a t r o i d axioms and which reduce t o t h o s e f o r d = 1 and t o p e r m u t a t i o n geometry axioms f o r d = 2. T h i s t a l k concerns a g e n e r a l i z a t i o n o f i n j e c t i o n geometries (namely 9-geom e t r i e s ) g i v e n i n [ 9 ] i n a v e r y a b s t r a c t way which i n c l u d e s a l s o p a r t i a l a p p l i c a t i o n ( o r c o a p p l i c a t i o n ) geometries ( c f . no. 2 ) . P r o f . M. Deza i n f o r m e d us d u r i n g t h i s conference t h a t t h e concept o f 9-geom e t r i e s i s claimed i n a c o n c l u d i n g remark (added i n p r o o f s ) i n [31 and t h a t i t i s going t o be developed i n [ 4 1 , where our 3-geometries a r e c a l l e d squashed geometries.

A d i f f e r e n t and v e r y e l e g a n t approach t o squashed geometries as "bou-

quets o f m a t r o i d s " i s g i v e n i n [81, where i n s t e a d o f o u r s e t 7 an a n t i c h a i n C i s c o n s i d e r e d s a t i s f y i n g s u i t a b l e axioms ( i n o u r language C i s t h e s e t o f maximal elements o f A ) ; t h a t approach seems t o be v e r y e f f i c i e n t . I n what f o l l o w s we s k e t c h a t h e o r y o f 9-geometries, i m p r o v i n g i n some cases p r o p e r t i e s o f i n j e c t i o n geometries s t a t e d i n [31.

2. DEFINITION OF 9-GEOMETRIES L e t us s t a r t w i t h t h e f a m i l i a r concept o f m a t r o i d . A m a t r o i d Mr(X), o r rank r on s e t X, i s a p a i r Mr(X) = (X,A)

DEFINITION 2.1.

where A i s a s e t o f subsets o f X, p a r t i t i o n e d i n t o A = A,,u...

UAr

w i t h Ar # 0,

s a t i s f y i n g t h e f o l l o w i n g axioms ( t h e elements A . E A . a r e c a l l e d t h e f l a t s o f r a n k 1

1

i o f M (X), O < i < r ) : r ( 1 ) A i s c l o s e d under i n t e r s e c t i o n ; ( 2 ) i f Ai E Ai

( 3 ) i f AifAi, such t h a t Ait,

A' 3

AU

,

A . E A , and Ai C_ A . t h e n i 6 j; J J J i < r and b E X \ A t h e n t h e r e e x i s t s an u n i q u e AitlE

i'

>_ A i u I b 1 ;

moreover A

it1

i s i n c l u d e d i n each A ' E A

Aitl such t h a t

[bl.

We r e c a l l now t h e d e f i n i t i o n s o f t h e s t r u c t u r e s mentioned i n $ 1 . DEFINITION 2.2

( [ 1 1). L e t N be a non empty s e t and X c N x N. A p e r m u t a t i o n

127

On a Generalization offnjection Geometries

geometry P ( X ) o f r a n k r on X, i s a p a i r P ( X ) = ( X , A ) where A i s a s e t o f subperr r w i t h A # 0, s a t i s f y i n g t h e m u t a t i o n s o f N, p a r t i t i o n e d i n t o A = A o u UAr r axioms ( 1 ) - ( 3 ) w i t h t h e r e s t r i c t i o n t h a t axiom ( 3 ) h o l d s f o r t h o s e b E X \ A . such

...

1

i s a subpermutation of N. (Note t h a t t h e o r i g i n a l d e f i n i t i o n g i v e n 2 . i n [ 1 1 concerns t h e p a r t i c u l a r case X = N ; 1.e. o u r d e f i n i t i o n i s s l i g h t l y more t h a t AiU t b l

general 1. DEFINITION 2.3

([5]). L e t N be a non empty s e t and X C N x N. A p a r t i a l ap-

p l i c a t i o n ( r e s p . c o a p p l i c a t i o n ) geometry o f r a n k r on X i s a p a i r ( X , A ) , where

A i s a s e t o f p a r t i a l a p p l i c a t i o n s ( r e s p . c o a p p l i c a t i o n s ) o f N, p a r t i t i o n e d i n t o A = A,

U...

UA

r

with A

r

# 0, s a t i s f y i n g t h e axioms ( 1 ) - ( 3 ) w i t h t h e r e s t r i c -

t i o n t h a t axiom ( 3 ) h o l d s f o r t h o s e b E X \ A . such t h a t A . u i b 1 i s a p a r t i a l a p p l i 1

1

c a t i o n ( r e s p . c o a p p l i c a t i o n ) o f N. DEFINITION 2.4 “ 3 1 ) .

d L e t d > 1 be an i n t e g e r , N a non empty s e t and Xc- N

.

An i n j e c t i o n geometry o f rank r on X i s a p a i r I ( X I = (X,A) where A i s a s e t o f r d i n j e c t i v e subsets o f N , p a r t i t i o n e d i n t o A A , U U A w i t h Ar # 0, s a t i s f y r i n g axioms ( 1 ) - ( 3 ) w i t h t h e r e s t r i c t i o n t h a t axiom ( 3 ) h o l d s f o r t h o s e b E X \ Ai d The number d w i l l be c a l l e d t h e such t h a t A . u t b l i s an i n j e c t i v e subset o f N

...

.

1

dimension o f I r ( X ) . We g i v e now t h e de i n i t i o n o f

3-geometries.

DEFINITION 2.5 ( c f . [9], and a l s o [ 3 ] , [ 4 ] , [ 8 ] where t h e d e f i n i t i o n i s g i v e n i n a s l i g h t l y d i f f e r e n t f o r m ) . An 9-geometry o f r a n k r on a s e t X, i s a quadruple

G ( X ) = (S,3,X,A) where S i s a non empty set, I i s a s i m p l i c i a 1 complex o f d i s t i n -

r

guished subsets o f S ( i . e .

Z C Z ’ E ~ i m p l i e s Z E I ) , A i s a subset o f 9 p a r t i t i o n e d

... u

A w i t h A # 0 and X = u A, s a t i s f y i n g t h e axioms ( 1 ) - ( 3 ) r r AEA w i t h t h e r e s t r i c t i o n t h a t axiom ( 3 ) h o l d s f o r t h o s e b € X \ Ai such t h a t i n t o A = A,u

A . U t b l E 3. 1

Ai a r e c a l l e d t h e f l a t s o f r a n k i o f t h e geometry G ( X I . r = A. v i b l f o r t h e s e t Ai+l mentioned i n axiom ( 3 ) . We s h a l l w r i t e A i+l 1 S A m a t r o i d M ( X ) i s a geometry Gr(X) = (S,Y,X,A) w i t h 9 = 2 and X = S, and The elements

r

AiE

2

c o n v e r s e l y . A p e r m u t a t i o n geometry Pr(X) = (X,A), w i t h XcN , i s a geometry Gr(X) = 2 = (S,3,X,A) w i t h S = N and 3 = IF c N2 : F i s a subpermutation o f N l , and convers e l y . P a r t i a l a p p l i c a t i o n ( r e s p . c o a p p l i c a t i o n ) geometries can be e a s i l y c h a r a c t e -

P.V, Ceccherini and N. Venanzangeli

128

r i z e d i n a s i m i l a r way between %geometries. An i n j e c t i o n geometry I r ( X ) = (X,d) d w i t h XsNd i s a geometry Gr(X) = (S,Y,X,A) w i t h S = N and 9 = I F 2 Nd : F i s i n j e c t i v e ) , and conversely. Several examples o f geometries G ( X I can then be dedx r ced from [ l ] , [5], [ 3 ] where examples o f permutation geometries and o f i n j e c t i o n geometries are given. We now g i v e some o t h e r examples. EXAMPLE 2.6.

Free 9-geometries.

Gr(X) = (S,9,X,A)

The f r e e geometry

d e f i n e d by assuming X = S, 9 a s i m p l i c i a 1 complex o f S, A

i

is

= t A E 9 : ( A ( = il,

ral.

O
EXAMPLE 2.7.

S t a r 9-geometries.

A s t a r geometry G (X) = (S,9,X,A)

1 center C i s defined by assuming CCS, A, = t C l , A , = t D ( t a 2 , i,j=l,...,t, i # j ) ,X =

and D.n 0 . = C 1

J

l,...,Dt

where C C D . c S J

t

t

Di,

5’=

with

u 2Di.

i=1

REMARK 2.8. S t a r %geometries are 9-geometries o f rank 1 w i t h IA

1

I > 1 and

conversely ( c f . Theorem 5 . 3 ) . EXAMPLE 2.9. metry w i t h A =

Truncations o f an 9-geometry. I f Gr(X) = (S,9,X,A) A,U

...u A r

(k)

Then Grlk) = (S,9,X,A

i s an 9-geo-

and i f 1 6 k < r , we can consider A ( k ) = A , u . . UAk. . . ) i s an 9-geometry o f rank k, c a l l e d t h e k - t r u n c a t i o n o f

Gr’

EXAMPLE 2.10.

Cotruncation o f an 9-geometry. If Gr(X) = (S,S,X,A)

i s an

9-geometry and i f Ah€ Ah (06 h c r ) i s i n c l u d e d i n some Are Ar, then we can consider =

2i = g,i(A,,)

=

(S,g,X,A) w i t h

G (A,) r

=

tAhti€Ah+i

i, U... ~ f i

c_

Ahti,

i=O,l,.

c f . 5.1 ( a ) .

= Gr,

EXAMPLE 2.11.

B i t r u n c a t i o n s o f an 9-geometry.

metry and i f A h € Ah, w i t h O < h c k c r and A h c A sider

-

-

A = A,

Ji

=

..,r-h].

Then G r ( A h 1 = ~i s -an~9-geometry , o f rank r - h , Note t h a t

: Ah

i.(A ) i h

=

{Ahti

E Ahti

: AhCAh+i,

k i=O,

u... uA;(-~i s an 9-geometry o f rank k-h.

EXAMPLE 2.12.

I n t e r v a l s o f an 9-geometry.

I f Gr = (S,Y,X,A)

i s an 9-geo-

f o r some A k € A k then we can con-

...,k-h}.

I f Gr

=

Then G = (S,Y,X,#)

(S,g,X,A)

i s an %geometry

and i f Ah€ Ah, Ak€ Ak w i t h Ahc Ak, O d h < k < r, then we can consider Ai(A =

{Ahti

E Ahti

: A h L AhtiCAk,

o f rank k-h on Ah.

i=O

,...,k - h l .

Then i i , u . . . u f i k-h

with

A ) = h’ k gives a m a t r o i d

129

On a Generalization of Injection Geometries

EXAMPLE 2.13.

Restrictions of

J-geometries.

l e t Gr(X) = (S,9,X,A)

be an 9-

geometry and l e t be S ' such t h a t A € S ' C X f o r some A E d Define A' = 1 1 1' LJ Z A ' , r ' = max t i : A i E A ' l . Then = tAEA: AcSIl, X I = u ,A, 9' = A€ A A' € 4 ' G r , ( X ' ) = (S'J',X',A') i s an 3-geometry o f rank r ' on X ' .

A s p e c i a l i n t e r e s t i n g case i s obtained when S 1 = X ' = U A f o r some f i x e d AE B Bc A such t h a t D n A l # 0; i n t h i s case r ' = max l i : d i n B f 01, A' = t A E A : A C E f o r some B C B } and 9' =

A€ D

2A.

3. D I R E C T SUMS OF 9-GEOMETRIES By s t a r t i n g from two g i v e n 9-geometries, i t i s p o s s i b l e t o c o n s t r u c t a new one (namely t h e i r d i r e c t sum), i n t h e standard way described below. DEFINITION 3.1. G",,

=

Q

of

9-geometries.

Let GIr,

= ( S ' , 9 ' , X 1 , d ' )and

( S " , ~ " , X " , ~ " ) be two 9-geometries; we can suppose w i t h o u t l o s s o f genera-

0. Assume:

l i t y t h a t S'n S" =

A. = [A!, 1

D i r e c t sum

1

uA" : A ! , € A ! , , A!,,€ i" 1 1 1

A = A, u.. . u A

r = r'

r'

t

A'.',,, i ' t 1

i " =

i},

r".

w i l l be c a l l e d t h e d i r e c t sum G '

Then (S,9,X,A)

r'

8

GI,, o f G,;

and G I , , . It i s

easy t o prove t h e f o l l o w i n g : THEOREM 3.2.

The d i r e c t sum G,;

b G'',

o f two

9-geometries i s an 9-geometry

o f rank r = r ' t r " . DEFINITION 3.3. = (S',9',X',A')

F u l l d i r e c t sum 6 o f i n j e c t i o n geometries. L e t G,;

and G'',

= (S",?',X",A")

dimension d. A c t u a l l y X ' c - S'

=

be two i n j e c t i o n geometries o f t h e

N o d and X " c - S" = NtId. We can suppose w i t h o u t l o s s

o f g e n e r a l i t y t h a t N'n N" = 0, so t h a t S'n S" = N =

N'UN",

-

s

A. = ( A ' u A;, 1 i' A = A, u...uAr, Then (S,?,X,A)

d

= N

-

, x

=

x'ux", T = r 1 E 2

: A i l € A ; , , A;,

r

~ame

0 . Assume:

: I i s injective),

E A;,,, i ' t i"= il,

= r ' t r".

w i l l be c a l l e d t h e f u l l d i r e c t sum

cr

= G,;

s

G I , , o f G;,

and GF,,.

P.V , Ceccherini and N. Venanrangeli

130

THEOREM 3.4.

The f u l l d i r e c t sum GAI & GFII o f two i n j e c t i o n geometries o f di

mension d i s an i n j e c t i o n geometry o f rank r = r ' t r " and dimension d. P r o o f . L e t Gr = (S,5',XP)=G', B G" be t h e d i r e c t sum o f G,; and G;,, c o n s i d r r" d ered as 9-geometries ( c f . Theorem 3.2). A c t u a l l y S = S ' U S " = N ' u N1IdcNd = 3 and 9 C T . I t f o l l o w s immediately t h a t

Er

i s an i n j e c t i o n geometry o f rank

r = r ' t r " and dimension d. F o r d = 2 we have COROLLARY 3.5.

The f u l l d i r e c t sum G L I

5 G;,

o f two p e r m u t a t i o n

( c o n s i d e r e d as i n j e c t i o n geometries o f dimension d.2) i t i s a p e r m u t a t i o n geometry.

o f dimension two, i . e . REMARK 3.6.

t e d meaning -

L e t GAI and G",,

o f (1 1.

geometries

i s an i n j e c t i o n geometry 0

be two " p e r m u t a t i o n geometries" i n t h e r e s t r i c -

Then t h e f u l l d i r e c t sum GAI

G;I,

i s a p e r m u t a t i o n geometry

( i n o u r meaning), b u t i t i s n o t a " p e r m u t a t i o n geometry" i n t h e meaning o f [l], 2 because ( w i t h t h e n o t a t i o n o f D e f i n i t i o n 2.2) XcN ,

4. REGULAR 9-GEOMETRIES

A geometry Gr (X)

= (W,9,XyA)

i s c a l l e d r e g u l a r i f each A Ed i s i n c l u d e d i n

.

Every s t a r 9-geametry i s r e g u l a r . The f o l l o w i n g G2(X) = (S,S,X,A) some A E A r r i s not regular : A, = tA,

X = S = ta,b,c,d),

A

9

=

(A,,

2

[A

=

2

t b ) , {c}, t d l ,

:[

all,

A;,

1

=

[A

= Ia,bl,

1

A ' = Ia,cl, 1

A" = ta,dl.}, 1

A = A, u A u A 1 2'

= Ia,b,c}},

A1,

A

A", 1

{b,cl,

A21. We n o t e t h a t A;' i s n o t i n c l u d e d

in A

2' The same example shows t h a t i f G (X) = ( S , S , X , A ) i s a geometry, t h e n 9 i s r n o t n e c e s s a r i l y t h e f a m i l y o f t h e independent s e t s o f a m a t r o i d on S . I t i s a l s o easy t o g i v e examples o f r e g u l a r Gr(X) w i t h t h e same p r o p e r t y (191).

A geometry G (X) = (W,Y,X,A) i s r e g u l a r i f and o n l y i f f o r r each A. E A , , w i t h i < r , t h e r e e x i s t s b e X \ A. such t h a t A. u [ ~ } 1 E . PROPOSITION 4.1. 1

1

1

P r o o f . Suppose G ( X I r e g u l a r , A. E Ai,

1

i< r . L e t A . c A E dr. We have r 1 1 r A.cA s i n c e i < r ( c f . Prop. 5.1. ( e ) ) . If b E A r \ A t h e n b E X \ A i and i r i' A . W b I E 9 , s i n c e A.u:bl C_ A r E 9 and 7 i s a s i m p l i c i a 1 complex. 1

1

On a Generalization of Injection Geometries Conversely, i f f o r each A.E Ai, 1

A.U 1

w i t h i < r. t h e r e e x i s t s b € X \ A. such t h a t 1

I b I E 9, t h e n A . i s i n c l u d e d i n some 1

times.

131

ArE

A : t h i s f o l l o w s by a p p l y i n g ( 3 ) r - i

r

0

P R O P O S I T I O N 4.2. A geometry Gr(X) i s r e g u l a r w i t h lArl

1 i f f Ar

=

=

[XI.

( I n t h i s case G ( X I i s e x a c t l y a m a t r o i d ) .

r P r o o f . I f A = ( X I , t h e n each A. E A. w i t h i < r i s i n c l u d e d i n A = X = u A r 1 1 r AEA s o t h a t G i s r e g u l a r w i t h IA 1 = 1. Conversely l e t G (X) be r e g u l a r w i t h r r r A = ( A I . I f X E X = u A, t h e n XE A. f o r some A. E A , , so t h a t X E A . c A AE A 1 1 1 1 r r r by r e g u l a r i t y . T h e r e f o r e X c Ar, i.e. X = A r

.

5. FIRST PROPERTIES OF .?'-GEOMETRIES PROPOSITION 5.1.

( a ) IA,1

L e t Gr

=

(S,S,X,A)

1 and t h e element A,EA,

=

be an 3-geometry. i s t h e minimum o f A .

( b ) I f Ah E Ah, A E A w i t h A c A and 0 s h < k-< r. t h e n t h e r e e x i s t s a k k h k k). c h a i n A h C Ah+l c...c A w i t h A E A ( s = h , h t l , k s s ( c ) Each A. E A . ( O < i s r ) i s i n c l u d e d i n a c h a i n A,C A C . . C A. w i t h

...,

1

A

S

E

AS (s=O,l,

1

1

...,i ) .

1

( d ) I f Gr i s r e g u l a r , each A.E A . ( O < i < r ) i s i n c l u d e d i n a c h a i n 1

A, c...c Ai

C...C

( e ) I f Aie where Ai

=

A, v { b }

=

1

w i t h A E A ( s = O ,...,r ) . r s s A i , A . € A . ( w i t h i , j = 0,1, A

J

...,r ) ,

J

and i f A . L A . , 1 J

then i C j ,

A . i f and o n l y i f i = j . J P r o o f . ( a ) L e t 1.) A = A . E A . . From ( 2 ) i t f o l l o w s t h a t i = 0. I f AAEA,, AEA 1 t h e n A,, C - A:. Then A, = A;; o t h e r w i s e i f bEA:\ A,, then from ( 3 ) i t f o l l o w s t h a t A

C A; w i t h AIEA1, contradicting (2). 1C A w i t h Ah+l~Ahtl. I f h t l = k, From (31, Ah v I b } = A ( b ) Let b E A k \ A h' htl - k t h e n Ahtl = Ak by (3). I f h t l < k , t h e n Ahtl C A and i t e r a t i o n o f t h e same a r g u k ment l e a d s t o a c h a i n A h C ... C Ak-l C "A, A,, w i t h A S e A ( h < s < k ) and A ; ( € A k .

S

A c t u a l l y k - 1 < r , so t h a t ( 3 ) i m p l i e s A;( = Ak, required. ( c ) Take h = 0 and k

=

and Ahc

r e g u l a r i t y . By ( c ) we have a c h a i n A, C...C

s=O,

Ar-lC

...,r ) .

Ar,

a c h a i n as

i i n (b).

( d ) If i = r, ( d ) f o l l o w s f r o m ( c ) . I f i c r , t h e n

Ai

... C A ~ - ~ kC iAs

C...C

s o t h a t we g e t a c h a i n

A.C

i

A

r

f o r some A

r

A. and b y ( b ) we have a c h a i n 1

A,C

...C

A.

1

C...C

Ar

(AS€ A

S'

E

A

r

by

P. V. Ceccherini and N . Venanzangeli

132

( e ) We have i c j by ( 2 ) . From Ai pose now A.

A ! w i t h Ai,

C

1 -

1

we g e t a c h a i n

A.

1

=

A,C

A . i t f o l l o w s o b v i o u s l y t h a t i = j. SupJ I f i = O , t h e n A, = A: by ( a ) . I f i f O , t h e n by ( c )

A;EAi.

...CAi-lCAi

C_ A;,

=

so t h a t c o n d i t i o n ( 3 ) i m p l i e s

v { b ] = A! where b E A . \ A . So A . C A! i m p l i e s A. = A ! . Ai-l 1 1 1-11 1 1 1

0

From 5.1 ( e l f o l l o w s COROLLARY 5.2. then AinA;

E A .

J

THEOREM 5.3.

L e t Gr be an 9-geometry.

I f Ai,

A; €Ai

w i t h Ai

# A;(O c i s r ) ,

with O s j c i - 1 . L e t Gr

=

(S,3',X,A)

be an

%geometry.

I f x i s i n c l u d e d i n some i n c l u d e d i n a unique A E A 1 1' A . ( j > O ) , t h e n x i s i n c l u d e d i n some Ai E Ai f o r e v e r y O < i c j . I n p a r t i c u l a r i f J G i s r e g u l a r , t h e n each x i s i n c l u d e d i n some A. E A . f o r e v e r y 0 < i c r . ( a ) Each X E X \ A,is

r

1

1

( b ) I f 0 < i < j < r , each A . E A . i s t h e u n i o n o f t h e elements o f A . which J J 1 are included i n A j' ( c ) I f Ai E A i ( O 6 i < r ) i s i n c l u d e d i n two d i s t i n c t element A ! A'.' E Aitl, ltl'

ltl

then A. = A ' n A:' = n A. 1 it1 it1 Ai 5 A €Aitl ( d ) The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : ( d l ) f o r a l l 0s i < r, each A . E Ai i s i n c l u d e d i n two d i s t i n c t elements o f 1

Aitl; ( d ) f o r a l l 0 c i < r , each A . E A . i s i n c l u d e d i n two d i s t i n c t elements o f 2 1 1 ( d 1 f o r a l l 0 s i < r , each A. E A. i s t h e i n t e r s e c t i o n o f elements o f A * 3 1 1 r' (d f o r a l l O c i < j 6 r , each A. E A . i s t h e i n t e r s e c t i o n o f t h e elements 4 1 1 o f A. c o n t a i n i n g A i ' J P r o o f . ( a ) Suppose x E X \ A , . Since X = U A, we have X E A . f o r some A.EA AE A J J j' Then x E A , v {XI = A C A , and A i s t h e unique element o f A i n c l u d i n g x. I f 1 - j 1 1 x E A . and O < i < j , t h e r e i s a c h a i n A,CA C . . . C Aic c A. ( A S € A s , O t s < j ) , J 1 J w i t h x E A1 = A, v {XI, so t h a t x E A cA 1 i' ( b ) I t i s enough t o prove t h a t i f x E A . \ A,, t h e r e e x i s t s some A. E A . such J 1 1 t h a t xEAi CA This f o l l o w s from ( a ) .

...

j'

5 Aitl

= A . E A ., w i t h i s j s ( i t l l - 1 by C o r o l l a r y 5.2, so t h a t n J J A! n Uyt1 by P r o p o s i t i o n 5.1 ( e ) . i it1 ( d ) ( d l ) * ( d 2 ) . From ( d i t f o l l o w s t h a t each A.E A . ( O < i < r ) i s i n c l u d e d 1 1 1

( c ) Ai

j = i and A

=

On a Generalization of Injection Geometries

133

and Ar-l i s i n c l u d e d i n two d i s t i n c t elements o f A Ar-1 r' ( d 2 ) * ( d 1. From ( d we have t h a t A . C A ' n A " w i t h A,; A" d i s t i n c t e l e 2 1 r r r = ments o f Ar. I f x ' EA; \ A:; and x" EA; \ ,A; t h e n Ai 2 A;+1 n AYtl w i t h A! 1+ I = A. v t x ' l and A! = A. v t x " 1 , A ' 1 l+l 1 it 1 # AYtl. i n a chain A.C 1

. e C

L

1 2 1 2 ( d 3 ) . From ( d 1 we have A. c A. "Aitl w i t h Ai+l # Ai+l and f r o m ( a ) 1 1 - 1+1 L we g e t Ai = A : nAit,. I f i t 1 = r t h e n ( d ) i s proved. I f i t 1 < r t h e n f r o m ( d ) 1 +1 3 1 11 12 2 21 and f r o m ( a ) we g e t A1 = Ait2 nAi+2 and Ai+2 = A. nA:f2 so t h a t it1 1t 2 hk A. = /--\ Ait2, and so on. i l ~ h k, c 2 (dl)

=,

(d4) * ( d

3

1. Obviously.

(dl) * ( d 4 ) . I n d u c t i o n on j - i . If j - i = 1, then, by ( a ) , ( d 1 i m p l i e s ( d ) . 1 4 Suppose t h a t each Ai E Ai i s t h e i n t e r s e c t i o n o f t h e elements o f A including j-1 A.. By t h e p r e v i o u s argument each such A . i s t h e i n t e r s e c t i o n o f t h e elements 1 J-1 o f A. i n c l u d i n g A * a l l such A . a r e p r e c i s e l y t h e elements o f A . i n c l u d i n g A J - j-1' J J i (because i f AiC A. f o r some A . E A . , t h e n t h e r e e x i s t s Aj-, E Aj-l w i t h J J J A.C A . C J\., by P r o p o s i t i o n 5.1 ( b ) ) . 1 J-1 J I f A E A h and A E A we say t h a t A and A a r e j o i n a b l e i f t h e y a r e i n h k k' h k A; c l u d e d i n some A E A . I f A h and A k a r e j o i n a b l e , we d e f i n e Ah v A k = A n EA A2AhU A t h e n Ah v A = A E A f o r some c >, max [ h , k l . k k c c

P R O P O S I T I O N 5.4.

I f A,,€

Ah, Ak E Ak a r e j o i n a b l e w i t h Ah v A

A nA = A . E A . t h e n h+k>Ii+c. h k 1 1 Proof. I n d u c t i o n on k - i 2 0 . I f k = i , t h e n A =

A

h

v Ai

=

Ah;

k

= A . C_

i

A

h

and A

c

= Ac E Ac and

k =

A v A = h k

i t follows htk = c t i . E A Then w i t h Ai 5 Ak-lCAk. k-1 k-1 Ah v Ak = Ac. Then c ' t c . I f c ' c c t h e n t d A c , ; i f x E A k \ A c , , t h e n

ifk & i + l , l e t be A

Ac, = Ah v Ak-l x

c_

) v Ix) = A v Ix) = = A v Ak = Ah v (A v {XI) = (Ah v A c h k- 1 k- 1 C' I n c o n c l u s i o n c ' t l > c . By t h e i n d u c t i o n h y p o t h e s i s h t k - 1 2 i + c ' , so

A so ~ t-h a~t A

~

Acl+l' that h

t

k %i t c'

THEOREM 5.5.

t

1> i

t

L e t Gr(X)

c. =

0

(S,S,X,A)

be a geometry o f r a n k r 6 3 . Then Gr(X)

satisfies the following condition: ( * ) i f Ah€

and A

k

$,

AkC

Aky A h n A k = A . w i t h h

are j o i n a b l e ,

1

t

k s i t r and i f AhU A E 7, t h e n Ah k

P.V. Ceccherini and N . Venanzangeli

134

Proof. We can suppose w i t h o u t l o s s o f g e n e r a l i t y t h a t h s k . The theorem i s t r u e f o r r = 1.

If r

=

-

2, t h e n a c t u a l l y h t k

i s 2. We can ObViOUSlY

assume t h a t Ah and A

k

a r e n o t i n i n c l u s i o n . I n t h i s case, t h e p o s s i b l e values o f h, k and i a r e : i = O , h = k = l . L e t A1, A ' € A be such t h a t AIUA' E 7 w i t h A nA' = A,€ A,. Then A1 # A' 1 1 1 1 1 1' and i f x E A ' \ A i t i s easy t o prove t h a t A v ( X Ii s an element o f A i n c l u d i n g 1 1 1 2 A1 and A;.

If r

= 3 then a c t u a l l y h t k -

i -<3. kk can obviously

assume t h a t Ah and A

k a r e n o t i n i n c l u s i o n . I n t h i s case, t h e p o s s i b l e values o f h, k and i, which a r e d i s t i n c t from t h e choice

i=O,

h = k = l a l r e a d y considered, a r e

Ii=O,

h = l , k=21 and

li=1, h=k=2) ,

we have A E A1, A2€A2, A1 n A2 = A, A U A € 9 . It 1 1 2 i s easy t o check t h a t t h e element A 3 € A 3 d e f i n e d by A v {XIwhere x E A \ A o , 2 1 c o n t a i n s A and A 1 2' When i = l , h=k=2, we have A A;EA2, A nA' = A € A and A 2 U A ' € 9 , Then i t 2' 2 2 1 1 2 i s easy t o prove t h a t t h e element A E A d e f i n e d by A3 = A v { X Iwhere 3 3' 2 xEA' \ A = A' \ A c o n t a i n s A and A' 2 2 2 1 ' 2 2' When i = O , h = l and k.2,

REMARK 5.6. rem 5.8).

When r ,4,

c o n d i t i o n ( * ) i s n o t n e c e s s a r i l y s a t i s f i e d ( c f . Theo-

I t can be proved t h a t t h e f i r s t values o f i, h and k , f o r which t h e

p r e v i o u s argument f a i l s , a r e I i = O , h=k=2). There e x i s t s an 9-geometry G ( X ) = ( S , 9 , X , A ) , namely a p e r m u t a t i o n 4 geometry o f rank 4, f o r which c o n d i t i o n ( * ) i s n o t s a t i s f i e d . 2 P r o o f . L e t be N = {1,2, 61. An element ( h , k ) E N w i l l b e w r i t t e n hk. L e t LEMMA 5.7.

...,

X

U l , 22, 33, 44, 45, 35, 26, 15), A,,

=

A.

I

=

I t h e set o f singletons o f X I

=

=

0, A,

tIxy1 : x y E X 1 ,

A = I t h e s e t of a l l t h e i n j e c t i v e p a i r s o f X I , 2 A:')=

t l l , 22, 33, 451,

A):'

= t l l , 22, 44, 351,

A:3)=

[33, 44, 11, 261,

A:4)

= I33, 44, 22, 151,

A:51=

Ill, 45, 261,

Ai6)

=

= IAJ,

(11, 35, 26),

A:7)

10 A = U A(i) 3 i=l 3

with:

= I33, 45, 261,

135

On a Generalization of Injection Geometries

) !A

= (33, 26,

151,

A )':

A"):

= (44, 35, 261,

= t44, 26,

151,

3

A = ." A(i) 4 1=1 4

with:

A:)'

ill, 44, 35, 26),

A:')=

= (11,

33, 45, 261,

A t 3 ) = f33, 44, 26, 15). L e t be

4 9

I1 c - N2 : I i s i n j e c t i v e 1 and A =

=

A

i'

i s a p e r m u t a t i o n geometry G ( X ) o f r a n k 4 4

I t i s easy t o check t h a t (S,y,X,A)

on

i =uo

x.

L e t us c o n s i d e r t h e elements o f A2 '. A 2 = ill, 221 and A; = I33, 441. Actual= 0 = A,, 2 + 2 - 0 6 4 , A p U A ; € 7 , b u t A and A ' a r e n o t j o i n a b l e , i . e . 2 2 G ( X I does n o t s a t i s f y c o n d i t i o n [ * I . 4 l y A2nA;

F o r each r >, 4 t h e r e e x i s t s an 9-geometry (resp. a p e r m u t a t i o n

THEOREM 5.8.

geometry) Gr(X) o f r a n k r, such t h a t c o n d i t i o n ( * ) i s n o t s a t i s f i e d . P r o o f . I n d u c t i o n on r 3 4 . For r = 4, t h e theorem reduces t o Lemma 5.7. Suppose t h a t t h e r e e x i s t s an 9-geometry ( r e s p . a p e r m u t a t i o n geometry) G;-l =

(S',S',X',A')

o f r a n k r - 1 2 4, f o r which c o n d i t i o n ( * ) i s n o t s a t i s f i e d ; i n o t h e r

A ' E A ' such t h a t : A ' n A ' = A!, h t k - i s r - 1, k k h k i A' UA' E 7' and A' and A ' a r e n o t j o i n a b l e . h k h k be an F g e o m e t r y ( r e s p . a p e r m u t a t i o n geometry) of L e t GI' = (S",7",X",d") 1 r a n k 1 w i t h A: = (A:). Then we c l a i m t h a t Gr = GAq1 b G; = (A,9,X,A) ( r e s p . words, t h e r e e x i s t

-

b G;) i s an 9-geometry ( r e s p . a p e r m u t a t i o n geometry) o f rank r such Gr = G ' r-1 t h a t c o n d i t i o n ( * ) i s n o t s a t i s f i e d . Indeed

Ah = A ' U A t E A h h'

A = A' U A: E A a r e such t h a t : k k k

A h n A k = (A; n A ; ( ) u A :

= A;UA:

€Ai,

5

k ( A ' U A " ) E A t h e n A,',uA'

h

t

k - i s r

-

I
and A a r e n o t j o i n a b l e , because i f h k C A ' E A ' , which i s i m p o s s i b l e . k -

AhUAk = (A;uAt)uA:€9(resp.E3), AhUAk

but A

0

REMARK 5.9. o f [31 f o r a l l r

=

From Theorem 5.8. 5

we g e t a

counterexample f o r P r o p o s i t i o n 2.3.

4. The p r o o f o f P r o p o s i t i o n 2.3.

i s i n c o r r e c t because t h e f l a t s

" A " and " B ' " a r e n o t n e c e s s a r i l y such t h a t A u B ' i s an i n j e c t i v e s e t .

We n o t e t h a t P r o p o s i t i o n 2.3. o f [ 3 ] i s t r u e f o r r 4 3 , by Theorem 5.6.

136

P. V. Ceccherini and N. Venanzangeli Other p r o p e r t i e s o f 3'-geometries,

s t a t e d i n [ 9 ] , w i l l be developed i n ano-

t h e r paper. ACKNOWLEDGEMENT. This r e s e a r c h was p a r t i a l l y supported b y GNSAGA o f CNR and by M P I . BIBLIOGRAPHY

[ l ]P.J. Cameron and M. Deza, On p e r m u t a t i o n geometries, J. London Math. SOC. ( 2 ) 20 (1979) 373-386. [ 2 ] P.V. Ceccherini and G. Ghera, A Vagner-Preston t y p e theorem f o r semigroups w i t h r i g h t i d e n t i t i e s , Quad. Sem. Geom. Comb. 1 s t . Mat. Appl. Univ. L ' A q u i l a 4, (1984). [ 3 ] M. Deza and P. F r a n k l , I n j e c t i o n geometries, J . Comb. Theory ( B ) 37 (1984) 31-40.

[ 4 1 M. Deza and P. F r a n k l , On squashed designs, ( t o appear).

[51 G. Ghera, Algebra e geometria d e l l e corrispondenze p a r z i a l i d i un insierne i n

se,

Tesi, Roma, 1 s t . Mat. " G . Castelnuovo",

l u g l i o 1983.

[ 6 1 M. Laurent, Geometries laminges: aspects a l g e b r i q u e s e t a l g o r i t h m i q u e s , These Univ. P a r i s V I I ( t o appear). I 7 1 6. Segre, I s t i t u z i o n i d i geometria s u p e r i o r e (a.a. 1963-641, Appunti d i P . V . C e c c h e r i n i Vol. 111: Complessi, r e t i , d i s e g n i , Roma, 1 s t . Mat. "G. Castelnuovo", 1965.

[81 M.C.

S c h i l l i n g , GBometries laminees e t bouquets de m a t r o i d s , P a r i s V I ( t o appear).

These,

Univ.

[ 9 ] N. Venanzangeli, Geometrie d i permutazioni, geometrie i n i e t t i v e e l o r o gener a l i z z a z i o n e , Tesi, Roma, 1 s t . Mat. "G. Castelnuovo", l u g l i o 1984.

Annals of Discrete Mathematics 30 (1986) 137-142 0 Elsevier Science Publishers B.V. (North-Holland)

137

A NEW CHARACTERIZATION OF HYPERCUBES P i e r V i t t o r i o C e c c h e r i n i and Anna Sappa O i p a r t i m e n t o d i Matematica "G. Castelnuovo" U n i v e r s i t i d i Roma "La Sapienza" C i t t i U n i v e r s i t a r i a , 00100 Roma, I t a l y

By u s i n g a theorem o f S. Foldes [ Z ] , we p r o v e t h a t a f i n i t e graph G i s a hypercube i f f i t i s connected, b i p a r t i t e , and t h e number o f geodesics between any two v e r t i c e s o f t h e graph GxK2 depends o n l y on t h e i r d i s t a n c e . Graphs o f t h e t y p e GxK, are also considered.

1. INTRODUCTION I n what f o l l o w s ,

a l l graphs w i l l be f i n i t e w i t h o u t l o o p s o r m u l t i p l e edges.

I f G = (V,E)

i s a graph and i f two v e r t i c e s x , y ~ V a r e j o i n e d by a path, t h e dis-

tance d(x,y) -

i s d e f i n e d as t h e number o f edges i n a g e o d e s i c ( s h o r t e s t p a t h ) be-

tween x and y . We denote by u ( x , y )

= v G ( x , y ) t h e number o f d i s t i n c t geodesics o f

G between x and y . I f x=y, we p u t d(x,y)=O and y ( x , y ) = l . We say t h a t a connected graph G i s a graph w i t h a g e o d e t i c f u n c t i o n i f t h e r e e x i s t s a map F:[O,l,..,

diam G I - N

such t h a t u ( x , y )

= F(d(x,y));in

t h i s case we

s h a l l say a l s o t h a t G i s F-geodetic. A s t u d y o f F - g e o d e t i c graphs i s developed i n

[8] i n a more general c o n t e x t . For each p o s i t i v e i n t e g e r n, t h e n-cube Q

i s d e f i n e d ( u n i q u e l y up t o i s o n morphism) as t h e graph whose v e r t i c e s a r e t h e subsets o f a s e t S w i t h n elements and t w o v e r t i c e s a r e j o i n e d by an edge i f and o n l y i f t h e y d i f f e r f o r e x a c t l y one element. I n o t h e r words t h e v e r t e x s e t o f Qn i s t h e s e t o f o r d e r e d n - p l e s o f 0 and 1; two v e r t i c e s a r e j o i n e d b y an edge i f and o n l y i f t h e y d i f f e r f o r e x a c t l y one d i g i t . A hypercube i s a graph isomorphic t o some

9,.

Several c h a r a c t e r i z a t i o n s o f hypercubes were g i v e n i n [ 1 1 - [ 5 1 ;

i n [51

p r e v i o u s c h a r a c t e r i z a t i o n s a r e a l s o summarized. We m e n t i o n h e r e t h e f o l l o w i n g c h a r a c t e r i z a t i o n g i v e n by S. Foldes. THEOREM 1.1 (S. Foldes [ 2 1 ) . A graph G i s a hypercube i f and o n l y i f

the

P.V. Ceccherini and A . Sappa

138

( 1 ) G i s connected and b i p a r t i t e , ( 2 ) G i s F-geodetic w i t h F ( k ) = k ! . Theorem 1.1 has been extended i n [ 6 ] t o a q-analogous r e s u l t . A q-hypercube Qq i s d e f i n e d as t h e graph whose v e r t i c e s a r e t h e subspaces o f a g r a p h i c space S n o f o r d e r q and dimension n-1, and two v e r t i c e s a r e j o i n e d b y an edge i f f one o f

them i s covered by t h e o t h e r i n t h e l a t t i c e o f a l l subspaces. For q = l , we have 1 Qn 9., I n t h i s paper we g i v e a c h a r a c t e r i z a t i o n o f hypercubes which i s based on theorem 1.1

and on t h e concept o f t h e t r a n s l a t i o n graph TG' o f a graph G '

(V',E').

The graph TG'

where K

i s t h e complete graph on two v e r t i c e s ( 1 and 21, ( c f . a l s o [ 7 1 ) .

2

i s t h e p e r m u t a t i o n graph ( G ' , i d V , ) ,

copy o f

X'E

V';

be a copy o f G ' = ( V ' , E ' )

more g e n e r a l l y ,

i . e . A" = { ~ " E V " : x ' E A ' I .

E

=

E ' uE"LI{(x' ,XI')€ V'XV":

= G'xK

2'

A'CV',

and l e t x " E V " denote t h e

t h e n A"cV" - w i l l denote t h e copy o f A ' ,

Then T G ' = (V,E) where V = V'uV" and X'

EV'l

The d e s c r i p t i o n o f TG' = (V,E)

V = V'xt1,ZI = t(x',h)

-

if

TG'

as t h e p e r m u t a t i o n graph ( G ' , i d V , 1 i s t h e

The d e s c r i p t i o n o f TG' = (V,E) f o l l o w i n g . L e t G" = (V",E")

i.e.

=

. as t h e graph G'xK

2

i s the following:

: x ' E V ' , h=1,2},

E = t((x',h),(y',k))EVxV

: [ ( x ' = y ' and ( h , k ) E E ( K 2 ) ) o r ( ( x ' , y ' ) E E '

and h=k) 11. We s h a l l prove t h e f o l l o w i n g THEOREM 1 . 2 A graph G ' i s a hypercube i f and o n l y i f

( 1 ) G ' i s connected and b i p a r t i t e , (2) G' x K

2

i s F-geodetic, for some F.

2. PROOF OF THEOREM 1.2. We s h a l l s t a r t w i t h t h e f o l l o w i n g lemma. LEMMA 2.1.

L e t G ' be a connected graph and l e t G = G ' x K 2 = TG' be t h e t r a n s -

l a t i o n graph o f G ' .

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t

( a ) G' i s F'-geodetic w i t h F ' ( k )

k!,

(b) G

i s F-geodetic w i t h F ( k ) = k ! ,

(c) G

i s F - g e o d e t i c for some F.

A New Characterization of Hypercubes Proof. O b v i o u s l y ( b ) - ( a ) , ( c ) . p e r m u t a t i o n graph ( G ' , i d V , ) (a)-(bl:

We use t h e d e s c r i p t i o n o f G = G'xK

o f G ' = (V',E'!

=

dG(x,y!!.

I f x = x ' E V ' and y = y " ~V " ,

2

= (V,E)

as t h e

( 5 1).

l e t x,y€ V. It i s enough t o suppose d ( x , y b 2 .

=uG,'x,y) = d G , ( x , y ) !

139

I f x,y€V',

then

S i m i l a r l y i f x , y ~ V " , then yG(x,y) l e t x " E V " and

Y'E

=

(x,y) G dG(x,y)!. Y

=

V ' be t h e c o p i e s o f x ' and y "

r e s p e c t i v e l y . We have d ( x , y ) = d (x',y'')= d , ( x ' , y ' ) t d ( y ' , y " ) = d ( x ' , y ' ) t l . G G G G G Moreover each geodesic o f G between x = x ' and y=y" i s o f t h e f o r m g ( x , y ) =

...,z',z" ,...,y " )

i s a geodesic o f G '

between x ' and y ' and i t s copy g " ( x " , y " )

=

i s a geodesic o f G"

between x " and y".

z"=x", and t h e case z ' = y ' , i . e . z"=y", a r e

(x',

where g ' ( x ' , y ' )

...,z ' , ...,y ' ) ( x " , ...,z", ...,y" )

= (x',

=

(The case z'=x', i . e .

n o t e x c l u d e d ) . I n o t h e r words, each geodesic g ( x , y ) i s o b t a i n e d e x a c t l y once f r o m a geodesic g ' ( x ' , y ' )

by choosing i n g ' ( x ' , y ' )

(z',z") between

bridge

Y~(x,Y) = Y

G

V ' and V " ) .

(c)-(b): X'E

The number o f such z ' i s d G , ( x t , y ' ) t l . So

, ( x ' , y ' ) .(dG , ( x ' , y ' ) + l )

= (dG,(x',y')+l)!

a c t u a l l y F(0) = F ( 1 )

=

a vertex z ' (as t h e basis o f t h e

= d G , ( x ' ,y '

1. (dG,(X',Y')+l)

=

dG(x,Y)!. =

1. When x , y c V w i t h dG(x,y)

= k>2,

there exist

V ' and y " E V " such t h a t d G ( x ' , y " ) = k . I f Y ' E V ' i s t h e copy o f y", we

have

d , ( x ' , y ' ) = d ( x ' , y O = k - 1 2 1 . With t h e same argument used f o r p r o v i n g G G ( a ) * ( b ) we o b t a i n f r o m ( c ) :

By u s i n g Theorem 1.1 we can now g i v e a new p r o o f o f t h e f o l l o w i n g w e l l known

result

.

LEMMA 2.2.

Qntl

=

Q,

x Kp.

-P-r o o f . We can use t h e d e s c r i p t i o n o f Qn x K 2 ( Q n , i d v , ) o f Q, = ( V ' , E ' ) ( 5 1 ) . The graph Q,

=

TQ

n

as t h e p e r m u t a t i o n graph

i s connected and b i p a r t i t e (theorem 1.11,

so t h a t TQ,

i s also

connected and b i p a r t i t e : i f A ' u B ' i s a p a r t i t i o n o f t h e v e r t e x s e t V ' o f Q (A'u 6") u ( A "

u B ' ) i s a p a r t i t i o n o f the vertex set V

and 6" a r e t h e c o p i e s o f A ' and 6 ' r e s p e c t i v e l y .

= V'U,V" o f TQ

n'

then n where A"

P.V. Ceccherini and A. Sbppa

140

By theorem 1.1 t h e graph Q

i s F ' - g e o d e t i c w i t h F ' ( k ) = k ! , so t h a t (Lemma n 2.1) Q xK i s F-geodetic w i t h F ( k ) = k ! . The graph TQ i s consequently a hypern 2 n cube by theorem 1.1, and t h e lemma i s proved s i n c e diam (TQ,) = diamg t 1 = n t l . n

I f Tn denotes t h e n - t h power o f t h e o p e r a t o r T ( t r a n s l a t i o n n of a graph), we have Q = T K2. ntl COROLLARY 2.3.

We g i v e now t h e L e t G ' a hypercube. By theorem 1.1,

Proof o f theorem 1.2.

condition ( 1 ) i s

s a t i s f i e d . Moreover G = TG' = G'xK

i s a hypercube by lemma 2.2, so t h a t ( b y the: 2 rem 1.1 again) G'xK2 i s F-geodetic w i t h F ( k ) = k ! , ( 0 k ~ G d i a m G), and c o n d i t i o n ( 2 ' ) i s satisfied. Conversely i f G ' i s a graph s a t i s f y i n g ( 1 ) and ( 2 ' 1 , then, by lemma 2.1, s a t i s f i e s c o n d i t i o n ( 1 ) and ( 2 ) o f theorem 1.1,

G'

so t h a t G ' i s a hypercube.

3. A GENERALIZATION OF LEMMA 2.1.

If G'

= (V',E')

( m 2 2 ) , t h e graph G = G'xK, V = V'x11,2

i s t h e complete graph on m v e r t i c e s m i s defined by

i s any graph and K

,...,m l

= (V,E)

= t(x',h):x'EV',

E = [((x',h),(y',k)EVxV:

h = 1,2,..

.,ml,

[ ( x ' = y ' and ( h , k ) E E ( K

in

1) o r ( ( x ' , y ' ) E E '

and h=k)]l.

An o t h e r d e s c r i p t i o n o f t h e same graph G = G'xK = ( V , E ) i s t h e f o l l o w i n g . L e t i i i im i G = (V ,E 1 be a copy o f G ' = ( V ' , E ' ) and l e t x E V denote t h e copy o f x ' E V ' , 1 ,m, where we assume G1 = G I , x = x ' . Then i=1,2,.

..

v

=

m u i=1

i

v ,E

m =

.

. u E' 1 =1

i ' u i ( x ,~J).v

i xvj :

X'EVI,

i,j=1,z

,...,m ; i + j l .

Lemma 2.1 i s i n c l u d e d f o r m=2 i n t h e f o l l o w i n g PROPOSITION 3.1.

Let G' =

(V',E'l

be a connected graph and l e t be G

=

G'xK

( m 2 2 ) . The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t ( a ) G' i s F'-geodetic w i t h F ' ( k ) = k ! (b) G

i s F-geodetic w i t h F ( k ) = k !

(c) G

i s F-geodetic f o r some F.

Proof. We can use t h e same argument as f o r Lemma 2.1. Indeed, i f x,yC V w i t h i i i x=x E V and y = y j ~ V (j i # j )t,h e n t h e o n l y e s s e n t i a l p o i n t i s t h a t G x I i , j ) = i = TG " T G ' .

m

A New Characterization of Hypercubes

COROLLARY 3.2.

141

A graph G ' i s a hypercube i f and o n l y i f

( 1 ) G ' i s connected and b i p a r t i t e ,

( 2 " ) G'xK

m

REMARK 3.3.

i s F-geodetic f o r some F and some m. P r o p o s i t i o n 3.1 e a s i l y y i e l d s a f a m i l y o f connected graphs which

a r e F-geodetic w i t h F ( k ) = k ! and which a r e n o t hypercubes (because t h e y a r e n o t bipartite). ACKNOWLEDGEMENTS.

T h i s r e s e a r c h was p a r t i a l l y supported by GNSAGA o f CNk

and by M P I . BIBLIOGRAPHY [ l ] L.R. Alvarez, U n d i r e c t e d graphs r e a l i z a b l e as graphs o f modular l a t t i c e s , Can. J . Math. 17 (1965) 923-932. [ 2 j S . Foldes, A c h a r a c t e r i z a t i o n o f hypercubes, D i s c r e t e Math. 17 (1977) 155-159. 131 J.M. Laborde, C h a r a c t e r i z a t i o n l o c a l e du graphe du n-cube, t o i r e s , Grenoble (1978).

Journee Combina-

[ 4 1 J.D. McFall, Hypercubes and t h e i r c h a r a c t e r i z a t i o n s , U n i v e r s i t y o f Waterloo Dept. o f Combinatorics and O p t i m i z a t i o n Research Report CORR 78-26 ( 1 9 7 8 ) . [ 5 1 J.D. 241.

McFall,

C h a r a c t e r i z i n g hypercubes, Annals D i s c r e t e Math. 9 (1980) 237-

[ 6 1 P . V . C e c c h e r i n i , A q-analogous o f t h e c h a r a c t e r i z a t i o n o f hypercubes as graphs, J . Geometry 22 (1984) 57-74.

[ 7 1 A. Sappa, C a r a t t e r i z z a z i o n e d i g r a f i t r a m i t e geodetiche, Tesi, Univ. d i Roma, D i p a r t i m e n t o d i Matematica (1984). [81 P . V . C e c c h e r i n i and A. Sappa, F-binomial c o e f f i c i e n t s and r e l a t e d combinat o r i a l t o p i c s : p e r f e c t m a t r o i d designs, p a r t i a l l y o r d e r e d s e t s o f f u l l b i n o m i a l t y p e and F-graphs, Annals D i s c r e t e Math. ( t h i s volume).

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Annals of Discrete Mathematics 30 (1986) 143-1 58 0 Elsevier Science Publishers B.V. (North-Holland)

143

F-BINOMIAL COEFFICIENTS AND RELATED COMBINATORIAL TOPICS: PERFECT MATROID DESIGNS, POSETS OF FULL BINOMIAL TYPE AND F-GEODETIC GRAPHS P i e r V i t t o r i o C e c c h e r i n i and Anna Sappa D i p a r t i m e n t o d i Matematica "G. Castelnuovo" Uni v e r s i t l d i Roma "La Sapi enza" C i t t a U n i v e r s i t a r i a , 00100 Roma, I t a l y

We i n t r o d u c e F-binomial c o e f f i c i e n t s as a n a t u r a l g e n e r a l i z a t i o n o f b i n o m i a l and q - b i n o m i a l c o e f f i c i e n t s . A g e n e r a l c a l c u l u s w i t h these numbers l e a d s t o u n i f y t h e a r i t h m e t i c a l p r o p e r t i e s o f ( f i n i t e ) p r o j e c t i v e and a f f i n e spaces and o f S t e i n e r systems S(t,k,v) i n t o those o f p e r f e c t matroid designs ( 5 2 ) . P a r t i a l l y o r d e r e d s e t o f f u l l b i n o m i a l t y p e ( 5 3 ) and graphs such t h a t t h e number o f geodesics between any two v e r t i c e s depends o n l y on t h e i r d i s t a n c e ( 5 4 ) a r e a l s o s t u d i e d by means o f t h i s f o r m a l c a l c u l u s .

1. F-BINOMIAL COEFFICIENTS Let N (resp.

Q ) denote t h e s e t o f non n e g a t i v e i n t e g e r s ( r e s p . r a t i o n a l

numbers) and l e t be N* = N \ ( O I , L e t F: N

+

Q*

=

Q\rO).

Q* be any f u n c t i o n such t h a t F ( 0 )

any f u n c t i o n such t h a t f ( 0 ) = 0, f ( 1 )

=

1 and f ( N * )

=

F ( 1 ) = 1 and l e t f : N

+

Q be

5 Q*.

F and f w i l l be f u n c t i o n s s a t i s f y i n g t h e above c o n d i t i o n s .

I n what f o l l o w s ,

Given an F, t h e a s s o c i a t e d f w i l l be d e f i n e d by: f(0)

0, f ( n )

F(n)/F(n-l) for n 1 1 .

=

Given an f, t h e a s s o c i a t e d F w i l l be d e f i n e d by: F ( 0 ) = 1, F ( n ) = f ( n ) F ( n - 1 )

i . e . F(n) = f ( n ) f ( n - l ) . . , f ( l )

Two such f u n c t i o n s F and f a r e t h e n m u t u a l l y associated,

f o r n 31.

and sometimes below t h e y

w i l l be used i n t e r c h a n g e a b l y . DEFINITION 1.1.

Given a p a i r ( F , f ) o f m u t u a l l y a s s o c i a t e d f u n c t i o n s and g i v e n

any i n t e g e r s k,n w i t h

OC

n

k < n , we s h a l l d e f i n e t h e F-binomial c o e f f i c i e n t ( o r

b i n o m i a l c o e f f i c i e n t ) I 1 as t h e r a t i o n a l number k F

f-

144

P. V. Ceccherini and A. Sappa

These numbers t u r n o u t t o be p o s i t i v e i n t e g e r s i n t h e f o l l o w i n g examples ( i n which f and F a l s o t a k e i n t e g e r v a l u e s ) .

EXAMPLE 1.2.

Binomial c o e f f i c i e n t s . Consider t h e p a i r o f m u t u a l l y a s s o c i -

ated functions

fl(t)

=

t

F ( t ) = t! 1

and

for a l l tEN.

Then n

in)= k F1

= Ik} i s t h e usual b i n o m i a l c o e f f i c i e n t

k fl

(O
EXAMPLE 1.3. Gaussian numbers. Given an i n t e g e r q > 1 , c o n s i d e r t h e p a i r o f m u t u a l l y associated f u n c t i o n s

f (t) = [ t l 9 9 where [ ]

9

[Ol,!

F ( t ) = [t],! 9

and [ ] ! a r e d e f i n e d by q

9

[OI

and

=

[ll

0,

=

=

9

[ll ! q

[tlq

1,

= q

t-1

t

... + q t l ,

[ t l q l= [ t l [ t - 1 1

= 1,

q

q

... [ llq,

t 22.

Then

n - n I k I F - ikIf 9 q

n

[,Iq i s

the q-binomial coefficient

EXAMPLE 1.4. Constant c o e f f i c i e n t s .

( g a u s s i a n number, c f . 141).

Given an i n t e g e r a z 1, c o n s i d e r t h e

p a i r o f mutually associated functions f(0) = 0,

f(1)

=

1,

f ( t ) = a;

F ( 0 ) = F ( 1 ) = 1,

F ( t ) = at - ’

(tz2).

Then n n I 1 = i l =1, n F OF

and

n

IkjF =

a

for

Usual b i n o m i a l i d e n t i t i e s and q-binomial

O < k
are p a r t i c u -

I45

F-Binomial Coefficients and Related Combinatorial Topics l a r cases o f F-binomial i d e n t i t i e s ( o b t a i n e d f o r F

F

=

1 g i v e now some o f them, w r i t t e n a s f - b i n o m i a l i d e n t i t i e s .

and f o r F

=

F r e s p . ) . We q

P R O P O S I T I O N 1.5. The f o l l o w i n g f - b i n o m i a l i d e n t i t i e s h o l d :

n ‘k’f

- f(n) - fo

n

-

Ikff

-

n-1

I k

{

If +

-

n-1 k If

f(n)

fo

f(n)-f(n-k) f(k)

n-1 ‘k-l’f

- f(n-k+l) n f(k) ‘k-l’f

(O< k tn),

n-1 (k-l’f

These f o r m u l a s suggest t h e f o l l o w i n g DEFINITION 1.6. Given a f u n c t i o n f and any i n t e g e r s k,n w i t h 0 < k
PROPOSITIONS 1.7. L e t

n,k

be as Def. 1.6. The f o l l o w i n g c o n d i t i o n s a r e

equivalent: (a)

i s independent o f n;

n,k

(b) f = f

dfn,k

(c)

9

f o r some q E Q * (where f

= qk

f o r some

f (n)-f (k) 9 9 f q(n-k) bf

n,l

f(n) =

=

i s f o r m a l l y d e f i n e d as i n Ex. 1 . 3 ) .

q E Q*. f

P r o o f . O b v i o u s l y ( c ) = + ( a ) . We have ( b ) -

q

=

9

n-1

+ . . . +q

..+q+l

qn-k-1,.

9

( c ) because

=

gf

n,k

=

A

n,k

k (b).

= qk. We prove now t h a t ( a )

L e t us p u t

q f o r a l l n, Then f ( 1 ) = 1 and f o r a l l i n t e g e r s n 3 2 we g e t Af

n,l

f(n-l)+f(l),

so t h a t t h e e q u a l i t y f ( n )

=

q

n-1

+

...

+ q+l f o l l o w s by

induction.

2 . PERFECT M A T R O I O DESIGNS A c o m b i n a t o r i a l geometry ( o r s i m p l e m a t r o i d ) M on a f i n i t e s e t S i s a m a t r o i d

P.V. Ceccherini and A. Sappa

146

0, S andevery s i n g l e t o n o f S a r e f l a t s o f M ( c f . [ 2 1 , [911. A p e r f e c t m a t r o i d d e s i g n (PMD) i s a c o m b i n a t o r i a l geometry M such t h a t e v e r y k - f l a t ( f l a t of rank k ) has t h e same number f ( k ) o f p o i n t s , k = O,l, ...,r k M

on S such t h a t

[lo],

(cf. f(0)

[ 1 2 ] , [ 1 4 1 ) . We s h a l l say t h a t f i s t h e s i z e - f u n c t i o n o f M. Obviously

n

0 and f ( 1 ) = 1, so t h a t f - b i n o m i a l c o e f f i c i e n t s t k l f can be

=

considered

(Osk(n4r-k MI. X PMDs i n c l u d e boolean s e t s 2 , p r o j e c t i v e spaces, a f f i n e spaces, t - ( v , k , l )

designs ( c f . [ 1 4 ] ) . PROPOSITION 2.1.

L e t M be a PMD on a f i n i t e s e t S w i t h s i z e - f u n c t i o n f .

I n t h i s case ( a ) M i s t h e m a t r o i d o f a l l subsets o f S i f and o n l y i f f = f 1' n n f f ( n ) = n, 1 ~ = 1( k ) ~ and = 1.

( b ) M i s t h e m a t r o i d o f f l a t s o f a p r o j e c t i v e space o f o r d e r q i f and o n l y n n-1 n i f f = f ( w i t h q a 2 ) . I n t h i s case f ( n ) = q t ... t q t l , 1 ~ = 1[ 1 ~ and 9

f ",k

k q

(Conversely, each o f t h e s e e q u a l i t i e s i m p l i e s f = f 1. 9 n-1. ( c ) IfM i s thematroid o f f l a t s o f an a f f i n e space o f o r d e r q, t h e n f ( n ) = q , f k 2k-n n t h e converse i s t r u e when q a 4. I n t h i s case { k ] f = qk(n-kl and = q - q . n-1 1. (Conversely, each o f these e q u a l i t i e s i m p l i e s f ( n ) = q

q

=

*

P r o o f . ( a ) i s obvious. For ( b ) ( r e s p . ( c ) ) , we have t o prove o n l y t h e " i f " p a r t . A s i m p l e c o u n t i n g argument shows t h a t e v e r y 3 - f l a t i s a p r o j e c t i v e ( r e s p . an a f f i n e ) p l a n e o f o r d e r q, so t h a t t h e r e s u l t f o l l o w s f r o m a w e l l known charact e r i z a t i o n o f p r o j e c t i v e ( r e s p . a f f i n e ) spaces by means o f planes, c f . [ Z ] ( r e s p .

[ll).

0

With an argument which i s s t a n d a r d f o r f i n i t e p r o j e c t i v e spaces ( c f . [14] one can prove t h e f o l l o w i n g PROPOSITION 2.2.

L e t M be a PMD w i t h s i z e - f u n c t i o n f.

( a ) The number o f k - f l a t s i n c l u d e d i n a r - f l a t ( w i t h O < k 6 r ) i s g i v e n by

- 'r, 1 " ' ' r , k - l A k, 1 * k, k-1

'

...f ( r - k + l )

f(r)

.

f(k). .f(l)

--

A ~ , ~ . . . A ~ , ~ - ~

'k, 1

"

'Ak,k-l

'k'f'

( b ) When a k - f l a t i s i n c l u d e d i n a r - f l a t , t h e number o f j - f l a t s between them

(01:k < j c r ) i s g i v e n by

147

F-Binomial Coefficients and Related Cornbinatorial Topics

I n p a r t i c u l a r B(O,j,r) B(k,ktl,r)

= cr(j,r)

and

"" f ( r - k ) A

=

k+l, k

( c ) When a k - f l a t A i s i n c l u d e d i n a r - f l a t 6, t h e number o f maximal c h a i n s

s g i v e n by

o f f l a t s between them ;(k,r)

...B ( r - 2 , r - l , r )

r ) B(ktl,kt2,r)

B(k,ktl

=

=

Ar,k.'.Ar,r-2 Ak

F(r-k)

t l ,k ' * ' A r -,lr - 2

where F i s a s s o c i a t e d t o f . L e t M be a PMD on a f i n i t e s e t S w i t h s i z e - f u n c t i o n f (and

PROPOSITIONS 2.3.

a s s o c i a t e d f u n c t i o n F ) and l e t a ( k , r ) , t i o n 2.2.

B(k,j,r),

;(k,r)

be d e f i n e d as i n p r o p o s-i

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

( 1 ) M i s t h e PMD o f a g r a p h i c space S o f o r d e r q ( i . e . a p r o j e c t i v e space o f o r d e r q ( p o s s i b l y a " l i n e " ) , when 9 2 2 , o r t h e boolean s e t PS, when q

r-k =(j-k~

(lb) B(k,j,r)

( l b ' ) B(k,k+l,r) ( l c ) ;(k,r)

=

Moreover, i f c o n d i t i o n ( 1 (2) f

=

f

q'

for all

= f r-k)

F(r-k)

a(k,r)

D < k.S r

for all

( l a ) a ( k , r ) = {;If

1);

c r k M;

0 6 k < ;i < r s r k M;

for all

for a l l

=

O s k i r s r k M;

O < k < r < r k M.

i s s a t i s f i e d , then r k q'

= [ ]

l3(k,j,r)

=

[;][I,,

;(k,r)

= [r-kl!

P r o o f . I t i s w e l l known ( c f . [ 3 ] , [ 1 6 ] ) t h a t ( 1 ) i m p l i e s ( l a ) - ( l c ) and ( 2 ) . (la)-(l).

By Prop. 2.2,

we g e t

A r , l * { 2r1 f r a ( 2 , r ) = A- 121f

231

s

P.V. Ceccherini and A. Sappa

148

(Ib')

=,

( 1 ) . By Prop. 2.2,

we g e t R ( k , k t l , r )

=

'r k 'ktl

4

r,k

=

'k+l,k'

.

f o r k = l we o b t a i n

a(k,r)

A

= A

r,l

= R(O,k,rl

= f(r-k)

and ( 1 ) f o l l o w s as above.

2,l

r - 'k'f'

r-0 I k-O'f

=

f(r-k)

,k

-

(lb)

=,

(la).

(Ic)

=)

( l b ) . I n each maximal c h a i n of f l a t s between a k - f l a t A and a r - f l a t

B t h e r e e x i s t s e x a c t l y one j - f l a t C ( 0 k ~c j < r ( r k M ) ; g i v e n such a C, t h e j o i n -

i n g of a maximal c h a i n between A and C and o f a maximal c h a i n between C and El i s a maximal c h a i n between A and B. {(C,$):C

The double c o u n t i n g argument, a p p l i e d t o t h e s e t between A and B), g i v e s :

i s a j - f l a t belonging t o a chain 0

F(r-k)

Nk,j,r)

=

REMARK 2.4.

F(j-k)F(r-j),

i.e.

B(k,j,r)

- {r-k j-kff.

F(r-k)

= F(r-jlF(j-kl

0

L e t M be t h e m a t r o i d o f f l a t s o f an a f f i n e space o f o r d e r q and +

define

a(k,r),

R(k,j,r)

and v ( k , r ) as i n Prop. 2.2. From ( 2 ) o f Prop. 2.3,

f o l l o w s t h a t f ( 0 ) = 0, f ( t ) = qt-', r-k a(k,r) = q REMARK 2.5. such t h a t

(k,r)

F ( t ) = qt(t-1)'2

r-1

[k-llq, B ( k , j , r )

=

[;:;Iq,

;(k,r)

it

and f o r k > O = [r-kl

q

!.

We g i v e now an o t h e r example o f a PMD M with s i z e - f u n c t i o n f

# F ( r - k ) . L e t S be a S t e i n e r system S(Z,k,n)

(e.g. t h e S t e i n e r

system whose b l o c k s a r e t h e l i n e s o f a p r o j e c t i v e ( r e s p . a f f i n e ) space S o f o r d e r q w i t h k = q t l ( r e s p . k = q ) ) and l e t M be t h e PMD ( o f r a n k 3 ) whose f l a t s a r e t h e empty s e t ,

the s i n g l e points,

t h e b l o c k s and t h e f u l l s e t o f p o i n t s . The s i z e

f u n c t i o n f i s such t h a t f ( 0 ) = 0, f ( 1 ) = 1, f ( 2 ) = k, f ( 3 ) = n; t h e a s s o c i a t e d f u n c t i o n F i s such t h a t F(0) = F ( 1 ) = 1, F ( 2 ) = k, F ( 3 ) = nk; moreover =

= 1,

A

2,l

= k-1,

A

3,l

=-

'-' . k

=

Therefore

2 whenever n f k - k + l ( i n t h e p r e v i o u s examples, whenever t h e space S i s n o t a p r o j e c t i v e plane).

149

F-Binomial Coefficients and Related Combinatorial Topics 3. POSETS OF FULL

BINOMIAL TYPE

These s t r u c t u r e s have been i n t r o d u c e d i n [111, ( c f . a l s o [71). We s h a l l use t h e n o t a t i o n o f $1.

(P,*.) be a p a r t i a l l y o r d e r e d s e t ( p o s e t ) w i t h minimum 0, and l e t a,b be elements o f P. I f a 4 b, t h e i n t e r v a l between them i s d e f i n e d by I ( a , b ) = Let

=

f c ~ P :a < c c b l . I f a < b a maximal c h a i n between a and b i s a c h a i n

a = a < a <...< an = b o f I ( a , b ) which i s n o t i n c l u d e d i n a l o n g e r c h a i n o f 0 1 I ( a , b ) ; t h e number n i s c a l l e d t h e l e n g t h o f t h e c h a i n . We s h a l l denote by ;(a,b) t h e number o f maximal c h a i n s o f I ( a , b ) , We s h a l l say t h a t

and we s h a l l assume

=

1 when a=b.

P i s a JD-poset i f i t s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n

(Jordan-Dedeking c h a i n c o n d i t i o n ) : i f a,b

(JD)

;(a,b)

E

f

w i t h a < b, a l l t h e maximal

c h a i n s o f I ( a , b ) have one and t h e same l e n g t h , denoted by d(a,b)

and c a l l e d t h e

d i s t a n c e between a and b ( o r t h e l e n g t h o f t h e i n t e r v a l ( I ( a , b ) ) . If a=b, we p u t d(a,b)

=

If

0.

P i s a JD-poset w i t h minimum 0, t h e n t h e rank o f an element a

d e f i n e d by r k a=d(O,a), and t h e rank o f We n o t e t h a t i f a,bEP

there exists a function F : f O , l , ;(a,b)

interval o f length n o f that

i s d e f i n e d by r k

w i t h a < b, t h e n d(a,b)<

We s h a l l say t h a t a JD-poset

a < b, t h e number

P

rk

P

P

P.

P ( w i t h minimum 0 ) has a c h a i n f u n c t i o n if

...,r k

PI

+

N such t h a t , f o r a l l a,b E 2 w i t h i.e.

every

has F ( n ) maximal c h a i n s . I n t h i s case we s h a l l say

p i s an F-chain p o s e t . A poset

DEFINITION 3.1. (1)

P

i s c a l l e d a poset o f f u l l b i n o m i a l t y p e i f

P has minimum 0,

(2) P

i s a JD-poset,

(3)

P i s a F - c h a i n poset, f o r some F.

If

f i s a JD-poset w i t h minimum 0, f o r each i n t e r v a l I ( a , b )

each i n t e g e r k, w i t h O s k s d ( a , b ) , Ik(a,b)

= I c E I(a,b)

We s h a l l say t h a t f : IO,1,

...,r k

+ .

P and f o r

of

we s h a l l p u t

: d(a,c) = k)

P has a s i z e f u n c t i o n _ i f t h e r e e x i s t s a f u c n t i o n

P)

is

= max t r k a: a € $ , .

o f maximal c h a i n s between them i s F ( d ( a , b ) ) ,

P

E

N such t h a t , f o r a l l i n t e r v a l s I ( a , b ) o f f ,

f,

I50

P.V. Ceccherini and A . Sappa IIl(a,b)I

= f(d(a,b)).

I n t h i s case we s h a l l say t h a t

P

i s an f - s i z e poset. The f o l l o w i n g p r o p o s i -

t i o n y i e l d s o t h e r e q u i v a l e n t d e f i n i t i o n s o f poset o f f u l l b i n o m i a l t y p e . PROPOSITION 3.2.

P

Let

be a JD-poset w i t h minimum 0; t h e f o l l o w i n g c o n d i -

t i o n s a r e e q u i v a l e n t , where f and F a r e m u t u a l l y a s s o c i a t e d f u n c t i o n s : (a)

i s a poset o f f u l l binomial type).

(Oc k c d(a,b)).

= t d(a,b)]

( b l IIk(a,b)I (c)

P

P i s an F-chain poset ( i . e . P i s an f - s i z e poset.

Proof. Let a , b E P , (a) * ( b ) . e x a c t l y one c E

acb.

L e t be O < k < d ( a , b ) .

I n each maximal c h a i n o f I ( a , b ) t h e r e e x i s t s

P such t h a t d(a,c)

= k.

I f c i s such an element o f I ( a , b ) ,

the

j o i n i n g o f a maximal c h a i n o f I ( a , c ) and o f a maximal c h a i n o f I ( c , b ) i s a maximal c h a i n o f I ( a , b ) . CL

The double c o u n t i n g argument a p p l i e d t o t h e s e t I ( c , o ) :

i s a maximal c h a i n o f I ( a , b ) , F(d(a,b)) = IIk(a,b)

I

d(a,c)

= k)

C E ~ ,

gives:

F ( k ) F(d(a,b)-k),

so t h a t ( b ) f o l l o w s .

( b ) * ( c ) . For k = l we o b t a i n I1 ( a , b ) I = I d(;yb))f = f ( d ( a , b ) . 1 ( c ) * ( a ) . A c t u a l l y I 1 ( a , b ) I = f ( d ( a , b ) ) . We now a p p l y i n d u c t i o n on d ( a , b ) ; 1 i f d(a,b) = 0,1, t h e n 1 = ;(a,b) = F ( d ( a , b ) ) . I f d(a,b),2, a p p l y t h e double coun t i n g argument t o t h e s e t { ( c , ~ ) : c E

U,U

i s a maximal c h a i n o f I ( a , b ) ,

d(a,c)=ll.

By t h e p r e v i o u s argument and b y t h e i n d u c t i o n h y p o t h e s i s , we have ;(a,bl

= (Il(a,b)I

EXAMPLE 3.3.

F(1) F(d(a,bl-l)

= f(d(a,b))

P= 2

I f X i s a f i n i t e s e t and i f

o f X o r d e r e d by i n c l u s i o n , t h e n

P

F(d(a,b)-l)

X

= F(d(a,bl).

i s the set o f a l l

0

subsets

i s a poset o f f u l l b i n o m i a l type, w i t h s i z e

f u n c t i o n f ( t ) = t and c h a i n f u n c t i o n F ( t ) = t ! ( 0 t ~c

1x1)

any such f u n c t i o n s f and F ( w i t h dom f = dom F = t O , l ,

...,n l ) ,

(cf.

[ 1 3 1 ) . Conversely

( c f . example 1.21, X can be c o n s i d e r e d as s i z e f u n c t i o n and c h a i n f u n c t i o n o f a boolean s e t p = 2 w i t h x x X 1x1 = n. We n o t e t h a t , i f P = I X I U ( o ) U ( , ) U...U ( k ) w i t h 2 4 k < 1x1 - 1, t h e n t h e

P i s n o t of f u l l b i n o m i a l t y p e : i f Y

poset

= d(Z,X)

z

to

x.

= 2, b u t t h e r e a r e

X

E ( 2 ) and

2 c h a i n s from 0 t o Y and

Z E(krl)l

1x1

-

t h e n d(0,Y)

=

(k-1) > 2 chains from

F-Binomial Coefficients and Related Combinatorial Topics EXAMPLE 3.4. dimension n-1

IS1

If X i s a f i n i t e p r o j e c t i v e space PG(n-1.q)

o f o r d e r q and

2 and i f ?' i s t h e s e t o f f l a t s o f X o r d e r e d by i n c l u s i o n , t h e n ?

i s a poset o f f u l l b i n o m i a l t y p e w i t h s i z e f u n c t i o n f ( t ) = [ t ]

and c h a i n f u n c t i o n q Conversely, any such f u n c t i o n s f = [ ] and

F ( t ) = [ t ] !. ( c f . [ 3 ] , prop. 3.3.). 9 h F = [ I ! w i t h q = p ( h h l , p p r i m e 3 2 1 and dom f = d a n F = 10.1 f-l

,...,n l

q ( c f . exaE

p l e 1 . 3 ) can be c o n s i d e r e d as t h e s i z e and t h e c h a i n f u n c t i o n s o f t h e poset o f t h e subspaces of a p r o j e c t i v e space X = PG(n-l,q) We n o t e t h a t , if with i

=

P

o f o r d e r q and dimension n-1.

i s t h e s e t of a l l t h e i - d i m e n s i o n a l f l a t s o f X = PG(n-l,q),

-l,O,...,k,n-1

where 1 4 k < n - 2 , t h e n t h e poset

P i s not o f f u l l binomial

t y p e : i f Y i s a l i n e and 2 i s a ( k - 1 ) - d i m e n s i o n a l f l a t , t h e n d(0,Y) b u t t h e r e a r e q + l c h a i n s f r o m 0 t o Y and q wk-' +

EXAMPLE 3.5. I f

$=

...+q+l > q + l

,,..., xla ,...,xnl ,...,x na 1

{xo,xl

= d(Z,X)

= 2

c h a i n s f r o m Z t o X.

i s any s e t w i t h a n t 1

elements (a,n a l ) , o r d e r e d by assuming xo as minimum and x

< x i f f i < h, t h e n ij hk i s a p o s e t o f f u l l b i n o m i a l t y p e w i t h s i z e f u n c t i o n f ( O ) = O , f ( l ) = l , f ( t ) ad with

2

t-1 chain functionF(O)=F(l)=l,F(t) = a ( 2 6 t < n ) . Conversely, any such f u n c t i o n s f and F ( c f . example 1.3.) can be c o n s i d e r e d as t h e s i z e and t h e c h a i n f u n c t i o n s o f a poset o f f u l l b i n o m i a l t y p e p as above. REMARK 3.6.

L e t M be t h e PMD o f t h e f l a t s o f an a f f i n e space X = AG(n-1,q) and l e t B

o f dimension n - 1 3 2 ,

not o f f u l l binomial type: t h e n d(0,Y)

z

to

= d(Z,X)

be t h e p o s e t o f t h e f l a t s o f X. The poset

P

is

i f Y i s a l i n e and Z i s an ( n - 3 ) - d i m e n s i o n a l f l a t ,

= 2 , b u t t h e r e a r e q c h a i n s f r o m 0 t o Y and q + l c h a i n s f r o m

x. Therefore,

if

M i s a PMD on a f i n i t e s e t X and if P i s t h e s e t of t h e f l a t s

o f M ordered by inclusion, then

P i s not necessarily a poset o f f u l l binomial

t y p e : t h e number o f maximal c h a i n s i n an i n t e r v a l I ( a , b ) does n o t depend o n l y on t h e l e n g t h o f t h e i n t e r v a l , b u t a l s o on t h e r a n k s o f a and b ( c f . a l s o Remark 2.5,

and Examples 3.3,

3.4).

The case when

P i s a poset o f f u l l b i n o m i a l t y p e

i s characterized by t h e f o l l o w i n g proposition. PROPOSITION 3.7.

L e t M be a PMD on a f i n i t e s e t X and l e t

t h e f l a t s o f M o r d e r e d b y i n c l u s i o n . The poset

P

P

be t h e poset o f

i s of f u l l b i n o m i a l t y p e i f and

o n l y i f M i s one o f t h e f o l l o w i n g PMD's: ( a ) M i s t h e t r i v i a l PMD o f r a n k 1 o r t h e t r i v i a l PMD o f r a n k 2 on X ( i . e .

M i s a t r i v i a l g r a p h i c space o f dimension 0 on 1 on X r e s p . ) ;

152

P.V. Ceccherini and A . Sappa (b) M i s t h e m a t r o i d 2

X

o f a l l subsets o f X, i . e . M i s t h e g r a p h i c space o f

o r d e r 1 and dimension 1x1-1;

( X I o f a l l f l a t s o f a p r o j e c t i v e space, o f dimenY q h a v i n g X as s e t o f p o i n t s .

( c ) M i s t h e m a t r o i d Mn-l s i o n n-1 and o r d e r q * 2 ,

P r o o f . I f M i s a PMD as i n ( a ) - ( c ) , t h e n t h e poset

P

o f i t s f l a t s i s a poset

o f f u l l b i n o m i a l t y p e ( c f . Ex. 3.3 and Ex. 3.4). L e t M b e a PMD on X such t h a t t h e p o s e t b i n o m i a l type. I f r k

M r 2,

B o f i t s f l a t s i s a poset o f f u l l

t h e n we a r e i n t h e case ( a ) . If r k

M > 2,

l e t f denote

t h e s i z e f u n c t i o n o f M. F o r any f l a t Y o f r a n k 2, we have t h a t lIl(O,Y)l

P

I n o t h e r words, t h e p o s e t

= ( Y I = f ( 2 ) = f(d(0,Y)).

has t h e same s i z e f u n c t i o n f t h a n t h e PMD M. Thus

c o n d i t i o n ( b ) o f Prop. 3.2 holds; i t means t h a t c o n d i t i o n ( l b ) o f Prop. 2.3 h o l d s . So c o n d i t i o n ( 1 ) o f Prop. 2.3 h o l d s too, and t h e r e s u l t ( b ) - ( c ) i s proved.

0

4. F-GEODETIC GRAPHS a l l graphs w i l l be f i n i t e w i t h o u t l o o p s o r m u l t i p l e edges,

I n what f o l l o w s ,

an:i a l l d i r e c t e d graphs w i l l be w i t h o u t d i r e c t e d c i r c u i t s . Any d i r e c t e d graph

p =

p(t) =

G'

(V,;)

=

i s o b v i o u s l y t h e Hasse diagram o f a poset

where x < y i f and o n l y if t h e r e e x i s t s a d i r e c t e d p a t h f r o m x

(V,<)

t o y; c o n v e r s e l y t h e Hasse diagram o f a poset p = (V,<)

6

= (V,:),

=

that

E

where ( x , y )

P(E) a r e

and

I f G = (V,E)

EE

i s a d i r e c t e d graph

i f and o n l y i f x i s covered by y. We s h a l l say

mutually associated.

(resp.

6

*

= (V,E))

i s a graph ( r e s p . d i r e c t e d graph) and i f two

v e r t i c e s x,yE V a r e j o i n e d by a p a t h ( r e s p . d i r e c t e d p a t h ) , t h e d i s t a n c e d ( x , y ) ( r e s p . i ( x , y ) ) i s d e f i n e d as t h e number o f edges i n a geodesic i . e . p a t h (resp. d i r e c t e d s h o r t e s t p a t h ) between x and y. We denote by (resp. b y

*

r (x,y)

+ = rG(x,y))

i n a shortest r(x,y) =rG(x,y)

t h e s e t o f d i s t i n c t geodesics o f G (resp.

of

between x and y; we p u t : v(x,y) q(x,x)

=

I

r(x,y)l

y(x,x)

= 1

diam G = max I d ( x , y )

and ;(x,y) and

=

;I

x,y)I +

d ( x , x ) = d x,x)

: x,y E

Vj,

d am

We s h a l l say t h a t a connected graph G =

E

V,E)

ifx # y; = 0;

= max ~+d(x,y) : x,y E W .

( r e s p . a d i r e c t e d graph

5

= (V,:))

El

153

F-Binomial Coefficients and Related Combinatorial Topics

...,diam

i s a graph w i t h a geodetic f u n c t i o n i f t h e r e e x i s t s a map F : ( O , l , (resp. F : ( O , l ,

*

...,diam

G)

N ) such t h a t u ( x , y ) = F ( d ( x , y ) ) ( r e s p .

+

GI

+

N

+

u(x,y) =

I n t h i s case we s h a l l a l s o say t h a t G (resp.

i) i s

= F(J(x,y))

f o r a l l x,y E V .

F-geodetic.

Note t h a t F(0) = 1 and t h a t F(n) # 0 f o r a l l n E dom F, so t h a t t h e as

sociated f u n c t i o n f can be considered as i n $1. Let

2

= (V,;)

be a d i r e c t e d graph and l e t x , y ~ V be such t h a t x - < y . Then t h e

i n t e r v a l I ( x , y ) i s defined by +

I(x,y) := I z E V :

x < z - ( y l , i . e . I ( x , y ) i s defined as i n P ( G ) ,

and t h e geodetic i n t e r v a l I g ( x , y ) i s d e f i n e d by := t z e V:

P(x,y)

Note t h a t I 9 (x,y)

C;(x,y) f o r some

ZE

5 I(x,y)

C; E r ( x , y ) ~ .

and t h a t I g (x,y) = t z e I ( x , y ) : i ( x , z )

= ;(x,y)-;(y,z)).

For any 1 < k < i ( x , y ) , l e t : = I z e I g ( x , y ) : ~ ( x , z )= k l

I;(x,y)

E

We say t h a t

= (V,:)

has a source

= (zE

I9 ( x , y ) : J ( Z , Y ) = i ( x , y ) - k l .

O E V i f f o r any

XE

V \ (01 t h e r e e x i s t s a d i r e c -

t e d path from 0 t o x.

A graph

6

= (V,i)

w i l l be c a l l e d a d i r e c t e d graph o f f u l l binomial t y p e ( w i t h

geodetic f u n c t i o n F ) i f : (a)

E

has a source 0,

( b ) f o r a l l x , y ~ Vw i t h x < y : (c)

Let

E

be t h e poset associated w i t h

(2)

= I(x,y),

G i s F-geodetic f o r some F.

PROPOSITION 4.1.

(1)

I 9 (x,y)

P has a minimum P i s a JO-poset

= (V,i)

6

be a d i r e c t e d graph and l e t P = P ( 6 ) = ( V , < )

(so t h a t

=

0 i f and o n l y i f

6(P)).

Then

has source 0;

i f and o n l y i f I g ( x , y )

I(x,y),

=

for a l l x , y ~ V

with x
6

P

i s a poset o f f u l l binomial type w i t h chain f u n c t i o n F i f and o n l y i f

i s a d i r e c t e d graph o f f u l l binomial type w i t h geodetic f u n c t i o n F. Proof. -

( 1 ) i s obvious. ( 2 ) :

s e t M(x,y) o f maximal chains o f G(x,y;of

E

P i s a JD-poset P i n I ( x , y ) i s the

O f o r a l l x , y ~ Vw i t h x c y

0

f o r a l l x , y ~ Vw i t h x < y t h e set

I(x,y) = Ig(x,y).

? ( x , y ) o f t h e geodesics

( 3 ) : P i s a poset

of

P.V . Ceccherini and A . Sappa

154

f u l l binomial type w i t h chain f u n c t i o n F

I

(M(x,y) Ig(x,y)

F(+d(x,y)) f o r a l l x < y i n V

=

= I(x,y)

and [;(x,y)(

= F(d(x,y))

t y p e w i t h g e o d e t i c f u n c t i o n F. The poset

p = (V,<),

o

O P has minimum 0, I' i s a JD-poset and

"G

has source 0, f o r a l l x < y i n V

E

i s a d i r e c t e d graph o f f u l l b i n o m i a l

0

where V = IO,x,y,z,tl

and O < x < z < y , O < t < y , i s n o t a

b u t t h e graph 6 ( p ) i s F-geodetic ( w i t h F = l ) ; n o t e t h a t Ig(O,y)

JD-poset, = {O,t,Yl

c I(0,y)

=

=

v.

PROPOSITION 4.2.

Let

"G

=

be a d i r e c t e d graph. The f o l l o w i n g c o n d i t i o n s

(V,i)

a r e e q u i v a l e n t (where F and f a r e m u t u a l l y a s s o c i a t e d ) : i s F-geodetic f o r some F,

(a)

( b ) f o r a l l x < y i n V and f o r a l l 1 g k c a ( x , y ) : (c) f o r a l l x < y i n V:

II~I=

k

*

The double c o u n t i n g argument

+

(z,<(x$y)): z ~ < ( x , y ) ,ger(x,y),

=

J(X,Y) k IF'

f(i(x,y)).

Proof, L e t x,y be elements o f V w i t h x < y . ( a ) - ( b ) . a p p l i e d t o t h e s e t Zk

9

( I (x,y)I = i

z(x,z) = k l

gives:

+

F ( d ( x , y l ) = I I z ( x , y ) l F ( k ) F(a(x,y)-k), (b) * (c) for k (c)

=)

=

9 d ( x ,y so t h a t I k ( x , y ) = I

1.

( a ) . We a p p l y i n d u c t i o n on a ( x , y ) 3 1 . I f a ( x , y )

= 1 = F ( 1 ) . Suppose d ( x , y ) h 2 ; by t h e i n d u c t i o n hypothesis,

IT(z,y)

1

= F(i(z,y))

t o t h e s e t Z1 g i v e s : = F(a(~,y)).

If

2

ated t o

E.

F.

1

I

=

we have 1 Therefore t h e double c o u n t i n g argument a p p l i e d

= F(G(x,y)-l).

I?(x,y)

1, t h e n I;(x,y) when Z E Z

= IIg(x,y) l F ( l ) F ( i i ( ~ , y ) - l ) = f ( a ( x , y ) )

1

F(a(x,y)-l)=

0

i s a d i r e c t e d graph, we s h a l l denote by G t h e u n d i r e c t e d graph a s s o c i -

PROPOSITION 4.3.

Let

= (V,:)

be a d i r e c t e d graph w i t h source O S V . Suppose

t h a t f o r each x E V t h e p a r i t y o f t h e l e n g t h o f any d i r e c t e d path f r o m 0 t o x depends o n l y on x; w r i t e p ( x ) =O i f i t i s even and p ( x ) = 1 i f i t i s odd. Then +

t h e u n d i r e c t e d graph G a s s o c i a t e d t o G i s connected and b i p a r t i t e . P r o o f . G i s connected s i n c e 0 i s a source. We v e r i f y t h a t G i s b i p a r t i t e by 0 1 i assuming V = V U V where V = I x E V : p ( x ) = il, i = 0 , l . We have t o prove t h a t i f ( x , ~ ) EE then p ( x ) # p ( y ) . We can suppose w i t h o u t loss o f g e n e r a l i t y t h a t (X,Y)E

E.

I f G(0,x) and G(0,y)

a r e geodesics f r o m 0 t o x and t o y resp.,

then

F-Binomial Coefficients and Related Combinatorial Topics

155

g(0,y) and G(0,x) u (x,y) are both d i r e c t e d paths from 0 t o y, so t h a t t h e i r lengths have t h e same p a r i t y p ( y ) . It f o l l o w s t h a t p ( x ) # p ( y ) . PROPOSITION 4.4.

L e t G = (V,E) be a connected b i p a r t i t e graph and l e t 0 be

any vertex o f G. Then by s t a r t i n g from 0, a n a t u r a l o r i e n t a t i o n can be d e f i n e d

on E, i n such a way t h a t

= (V,:)

i s a d i r e c t e d graph w i t h source 0 (and w i t h o u t -*

d i r e c t e d c i r c u i t s ) . It f o l l o w s t h a t

$=

where x
E

Proof. I f x,y E V w i t h d(0,x)

p(G)

=

(V,C) i s a poset w i t h minimum 0,

a d i r e c t e d path from x t o y .

= d(O,y),

then x and y cannot be adjacent be-

cause G i s b i p a r t i t e . Let ( x , y ) be an edge o f G. We have e i t h e r d(0,y) = d(O,x)+l o r d(O,x)

= d(0,y)tl.

Orient t h e edge from x t o y i n the f i r s t case, from y t o +

x i n t h e second. It i s easy t o show t h a t , whenever t h e r e i s an edge (x,y) EE, then (x,y)

i s t h e o n l y d i r e c t e d path from x t o y, and t h a t whenever t h e r e e x i s t s

a d i r e c t e d path from x t o y then t h e r e i s no d i r e c t e d path from y t o x; indeed i s a d i r e c t e d path, then we have

(by i n d u c t i o n on i ) , i f (xo,x l,...,xi) (xo,xl)

,..., ( X ~ - ~1 , X . and ) E ~d(O,xi)

THEOREM 4.5.

= d(O,xo) t i.

0 +

+

L e t G = (V,E) be a connected b i p a r t i t e graph and l e t G = (V,E)

be the d i r e c t e d graph obtained by s t a r t i n g from a given vertex OEV as i n Prop.

4.4. Then, whenever x and y are elements o f

V w i t h x < y , the f o l l o w i n g c o n d i t i o n s

are e q u i v a l e n t : (a

p(x,y) i s a d i r e c t e d path from x t o y i n

(b

p(x,y)

(C

p(x,y) i s a geodesic from x t o y i n G.

t,

i s a geodesic from x t o y i n 6, *

Proof.

be any d i r e c a) * ( b ) . Assume x = 0 f i r s t . L e t p(0,y) = (O=x",,.,,x.=y) 1

t e d path o f lenght i from x t o y i n

5.

We have d(O,y)=i, by t h e i n d u c t i o n argu-

ment sketched a t the end o f t h e p r o o f o f Prop. 4.4.

Therefore p(0,y) = p ( x , y ) i s

a geodesic i n G. Assume now 0 < x < y . L e t p(0,x)

and p ( x , y ) be any d i r e c t e d paths i n

8

from

0 t o x and from x t o y resp. By g l u e i n g p(0,x) and p(x,y) we get a d i r e c t e d path p(0,y)

in

'G.

For t h e previous case p(0,y)

i s a geodesic i n G. Thus i t s subpath

p(x,y) must a l s o be a geodesic i n 6.

(b)

=)

( a ) . I n d u c t i o n on i = d ( x , y ) .

x < y . so t h a t p(x,y)

When i = l , we have p(x,y) = ( x , y ) ~ isince *

i s a d i r e c t e d path i n G. Assume now i Z 2 and suppose t h a t

P.V. Ceccherini and A . Sappa

156

any geodesic o f G o f l e n g t h j w i t h 1 6 j < i i s a d i r e c t e d p a t h i n

,...,x 1. - 1 ,xi=y)

p ( x , y ) = ( x =x,x

Then p ( x , y ) i s a d i r e c t e d p a t h i n obviously.

(b)*(c).

I f p(x,y)

The geodesic

i s o b t a i n e d by g l u e i n g t h e geodesics p(x.xi-,)

and ( X ~ - ~ , Xo f~ G. ) These a r e b o t h d i r e c t e d p a t h s i n

(c) *(a)

6.

6

by t h e i n d u c t i o n h y p o t h e s i s .

E.

...,x .1= y ) i s a geodesic i n G, t h e n p(x,y) i s 0 1' (because ( b ) = . ( a ) ) , and i t must be a geodesic i n because = ( x =x,x

6,

a d i r e c t e d path i n

+

i n G would be a p a t h o f G s h o r t e r t h a n p ( x , y ) ,

any s h o r t e r d i r e c t e d p a t h p ' ( x , y )

which i s i m p o s s i b l e s i n c e p ( x , y ) i s a geodesic i n G. COROLLARY 4.6. L e t G

(V,E)

=

0

be t h e

be a connected b i p a r t i t e graph and l e t

d i r e c t e d graph o b t a i n e d by s t a r t i n g f r o m a g i v e n v e r t e x O E V as i n Prop. 4.4. Then ( 1 ) whenever x , y ~ V w i t h x r y , we have d ( x , y ) that

= +d(x,y), r ( x , y )

= ?(x,y)

so

r(x,y) = t(x,y); ( 2 ) G i s F-geodetic i f and o n l y i f

Proof.

E

i s F-geodetic f o r a l l O E V .

( 1 ) i s obvious. For ( 2 ) i t i s enough t o n o t e t h a t ;(x,y)

= r ( x , y ) when we

assume 0 = x. COROLLARY 4.7.

Let G

=

be a connected b i p a r t i t e graph. Then t h e f o l -

(V,E)

l o w i n g c o n d i t i o n s a r e e q u i v a l e n t , where F and f a r e m u t u a l l y a s s o c i a t e d f u n c t i o n s : ( a ) G i s F-geodetic,

IIzEV:

d(x,y)

( b ) x,yEV,

Otkcd(x,y)*

( c ) x,yEV

*

REMARK 4.8.

The statement o f C o r o l l a r y 4.7 also h o l d s i f G i s n o t b i p a r t i t e ;

~ [ Z E V :d(x,z)

d ( x , z ) = k, d(z,y)

= 1,

d(z,y)

= d(~,yl-kl(=

d(x,y)-llI

I

= f(d(x,y)).

)f,

0

i t can be proved by a d i r e c t argument ( c f . [ 1 6 1 ) . T h i s can a l s o be o b t a i n e d f r o m -t

t h e p r o o f o f Prop. 4.2, r(x,y), (zEV:

d(x,y)

by r e p l a c i n g g(x,y),

+

r(x,y),

+

d(x,y)

(and by exchanging consequently t h e s e t I!(x,y)

r e s p . w i t h g(x,y), and t h e s e t

d ( x , z ) = k, d ( z , y ) = d ( x , y ) - k ) ) .

EXAMPLE 4.9.

The complete graph Kn, a t r e e G , t h e ( 2 k t l ) - c i r c u i t G a r e F-geo-

d e t i c graphs w i t h F = l . An F-geodetic graph w i t h F-1 i s c a l l e d g e o d e t i c - g r a p h ( c f . [161, [ l a ] ) . The i l k - c i r c u i t G i s F - g e o d e t i c w i t h F ( t ) = l , f o r t < k , and w i t h F ( 2 ) = 2 otherwise. EXAMPLE 4.10.

The complete b i p a r t i t e graph Kn,,=(V,E)

with V =

V'UV",

157

F-Binomial Coefficients and Related Combinatorial Topics

IV'(

=

IV"I

= n i s F-geodetic w i t h F ( 0 ) = F ( 1 ) = 1, F ( 2 ) = n.

A hypercube i s an F-graph w i t h F ( t ) = t ! . Conversely any con-

EXAMPLE 4.11.

n e c t e d b i p a r t i t e F-graph w i t h F ( t ) = t ! i s a hypercube ( c f .

= t (0

The graph Kn x Km i s F - g e o d e t i c w i t h F ( t

EXAMPLE 4.12. (cf. (51,

131).

< t s 2 cn,m)

[151 1 -

I f Qn i s t h e n-cube, t h e graph Qn x K

EXAMPLE 4.13.

s

m

F-geodetic

with

F ( t ) = t ! ( O s t < n t l ) ( c f . [51, Prop. 3.1).

i s F - g e o d e t i c f o r some F

I f G i s a connected graph and i f GxK,

EXAMPLE 4.14.

and some m x 2 , t h e n F ( t ) = t ! . Moreover, i f G i s b i p a r t i t e , t h e n G i s a hypercube (and GxK i s a l s o a hypercube i f and o n l y i f m = 2 ) ( c f . [ 5 ] , Prop. 3.1, Cor. 3.2) m EXAMPLE 4.15. L e t G be t h e d i r e c t e d graph whose v e r t i c e s a r e t h e subspaces ~o f a g r a p h i c space o f dimension n and o f o r d e r q 3 1 and ( x , y f l a t x i s covered b y t h e f l a t y. Then

6

i s F-geodesic w i t h F ( t

i s an edge i f t h e =

[tlq!( c f .

[31,

Prop. 3.3).

REMARK 4.16.

E

L e t G be t h e u n d i r e c t e d graph a s s o c i a t e d t o t h e d i r e c t e d graph

c o n s i d e r e d i n t h e Ex. 4.15.

I n o t h e r words, G i s t h e q-analogue o f Qn-,.

Note

t h a t when q a 2, G i s n o t F-geodesic: i f x,y a r e two p o i n t s and z i s a p l a n e cont a i n i n g x, t h e n d(x,y)

= d(x,z)

t h a t t h e d i r e c t e d graph

5

graph o f Example 4.15;

=

2, b u t 2 = v(x,y)

#

u ( x , z ) = q t l . Note a l s o

o b t a i n e d f r o m G by s t a r t i n g f r o m t h e empty f l a t i s t h e

when we s t a r t from a f l a t O f 0 , t h e n

E

i s F - g e o d e t i c i f and

only i f q=l. EXAMPLE 4.17. o f Ex, 3.5.

Then

Let

6

6

=

6(P) be

t h e d i r e c t e d graph a s s o c i a t e d t o t h e p o s e t

p

i s F-geodetic a c c o r d i n g l y t o Prop, 4.1 ( b u t G i s n o t F-geode-

tic). ACKNOWLEDGEMENT.

T h i s r e s e a r c h was p a r t i a l l y supported by GNSAGA o f CNR and by

MPI.

BIBLIOGRAPHY [ 1 1 F. Buekenhout, Une c h a r a c t e r i z a t i o n des espaces a f f i n s basee s u r l a n o t i o n de d r o i t e , Math. Z. 111 (1969) 367-371. [ 21 P . V . C e c c h e r i n i , 78-98.

S u l l a nozione d i s p a z i o g r a f i c o , Rend.

Mat.

( 5 ) 6 (1967)

P.V , Ceccherini and A . Sappa

158

[ 3 ] P.V. C e c c h e r i n i , A q-analogous o f t h e c h a r a c t e r i z a t i o n o f hypercubes as graphs, J. Geometry 22 (1984) 57-74. [ 4 1 P.V. Ceccherini, A. Dragomir, Combinazioni g e n e r a l i z z a t e , q - c o e f f i c i e n t i b i n o m i a l i e spazi g r a f i c i , A t t i Convegno Geometria Combinatoria e sue a p p l i c a z i o n i (Peruaia. Settembre 1970) 137-158.

-~

[ 5 1 P.V.

Ceccherini, A. Sappa, A new c h a r a c t e r i z a t i o n o f hypercubes,Annals D i s c r e t e Math. ( t h i s volume).

[6

1 L.

[7

1 L.

C e r l i e n c o , F. P i r a s , C o e f f i c i e n t i b i n o m i a l i g e n e r a l i z z a t i , Fac. S c i . C a g l i a r i 52 (1982) 47-56.

Rend.

Sem.

C e r l i e n c o , F. Piras, G-R-Sequences and i n c i d e n c e coalgebras o f p o s e t s o f f u l l b i n o m i a l t y p e , ( t o appear).

[ 8 ] R.J. Cook, D.G. [ 9 ] H. Crapo, G.C.

Pryce, U n i f o r m l y g e o d e t i c graphs, t o appear. Rota, C o m b i n a t o r i a l geometries, MIT Press, Cambridge (1970).

[ 1 0 1 M. Deza, N.M. S i n g h i , Some p r o p e r t i e s o f p e r f e c t m a t r o i d designs, Annals D i s c r e t e Math. 6 (1980) 57-76. [ 11 ] P. D o u b i l e t , G.C. Rota, R.P. SRanley, On t h e f o u n d a t i o n s o f C o m b i n a t o r i a l t h e o r y V I : t h e i d e a o f g e n e r a t i n g f u n c t i o n , i n G.C. Rota ( e d . ) , F i n i t e Oper a t o r Calculus, Academic Press, New York (1975) 83-134. [ 12 1 J. Edmonds, U.S.R. Murti, P. Young, E q u i c a r d i n a l m a t r o i d s and m a t r o i d d e s i gns, i n "Combinatorial Mathematics and i t s A p p l i c a t i o n s " , Second Chapel H i 11 Conference ( 1 9 7 0 ) .

[ 1 3 1 S. Foldes, A 159.

c h a r a c t e r i z a t i o n o f hypercubes, D i s c r e t e Math. 17 (1977) 155-

[ 14 1 B.L. R o t h s c h i l d , N.M. S i n g h i , C h a r a c t e r i z i n g k - f l a t s i n geometric designs, J. Comb. Theory A 20 (19761, 398-403. [ 15 1 A. Sappa, C a r a t t e r i z z a z i o n e d i g r a f i t r a m i t e geodetiche, Tesi, Univ. d i Roma, D i p a r t i m e n t o d i Matematica (1984). [ 16

I R.

S c a p e l l a t o , On g e o d e t i c graphs o f diameter two and some r e l a t e d s t r u c t u r e s ( t o appear).

[ 1 7 1 B. Segre, L e c t u r e s on modern geometry. With an Appendix by L. Lombardo-Rad i c e , Cremonese, Roma (1961).

[ 1 8 ] J.C. Stempe, 266-280.

Geodetic graphs o f diameter two, J. Comb. Theory

B 17 (1974)

Annals of Discrete Mathematics 30 (1986) 159-170 0 Elsevier Science Publishers B.V. (North-Holland)

159

POLYNOMIAL SEQUENCES ASSOCIATED WITH A CLASS OF INCIDENCE COALGEBRAS. Luigi Cerlienco Dip. di Matematica Univ. di Cagliari 09100 Cagliari Italy

Giorgio Nicoletti Dip. di Matematica Univ. di Bologna 40127 Bologna Italy

+

Francesco Piras Dip. di Matematica Univ. di Cagliari 09100 Cagliari Italy

A few special sequences of polynomials associated with both automorphisms and hemimorphisms of a particular class of coalgebras as well as their links with locally finite posets of binomial type are analysed.

50 *

The purpose o f this paper is to briefly study a class of coalgebras, which we call c o a Z g e b r a s of binorniaZ t y p e . These are the coalgebras C having a countable basis (bi) such that i i 0 ~ bi. =i6 Abi = j ~ ohj bj@bi-jy i where h. are integers and hE=l. 1

Coalgebras of binomial type are the background for some recent work in combinatorics beginning with polynomials o f binomial type [17].We show that, under mild conditions, the structure constants hj behave much like binomial coefficients, and that the analog of sequence of polynomials of binomial type is obtained in any coalgebra of binomial type from coalgebra morphism. The one new coalgebraic notion introduced in this work is the notion of k e m i m o r p h i s m , namely, a linear map f on C into itself such that A*f = (1ef)oA. The image of the distinguished basis (bi) under a hemimorphism generalizes sequences of polynomials introduced by Goldman and Rota [ 1 4 ] . We believe the general setting of coalgebras (and Hopf algebras) to be suited to a variety o f combinatorial problems which we propose to study in future publications.

§ I* 1.1. Let C:=(V,A,E) be a coalgebra over a field K of characteristic zero. Here V is a K-vector space and e:V-K fcounitland A:VNVBV f c o m u Z t i p Z i c a t i o n o r diagonaZizationl are linear maps such that the following diagrams commute: Research partially supported b y "Fondi Ministeriali per la Ricerca 40% e 60%".

L. Cerlienro, G. Nicoletti and F. Rras

160

vA ' ev K@V

-

h

V-

V@V /A@ I

(coassociativity)

V@V@V

IQA

- €@I

I @ E

V@V

VOK

i s t h e c a n o n i c a l isomorphism and I t h e i d e n t i c a l map).

($

v@w"w@v (twistI f , moreover, we have A = T o A , where T : V 0 V - V @ V y - o p e r a t o r ) , C i s s a i d t o be c o c o m m u t a t i v e . A s u s u a l , we s h a l l d e n o t e by C*=(V4,m,u) t h e d u a l K-algebra of C ; w e have U=E* and m=A*o j , where j:v'@v"-(V@V)" i s t h e c a n o n i c a l embedding. A l i n e a r map f:V-V i s an endomorphis m o f C i f (3) Aof = ( f 8 f ) o A (4) €Of = E . A l i n e a r map f : V - V

i s s a i d t o be a r i g h t he m im or phis m of C i f = ( I e f ) 0 A. L e f t hemi morph is ms a r e d e f i n e d i n a s i m i l a r way. I f C i s cocommutativ e , e a c h r i g h t hemimorphism i s a l s o a l e f t hemimorphism, and c o n v e r s e l y . L e t u s d e n o t e by & m ( C ) t h e s e t of a l l r i g h t hemimorphisms of t h e c o a l g e b r a C . Hem(C) i s c l o s e d under l i n e a r c o m b i n a t i o n and f u n c t i o n a l c o m p o s i t i o n . Hence Hem(C) i s an a l g e b r a . Moreover, i f f" i s t h e d u a l map of feHem(C), t h e n f o r every arV w e have: (5)

p(a) = =

A o f

f'(m(a@l))

= ( f ' o m ) ( a @ l ) = (f'O(A*oj))(a@Jl) =

( ( A o f ) * o j ) ( a B l ) = ( ( ( I @ f ) o A ) * o j ) ( a @ l ) = ( A * o ( I @ f f o j ) ( a @ 1) =

(because of ( f e g f o j =

=

jo(P@g") )

( ( A * o j ) o ( f @ f " ) ) (a el)

=

(A* o j ) ( a @ f*(l))

= m(a@ f C ( l ) ) ,

i . e , , u s i n g aB i n s t e a d of m(a @ 6) : F(a) = a*f"(l) w i t h l=u(lK)EV, (6) which becomes f * ( a ) = ? ( l ) . a i n t h e c a s e of l e f t hemimorphisms.

The e l e m e n t f'(l) ind(f) :

w i l l be c a l l e d t h e i n d i c a t o r o f f and d e n o t e d by

i n d ( f ) := f'(1)

(7)

.

A s a consequence of ( 6 ) we o b t a i n :

Prop.1. 17')

The map ind : &m(C)C* f c----ci n d l f )

i s a monomorphism o f a Z g e b r a s .

( N o t i c e , however, t h a t ( 7 ' )

i s an antimonbmorphism i n t h e c a s e o f

161

Polynomial Sequences and Incidence Coalgebras

.

l e f t hemimorphisms) P r o o f , I t r e m a i n s t o p r o v e t h a t ind(f0:) we have = g'(f"(1) (fogf (1) = (g*of')(l)

= ind(f)-ind(g); in fact, = P(l)*g*(l).

1.2. I n t h e f o l l o w i n g w e s h a l l a l w a y s assume t h a t C = ( V , A , E ) a c o u n t a b l e b a s i s (bi)iEN such t h a t

i

Abi =

i

jio hj bj@bi-j,

D has

0

h 0 =1

I f t h i s i s t h e c a s e , C i s s a i d t o be a c o a t g e b r a of b i n o m i a l t y p e . I t i s t h e n a s t r i c t l y g r a d e d c o a l g e b r a . I n p a r t i c u l a r , bo i s t h e u n i q u e g r o u p - l i k e e l e m e n t ( i . e . Abo=bo@3bo) and bl i s a p r i m i t i v e e l ement ( i . e . Abl=bl@bo + b o @ b l ) . The d u a l a l g e b r a C' i s n a t u r a l l y endowed w i t h a t o p o l o g i c a l s t r u c t u r e ( t h e s o - c a l l e d f i n i t e t o p o t o g y ) assuming t h e f a m i l y Ui = IBEV

1 ( V j ) ( j s i =$ B ( b1. ) = O l ,

ieN

a s a b a s e f o r a s y s t e m o f n e i g h b o u r h o o d s o f z e r o . With t h i s t o p o l o g y , e a c h e l e m e n t a o f C y c a n be r e p r e s e n t e d a s f o l l o w s : i n i a = 1 a.b := l i m .Z a . b i>o 1 n-m i=o 1 where a := < a l b i > : = a ( b i ) E K i and bi: V-K i b -6. j 3 i T h i s i s sometimes e x p r e s s e d by s a y i n g t h a t ( b ) i E N i s a p s e u d o - b a s i s o f v". With r e f e r e n c e t o a f i x e d b a s i s ( b i ) and i t s d u a l p s e u d o - b a s i s ( b i ) , we c a n r e p r e s e n t e a c h c o u p l e o f e l e m e n t s i " i a = L aib EC' Z a bi E C , i=o i>o i i r e s p e c t i v e l y a s a c o l u m n - v e c t o r ( a ) o f e n t r i e s a and a s a row-vecT h u s , we may w r i t e t o r ( a , )o f e n t r i e s a 1 i' i .(a) = : < a l a > = ( a i ) * ( a ) .

a =

Moreover, f o r a n y g i v e n l i n e a r map f : V - V l e t us define the repres e n t i n g m a t r i x M f f ) , i n t h e same way a s i n t h e f i n i t e - d i m e n s i o n a l c a s e , t o be t h e ( N r N ) - m a t r i x whose ( r , s ) - e n t r y i s g i v e n by < r l f ( s > := < b r l f ( b s ) > = < p ( b r ) l b S >

.

Of c o u r s e t h e same m a t r i x a l s o r e p r e s e n t s t h e d u a l map f*: M(f)=M(f*). N o t i c e t h a t t h e i - t h column f ( b i ) o f M(f) h a s a f i n i t e s u p p o r t . Using t h e above n o t a t i o n a l c o n v e n i e n c e s , w e may e a s i l y d e d u c e t h e following : Prop. 2 .

A l i n e a r map

g:C*--tCu

i s a continuous ( r e l a t i v e t o the

L. Cerlienco. G. Nicoletti and F. Piras

162

f i n i t e t o p o l o g y ) i f and onZy i f t h e r e e x i s t s a l i n e a r map f : C - + C s u c h t h a t g=f”.

Proof. In fact g is continuous if and only if its representing matrix M(g) has columns with finite support. 0 i i i If cr=~a.b , B = c ~ . b and y=zy.b :=m(a@B), we have 1 1 i i y i = 2 h. a . B . (10) j=o J J 1-j s o that hi’s are also the structure constants of the algebra C* relative toJ the pseudo-basis (b’). This, together with Prop.1 and 2, imp1 ie s : Prop. 3 . I f C i s a c o a Z g e b r a of b i n o m i a l t y p e , t h e n map ( 7 ’ ) i s a n i s o m o r p h i s m of a l g e b r a s . Proof. In fact, for every @EC* the representing matrix of the lin-

ear map m(-@g),

because of (lo), has columns with finite support.

1.3. Commutativity of diagrams (1) and (2) gives.rise to the following identities relative to structure constants hl: I i . h. h’ = (11) J

r

(12) Obviously, C is cocommutative if and only if i

i

h. = himj. (13) J The structure constants hi of a coalgebra of binomial type can all be expressed in terms &f h i alone, which will be more simply denoted by ni:=ht. In fact, we have the following: Prop. 4 . ~E{z,z,

t h e n f o r e v e r y i and e v e r y

f o r lrjss-1 b u t n s = O , 5e haue:

I f rl .#O

..., s-11

a)

n

b) i f

el

P

=O f o r some p > s , t h e n

i

i+l

J

j

h:h

*

I f i n s t e a d q.#O

...

.hi+s-j

=

j

and n i ! : =

3

j

+1

i

ti.i 11

rli-l!

oi

)

+j - 1

=...=h;

=O;

0.

f o r e v e r y j, t h e n h j =

3 q0!:=1

h!=hP

ni! = 0

,

i’‘ lli-i*

(where

f o r e v e r y i and e v e r y j . I n such a c a s e ,

C i s cocommutative. i-1 Proof. For r-1, (11) gives hi = hj-l ni/nj. From this, we deduce by recurrence both a) and b). Let us first prove c) when j=1. From (11) and a) we get

163

Polynomial Sequences and Incidence Coalgebras

and t h e n , s i n c e n =O, c ) f o l l o w s from tge case s t a t e m e n t i s now t r i v i a l .

..

i i+l i+s-li+s-l=h . hl . .hl -0. When j > l , and from b j . The second p a r t o f t h e

nini+b".n J=

0

Thus, e a c h c o a l g e b r a o f b i n o m i a l t y p e C i s c h a r a c t e r i z e d , r e l a t i v e Accordingly, we s h a l l t o t h e b a s i s ( b i ) , by t h e sequence TI=(, ) n nc w r i t e C,=(V,A,,E) i n s t e a d o f C=(V,A,c). F u r r h e r m o r e , C, i s s a i d t o be o f full b i n o m i a l t y p e i f q l = l and q i # O f o r e v e r y i > O .

.

P r o p . 5 . Any t w o c o a l g e b r a s o f f u l l b i n o m i a l t y p e , s a y C and CA, ll are isomorphic a s coalgebras:

c

(14)

n

bi

a c----t

qi

!/Ai

.f b i .

Proof. T r i v i a l . Here a r e some examples: i i i 1) C o a l g e b r a o f p o l y n o m i a l s : C,,=K[xl, b . = x , h . = ( . ) , nn=n. C; i s t h e a l g e b r a o f d i v i d e d power s e r i e s . I n t h e l f o l l o w l n g J we s h a l l d e n o t e t h i s coalgebra w i t h CN. i 2 ) C o a l g e b r a o f d i v i d e d powers: C =K[x], h . = n . = l . C* i s t h e a l g e b r a 11 1 1 rl of f o r m a l power s e r i e s . [i] ! 3 ) q - e u l e r i a n c o a l g e b r a : C =K[x], h i. = ( i. ) = (Gaussian rl J J [jlq! [i-jlq! 2

:= l + q + q + . . . + q c o e f f i c i e n t s ) and n i = [ i J algebra of formal e u l e r i l n series.

i-1

.

C*

n

i s s a i d t o be t h e

C o a l g e b r a s l i k e t h e s e have a s i g n i f i c a n t c o m b i n a t o r i a l c o u n t e r - p a r t . Let 5' be a l o c a l l y f i n i t e p a r t i a l l y o r d e r e d s e t ( f o r s h o r t , 1 . f . poset) t h a t s a t i s f i e s t h e following f u r t h e r conditions: a ) a l l maximal c h a i n s i n a g i v e n i n t e r v a l [x,y] o f 9 have t h e same c a r d i n a l i t y ( e q u a l t o " l + l e n g t h [ x , y ] " ) (Jordan-Dedekind c h a i n c o n d i t ion) ; b ) a l l i n t e r v a l s o f l e n g t h n i n 9 p o s s e s s t h e same number, s a y B n , o f maximal c h a i n s ; c ) t h e r e e x i s t s i n '7 o n l y one minimal e l e m e n t . A f t e r [ 1 2 ] , t h e s e p o s e t s a r e s a i d t o be 1.f. p o s e t s of full b i n o m i a l t y p e . With e v e r y 1 . f . p o s e t o f f u l l b i n o m i a l t y p e o f i n f i n i t e l e n g t h one c a n a s s o c i a t e a c o a l g e b r a of f u l l b i n o m i a l t y p e C,=(K[X],A,,E) - t h e s o - c a l l e d maximally r e d u c e d i n c i d e n c e c o a l g e b r a o f ( 7 - by d e n o t i n g w i t h b i t h e r e s i d u a l c l a s s o f a l l i n t e r v a l s o f t h e same l e n g t h i i n 9 and assuming n i = B i . Thus, e a c h s t r u c t u r e c o n s t a n t hf g i v e s t h e number h l = Bi/B,Bi-j o f e l e m e n t s o f r a n k j i n a n y i n t e r v a l o f l e n g t h i . I n $ h i s way, t h e c o a l g e b r a s c o n s i d e r e d above c o r r e s p o n d r e s p e c t i v e l y t o t h e following posets: a) t h e l a t t i c e of a l l f i n i t e s u b s e t s o f a c o u n t a b l e s e t ; b) t h e c o u n t a b l e c h a i n ; c ) t h e l a t t i c e of a l l f i n i t e - d i m e n s i o n a l subspaces of a v e c t o r space of dimension w o v e r GF(q)

.

164

L. Cerlienco, G. Nicoletti and F. Bras

12.

I n t h i s s e c t i o n we s h a l l show how b o t h automorphisms and hemimorphisms o f a c o a l g e b r a o f f u l l b i n o m i a l t y p e C, are a s s o c i a t e d w i t h s p e c i a l s e q u e n c e s o f p o l y n o m i a l s , whose g r e a t i n t e r e s t i s well-known ( a t l e a s t i n the p a r t i c u l a r c a s e of t h e coalgebra of polynomials). 2.1. Let u s b e g i n by g e n e r a l i z i n g t h e n o t i o n o f p o l y n o m i a l s e q u e n c e o f b i n o m i a l t y p e ( s e e (171 ) , I n o r d e r t o s t u d y a n a l i t i c a l l y a c o a l g e b r a C = ( V , A , E ) g i v e n i n some i n t r i n s i c way, i t i s c l e a r t h a t we may a r b i t r a r i l y c h o o s e any b a s i s ( v i ) o f V . Then, a l l we have t o know i s t h e v a l u e o f s t r u c t u r e con-

~. s t a n t s T~J r, c i o c c u r r i n g i n A V . = . L T J v~. @ v and E ( ~ . ) = E How1 i s h f h i A g 'but a usefuf t o o l . Thus, e v e r , t h e chosen b a s i s (vi) i t may happen t h a t t h e a n a l y s e s r e g a r d i n g C c a r r i e d o u t u s i n g two d i f f e r e n t b a s e s ( v i ) , (v;) c a n n o t be compared t o each o t h e r by means o f t h e map v i - v i . T h i s remark j u s t i f i e s t h e f o l l o w i n g d e f i n i t i o n . ,E) be a c o a l g e b r a o f ( f u l l ) b i n o m i a l t y p e and l e t ( b i ) L e t C,=(V,a b e a b a s i s l i x e d on i t . A new b a s i s (b:) o f V i s s a i d t o be an q - b a s i s of C i f t h e t h e map rl f : C -c (15) 0

b.

1

-

rl

b:

i s a n automorphism o f c o a l g e b r a s , t h a t i s

.

A,b!

(16)

= j=O

1

.

I'![J q

b!gbi-j. 3

C o n s i d e r t h e isomorphism $I:

c -cN n

bi-

ni!/i!

x

i

from C n t o t h e c o a l g e b r a o f p o l y n o m i a l s C N . We s h a l l s a y t h a t a s e q u e n c e p i ( x ) of p o l y n o m i a l s i s n-nomial i f t h e r e e x i s t s a n - b a s i s such t h a t p i ( x ) = $ ( b i ) . I t i s simple t o prove t h a t : (b:) i n C, P r o p . 6 . A polynomial sequence p i ( x j d ( [ x ] , onZy i f t h e f o l l o w i n g s t a t e m e n t s h o l d : 1 ) degfpil = i; 2 1 p o ( x l = I; 3) p i l o ) = 0 f o r every i f 0 ;

;EN,

i s q-nornial i f and

0 The i n t e r e s t i n q-nomial s e q u e n c e s o f p o l y n o m i a l s i s due t o t h e f a c t t h a t t h e y e n a b l e u s t o c a r r y o u t n - a n a l o g o f umbra1 c a l c u l u s a l o n g t h e l i n e s f o l l o w e d b y Rota and o t h e r s [17] , [18] ( s e e a l s o [ 8 ] , 191 ,[14). The f o l l o w i n g p r o p o s i t i o n s p r o v i d e u s w i t h a u s e f u l t o o l i n o r d e t t o g e t 17-nomial s e q u e n c e s . L e t C, be a c o a l g e b r a o f f u Z Z binomiaZ t y p e a n d Zet be a morphism of coaZgebra8. T h e n t h e r e p r e s e n t a t i v e m a t r i x i s c o m p l e t e l y d e t e r m i n e d by f"(bl):

Prop. 7 . f:C,,-C, M(f)

165

Polynomial Sequences and Incidence Coalgebras

w h e r e t h e i - t h power i s c a l c u l a t e d i n C , .

Proof.

With a straightforward calculation, from (3) we get t 0 r+s .[ 1 = j g o 151 <slflt-j>; < 0 / f / t > =t6 n rl which imply (18). i If a = .l Zd 0 a.b , B = .1)Z 1 Bib i EC,,* the element 1.3Z0 ( a1. / n1.! ) B iE C is ~ said 1 to be the c o m p o s i t i o n of a and Prop. (191

8.

The map

13

and denoted by

a0B.

--

A

Aut(C,l

C;

f ' f*(bl) i s a n i s o m o r p h i s m o f t h e g r o u p A u t l C , ) o f - t h e a u t o m o r p h i s m s of t h e of t h e e l e m e n t s c o a l g e b r a C , on t h e c o m p o s i t i o n a l g r o u p ( C G , o ) a=ZaibicC; s u c h t h a t ao=O#al. Proof. Because of (18), map M(fog)=M(f)xM(g) it follows:

(fog)'

1 (b )

=

19) is a bijection. Moreover, from

i$o
2.2. We come now to sequences of polynomials associated with hemimorphisms. For the sake of simplicity, in the remainder of this section we assume that the underlying vector space of the coalgebra of binomial type (not necessarily of full binomial type) C, is V=K[x] with its canonical basis bi=xi: i i i i-j A x = Z 11. xj8x . rl j=o J and Moreover, let,us identify the linear dual of K[x] with K[[x]] denote also b1 by X I ; thus, in general, the "series" x i is not the i-th power 6 f x1 in C;. i i Consider in :C the element < = x . (zeta-function) I and let u=.Z LJX 130 1

1+0

(Miibius function) be its multiplicative inverse: r,u =l. We have: P r o p . 9. and l e t

L e t f : C,+C D

'n

rl

be a h e m i m o r p h i s m s u c h t h a t i n d ( f l = f * ( l ) = u

( x i := f l x n i =

.;

z=o

hl

i

' I ~ -x ~

EC

n

.

T h e n t h e s e q u e n c e ( P i x , y i l n e N of homogeneous p o l y n o m i a l s d e f i n e d by a P (x,ll=p (xi satisfzes the identities: n P ( x , y J = Z hn P ( x , z l P Iz,yl ( 2 0) n k=o k k n-k (21)

More generally:

P (1,O)

n

= 1.

L. Cerlienco, G. Nicoletti and F. Piras

166

i

If u = . E

Prop. 10.

u.x i s an e l e m e n t o f ,C; g:C,-C i s t h e hemiz A n i n x morphism of i n d i c a t o r u, s 1xJ i s t h e p o l y n o m i a l g ( x I = L h a n i=o i n - i a n d S ( x , y l t h e homogeneous p o l y n o m i a l such t h a t S ( x , l ) = s ( x J , t h e n n n ue have n fz,y) S ( x , y ) = c hn P ( x , z ) S (22) k=o k k n-k n uhere Pk I x , y l i s a s i n Prop. 9 . 230

I n o r d e r t o p r o v e t h e p r e v i o u s p r o p o s i t i o n s , w e need t h e f o l l o w i n g f o u r lemmas. n r r n-r Lemma 1 . F o r a r b i t r a r i l y g i v e n polynomiaZs P ( ~ , y ) = ~ p: n~ x y n n r r n-r , the identity and S ( ~ , y ) = , s~n ~x y n 4 (23) qw k =z r hnk p rk s q n --kk = 6 rq s qn ,

is e q u i v a l e n t t o ( 2 2 1 . Proof.

igcittity

Conside6 t h t

n

k=o r = o t = o A ( k , r s t ) = Formula ( 2 2 ) becomes : E

Z

(24)

tzo

n-t t+r rfo kfr A(ksr,r+t-k)

(because of ( 2 4 ) ) r r n-r

n - t t+r r+t-k = r i o ~ -nt n=E ~ o r =z~ o k z= r h kn p kr s n-k n

=

rso {Sn-hr

n

Pr

n-t

- t =E l r z o '

n - r IxrYn-'

so

r n-r-t

X Y

z

t -

t+r I: hn pr s r + t - k k = r k k n-k

-

r n-r-t t y

z

Putting r+t=q, t h i s i s equivalent t o ( 2 3 ) . n r r n-r be a sequence o f p o l y n o m i a l s Lemma 2 . L e t P n I X , ~ ) = ~ $p n~ x y Let us consider the matrices P=(prl

s a t i s f y i n g ( 2 0) and 1 2 1 ) . &(hn) k

(251

n

( w i t h hn=o f o r n < k ) . Then k t . tl?.P

Thus, t h e sequence P (x,yJ n c i e n t s hn.

=

Pa h

=o. 0

and

= I.

i s compZetely determined by t h e c o e f f i -

k

Proof.

From (ZO), when y = o and z = 1 , we g e t xn

n

=

kio

This, together with pn(x,l)

hn P ( x , l ) . k k n = kgo P: Xk,

e x p r e s s e s t h e change o f b a s e s

xn-P

n

( x , l ) i n K[x], t h a t i s ( 2 5 ) ~

167

Polynomial Sequences and Incidence Coalgebras Lemma 3 .

I n t h e h y p o t h e s e s o f Lemma 2 . ,

the identity

(26)

i s e q u i v a l e n t t o / 2 3 ) , and t h e n t o ( 2 2 ) .

r Proof. ( 2 3 ) i m p l i e s ( 2 6 ) . I n f a c t , i f we m u l t i p l y (23) by ht and sum w i t h r e s p e c t t o t h e i n d e x r , t h e n w e g e t : k ,qn hqt = r = t k = r hn k p kr ,qqk n-k hrt = k = t hnk sq-k n-k r = 1 t p i hf =

2

2 9

(because o f (25)) hn s q - k g k = hn sq-t k = t k n-k t t n-t' C o n v e r s e l y , (26) i m p l i e s (23) : 6: s: = ( b e c a u s e o f ( 2 5 ) ) = sn 9 k& 9 =

3

h:

pk r =

1

k=r

hn ,q-k r k n-k 'k'

cr

n

~ b e yan a r ~b i t r a r- y s e~q u e n c e o f Lemma 4 . Let P (x,y)= z p k ~ n k=o n p o l y n o m i a Z s . T h e n , ( 2 0 ) a f 2 3 ) I ( ( 2 5 1 and (26)) w i t h s k = p k n n P r o o f . A s i n Lemma 1 . a n d Lemma 3 .

. 0

n - b s~e r v e t h a t P r o o f o f P r o p . 9 . C o n s i d e r t h e c o e f f i c i e n t s pA=hi+u ~. . .O t h e h y p o t h e s i s c p = 1 i s e x p r e s s e d by ( 2 5 ) . T h e r e f f i r e , owing t o Lemma 4 . , i t i s s u f f i c i e n t t o p r o v e f o r m u l a ( 2 6 ) . I n f a c t we h a v e : qwt = h: hn-t = ( b e c a u s e o f ( 1 1 ) ) = hn hq ht Pn-t q - t n-q q t n-q - ht P n * By a s i m i l a r a r g u m e n t , one c a n p r o v e P r o p . 1 0 ~

C o n v e r s e l y , i f we c a l l a n y s e q u e n c e o f homogeneous p o l y n o m i a l s P (xd ( r e s p e c t i v e l y , S n ( x , y ) ) s a t i s f y i n g (20) and ( 2 1 ) ( r e s p e c t i v e l y , ( 9 2 ) ) r e l a t i v e t o s u i t a b l e c o e f f i c i e n t s hf a GoZdman-Rota-sequence ( r e s p e c t i v e l y GoZdman-Rota- A e f f e r - s e q u e n c e ) , i t i s p o s s i b l e t o p r o v e t h a t : Prop. 1 1 . The c o e f f i c i e n t s h: (hn#Ol a s s o c i a t e d w i t h a n y Goldman- R o t a - s e q u e n c e can b e a s s u m e d a s s t g u c t u r e c o s t a n t s o f a c o a l g e b r a o f binomial type. P r o o f . We h a v e t o p r o v e t h a t (20) i m p l i e s b o t h (11) a n d ( 1 2 ) . We h a v e i h i h!-r = 1 A i h k hk-r = ( b e c a u s e o f Lemma 2 . ) r J-r k=r k r j-r =

i

z

{

i

z

k = r n=k

p

k

n

i

hn}h:

hk-r = ( b e c a u s e of ( 2 6 ) ) j -r

kzr

ic h i h k k-r = h: h: n h k. - r k-r = . -r k = r n = k n j - r h r 'n-r n=r J - r Pn-r ( b e c a u s e o f Lemma 2 . ) . . i = c h: . hrn 6 nj --rr = h i h: , n=r Moreover, (12) i s a s t r a i g h t f o r w a r d conset h a t i s f o r m u l a (11); q u e n c e o f ( 2 1 ) and Lemma 2 .

= iz

L. Cerlienco. G . Nicoletti and F. Piras

168

In conclusion, we give a combinatorial interpretation to both Goldman- Ro ta-sequences and Goldman-Ro ta-Sheffer-sequence s. Prop. 12. L e t C, b e t h e m a x i m a l l y r e d u c e d i n c i d e n c e c o a l g e b r a o f a 1 . f . p o s e t of f u l l b i n o m i a l t y p e 9. I n r e s p e c t o f t h i s c o a l g e b r a C q J l e t u s c o n s i d e r t h e Goldman-Rota-sequence Pnlx,yl and t h e Goldman-Rota- S z e f f e r - s e q u e n c e a s s o c i a t e d w i t h t h e s e r i e s c=;.,~ x n . F u r t h e r ?O more, l e t pn(x)=Pn(x,l) a n d s n ( x 1 = ~ ( x , l ) . Then e a c h z n t e r v a l of l e n g t h n i n T has p n ( x l a s its c h a r a c t e r i s t i c p o l y n o m i a l and s , ( x l a s i t s l e v e l number i n d i c a t o r ( t h a t i s , t h e c o e f f i c i e n t s of S n ( x ) a r e t h e l e v e l numbers of s e c o n d k i n d o f t h e g i v e n i n t e r v a l ) .

Proof. See [7]. REFERENCES: Abe, E., Hopf Algebras (Cambridge Univ. Press, Cambridge, 1980). Aigner, M., Combinatorial Theory (Springer-Verlag, New York, 1979). Allaway, W.R., A Comparison o f two Umbral Algebras, J.Math. Anal.App1. 85 (1982) 197-235. Andrews, G.E., On the Foundations o f Combinatorial Theory V: Eulerian Differential Operators, Studies in Appl.Math. 50 (1971) 345-375, Cerlienco, L . and Piras, F . , Coalgebre graduate e sequenze di Goldman-Rota, Actes du Sdminaire Lotharingien de Combinatoire, Publ. de l’IRMA, Strasbourg, 230/S-09 (1984) 113-125. Cerlienco, L . and Piras, F . , Aspetti coalgebrici del calcolo umbrale, Atti del Convegno “Geometria combinatoria e d’incidenza: fondamenti e applicazioni”, Rend.Sem.Mat. Brescia 7 (1984) 205-217. Cerlienco, L. and Piras, F., G-R-sequences and incidence coalgebras of posets o f full binomial type, J.Math.Anal.App1. (to appear). Cerlienco, L., Nicoletti, G. and Piras, F., Automorphisms of graded coalgebras and analogs of the umbra1 calculus, Actes du Sdminaire Lotharingien de Combinatoire. Publ. de l’IRMA, Strasbourg, 230/S-09 (1984) 126-132. Cerlienco, L., Nicoletti, G. and Piras, F., Coalgebre e Calcol o Umbrale, Rend.Sem.Mat.Fis. Milano (to appear). Cerlienco, L., Nicoletti, G. and Piras, F., Umbral Calculus, Actes du Sdminaire Lotharingien de Combinatoire, Publ. de l‘IRMA, Strasbourg, 266/S-11 (1985) 1-27. 1113

Comtet, L., Advanced Combinatorics (Reidel P.C., Boston, 1974).

Polynomial Sequences and Incidence Coalgebras

169

Doubilet, P., Rota, G.-C. and Stanley, R . P . , On the Foundations o f Combinatorial Theory VI: The Idea of Generating Function, in: Rota, G.-C. (Ed.), Finite Operator Calculus (Academic Press, New York,1975) 83-134. Garsia, A.M. and Joni, S.A., Composition Sequences, Comm. Algebra 8 (1980) 1195-1266. Goldman, G.R. and Rota, G.-C., On the Foundations o f Combinatorial Theory IV:’ Finite Vector Spaces and Eulerian Generating Functions, Studies in Appl.Math. 49 (1970) 239-258. Ihrig, E.C. and Ismail, M.E., A q-umbra1 Calculus, J.Math.Ana1. Appl. 84 (1981) 178-207. Joni, S.A. and Rota, G.-C., Coalgebras and Bialgebras in Combinatorics, Studies in Appl.Math. 61 (1979) 93-139. Mullin, R. and Rota, G.-C., On the Foundations o f Combinatorial Theory 1 1 1 : Theory o f Binomial Enumeration, in: Harris (Ed), Graph Theory and its Applications (Academic Press, New York, 1970) 167-213. Rota, G.-C., Kahaner, D. and Odlyzko, A., On the Foundations o f Combinatorial Theory V I I I : Finite Operator Calculus, J.Math.

Anal.App1. 42 (1973) 684-760.

I 191

Taft, E.J., Non-cocommutative Sequences o f Divided Powers, Lecture Notes in Math. 933 (Springer-Verlag, New York, 1980) 203-209.

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Annals of Discrete Mathematics 30 (1986) 171-184 0 Elavier Science Publishers B.V. (North-Holland)

171

R-REGULARITY AND CHARACTERIZATIONS OF THE GENERALIZED QUADRANGLE P(W(s),( - ) ) M.

De S o e t e a n d J . A .

Thas

S e m i n a r of G e o m e t r y S t a t e U n i v e r s i t y o f Ghent Krijgslaan 281 B-9000 Gent-Belgium

I n a generalized quadrangle of order ( s , s + 2 ) , s # 1, r e g u l a r p o i n t s c a n n o t o c c u r . T h e r e f o r e

we i n t r o d u c e t h e n o t i o n o f R - r e g u l a r i t y f o r p o i n t s and l i n e s i n t h o s e g e n e r a l i z e d quadrangles o f order, ( s , s t 2 ) which c o n t a i n a s p r e a d R U s i n g t h e s e new c o n c e p t s we g i v e t h r e e c h a r a c t e r i z a t i o n theorems f o r P(W(s),(m)). I. INTROUUCTION 1. DEFINITIONS

A f i n i t e generaZized quadrangle i s a n i n c i d e n c e s t r u c t u r e S = (P,B,I)

Zines r e s p . ,

where P a n d B are s e t s o f e l e m e n t s c a l l e d points a n d w i t h a symmetric i n c i d e n c e r e l a t i o n I which s a t i s f i e s

the f o l l o w i n g axioms :

( i ) each p o i n t i s i n c i d e n t with l+t l i n e s ( t 1 ) a n d two d i s t i n c t p o i n t s a r e i n c i d e n t w i t h a t most one l i n e ;

( i i ) e a c h l i n e i s i n c i d e n t w i t h l t s p o i n t s (s > 1 ) a n d t w o d i s t i n c t l i n e s a r e i n c i d e n t w i t h a t most one p o i n t ; ( i i i ) f o r e a c h n o n - i n c i d e n t p o i n t - l i n e p a i r (x,L), t h e r e e x i s t s a u n i q u e p a i r (y,M) E P X B s u c h t h a t x I M I y I L . We c a l l s a n d t t h e parametersof t h e g e n e r a l i z e d q u a d r a n g l e , a n d ( s , t ) ( o r s i f s = t ) i s t h e o r d e r of S . T h e r e h o l d s IPI = v = ( l + s ) ( l + s t ) IBI , = b = ( l + t ) ( l + s t ) and s + t I s t ( s t l ) ( t + l ) 1 1 2 1 . M o r e o v e r t h e r e i s a point-line d u a l i t y f o r g e n e r a l i z e d q u a d r a n g l e s o f o r d e r ( s , t ) ; t h i s means t h a t i n a n y d e f i n i t i o n o r t h e o r e m t h e w o r d s p o i n t a n d l i n e a n d t h e p a r a m e t e r s s a n d t may b e interchanged. If t h e p o i n t s x , y ( r e s p . l i n e s L,M) a r e c o l l i n e a r ( r e s p . c o n c u r r e n t ) we write x y (resp. L M ) . The l i n e d e f i n e d by d i s t i n c t

-

-

c o l l i n e a r p o i n t s x , y i s d e n o t e d by x y ; t h e p o i n t d e f i n e d b y d i s t i n c t c o n c u r r e n t l i n e s L,M i s d e n o t e d b y LM o r L n M .

M.de Soete and J.A . Thas

112

L e t S b e a g e n e r a l i z e d q u a d r a n g l e of o r d e r ( s , t ) . F o r x E P , we d e f i n e t h e s t a r of x as x'

{z

E

P II z

-

X I ; remark t h a t x

The t r a c e of two d i s t i n c t p o i n t s ( x , y ) i s t h e s e t x 1

.

1

1

E

x

1

.

y l and i s

-

d e n o t e d by { x , y } There holds I{x,yI I = s + l i f x y and 1 I{x,y} 1 t + l i f x f y . More g e n e r a l l y , f o r A C P we d e f i n e n{xl

A'

I1 x

E

A}.

F o r x # y we d e f i n e t h e s p a n o f ( x , y ) as t h e 1

set {x,y}ll { u E P II u E z', Vz E { x , y } I . If x f y , t h e s e t {x,y}" i s a l s o c a l l e d t h e h y p e r b o Z i c l i n e d e f i n e d by x a n d y . We 11 have I [ x , y } I s + l i f x y a n d I { x , y }11 I < t + l i f x f y . A p a i r

-

of d i s t i n c t p o i n t s ( x , y ) i s r e g u l a r p r o v i d e d x - y o r x f y a n d l{x,y}l11 = t + l . A p o i n t x i s reguZar provided ( x , y ) i s r e g u l a r f o r all y P , y # x . If S h a s a r e g u l a r p a i r o f r i o n - . c n l l i n , - a r p o i n t s , then s 1 or s t 1 9 1 . A triad of p o i n t s i s a t r i p l e o f p a i r w i s e n o n - c o l l i n e a r p o i n t s . A p o i n t u i s c a l l e d a c e n t e r of t h e t r i a d 1 ( x , y , z ) i f f u E x i n yl n z '. A s p r e u d o f S i s a s u b s e t R o f B s u c h t h a t e a c h p o i n t o f S i s i n c d e n t w i t h e x a c t l y one l i n e o f R . T h e r e holds I R I st+l. 2 . THE MODELS W ( q ) , T 2 ( 0 ' ) , T;(O)

( a ) The p o i n t s of P G ( 3 , q

AND P ( S , x )

together with t h e t o t a l l y i s o t r o p i c

l i n e s with respect t o a symplectic p o l a r i t y , define a generalized 2 q u a d r a n g l e W(q) w i t h p a r a m e t e r s s = t = q , v = b = ( q + l ) ( q t l ) .

We r e m a r k t h a t t h e l i n e s of W(q) a r e t h e e l e m e n t s o f a l i n e a r l i n e complex i n PG(3,q) [ 1 0 1 .

All p o i n t s o f W(q) a r e r e g u l a r ; t h e l i n e s of W(q) a r e r e g u l a r i f f q i s e v e n [ 9 1 . We n o t e a l s o t h a t W(q) i s s e l f - d u a l

i f f q i s even

I 9 1. ( b ) Let 0' be an oval [

4]

o f PG(2,q)

H , w h e r e H i s embedded i n

P. Define p o i n t s as ( i ) t h e p o i n t s o f P \

(ii) the p l a n e s X o f P w i t h IX n 0'1 = 1, ( i i i ) a new symbol ( m ) . The l i n e s a r e ( a ) t h e l i n e s o f P w h i c h a r e n o t c o n t a i n e d i n H a n d meet O f , PG(3,q)

H,

a n d ( b ) t h e p o i n t s of 0'. The i n c i d e n c e i s d e f i n e d as f o l l o w s . A p o i n t o f t y p e ( i ) i s o n l y i n c i d e n t w i t h l i n e s of t y p e ( a ) a n d t h e i n c i d e n c e i s t h a t o f P . A p o i n t of t y p e ( i i ) i s i n c i d e n t w i t h t h e l i n e s of t y p e ( a ) a n d ( b ) c o n t a i n e d i n i t . The p o i n t

(m)

i s incident

w i t h n o l i n e of t y p e ( a ) b u t w i t h a l l l i n e s of t y p e ( b ) . The obtained s t r u c t u r e i s a generalized quadrangle with parameters 2 s = t q , v = b = ( q + l ) ( q t 1 ) a n d i s d e n o t e d by T 2 ( O ' ) . T h e s e q u a d r a n g l e s a r e due t o J . T i t s

[4].

The q u a d r a n g l e T 2 ( 0 ' )

i s i s o m o r p h i c t o W(q) i f f q i s e v e n a n d 0' i s a n i r r e d u c i b l e c o n i c

1 9 1 . F u r t h e r , a l l p o i n t s of t y p e ( b ) are r e g u l a r , and t h e p o i n t i s r e g u l a r i f f q i s even [ 9 l .

(m)

R-Reguiaritv of the Generalized Quadrangle P( W ( d . (-I)

173

( c ) L e t 0 b e a c o m p l e t e o v a l ( i . e . a ( q + 2 ) - a r c 1101) o f t h e p l a n e a t i n f i n i t y o f A G ( j , q ) , q e v e n . The p o i n t s o f t h e s t r u c t u r e T ; ( O ) a r e t h e p o i n t s o f A G ( 3 , q ) ; t h e l i n e s o f T;(O) a r e t h e l i n e s of AG(3,q) i n t e r s e c t i n g t h e p l a n e a t i n f i n i t y i n t h e p o i n t s of 0 ; t h e i n c i d e n c e i s t h a t o f AG(3,q). Then T;(O) i s a g e n e r a l i z e d q u a d r a n g l e o f o r d e r ( q - l , q + l ) . It w a s f i r s t d i s c o v e r e d by R.W. A h r e n s a n d G. S z e k e r e s [ 1 ] a n d i n d e p e n d e n t l y by P4. H a l l J r . I 5 1 . A p a i r o f n o n - c o n c u r r e n t l i n e s (L,M) o f T;(O) is r e g u l a r i f f t h e l i n e s a r e p a r a l l e l 1 9 1. ( d ) I n [ 7 1 S . E . Payne g i v e s a n i m p o r t a n t c o n s t r u c t i o n o f g e n e r a l i z e d q u a d r a n g l e s o f o r d e r (s-l,s+l). C o n s i d e r a g e n e r a l i z e d q u a d r a n g l e S = ( P , B , I ) o f o r d e r 5,s > 1, w i t h a r e g u l a r p o i n t x . 1 D e f i n e PI a s t h e s e t P \ x I n B' t h e r e a r e t w o t y p e s o f e l e m e n t s : t h e e l e m e n t s of t y p e ( a ) a r e t h e l i n e s of B which are m t i n c i d e n t 11 with x, the elements of type ( b ) are t h e hyperbolic l i n e s {x,y} , y t. x . Now w e d e f i n e t h e i n c i d e n c e r e l a t i o n . If y E P ' , L E B ' w i t h L a l i n e o f t y p e ( a ) , t h e n y 1' L i f f y I L ; i f y E I" a n d L E B' w i t h L a l i n e o f t y p e ( b ) t h e n y I ' L i f f y E L . Then t h e s t r u c t u r e S ' = ( P f , B f , I f ) i s a g e n e r a l i z e d q u a d r a n g l e o f o r d e r (s-l,s+l) a n d i s d e n o t e d by P(S,x). I n t h e even c a s e t h e g e n e r a l i z e d quadrangle P ( W ( q ) , x ) , x a p o i n t o f W(q), i s i s o m o r p h i c t o a T;(O) ( h e r e 0 i s a n i r r e d u c i b l e c o n i c t o g e t h e r w i t h i t s n u c l e u s ) [ 9 1. The g e n e r a l i z e d q u a d r a n g ! e P(T2(01),(m)), w i t h T 2 ( 0 ' ) as i n ( b ) a n d q e v e n , i s i s o m o r p h i c t o T;(O) where 0 = 0' U { n } w i t h n t h e n u c l e u s o f 0' [9]. I n P ( W ( q ) , x ) , q o d d , a p a i r of n o n - c o n c u r r e n t l i n e s ( L , M ) i s r e g u l a r i f f o n e o f t h e f o l l o w i n g c a s e s o c c u r : (i) L a n d M a r e l i n e s o f t y p e ( b ) , ( i t ) L a n d M a r e c o n c u r r e n t l i n e s o f W(q) ( b u t a r e n o t c o n c u r r e n t i n P ( w ( q ) , x ) ) ; i n P ( w ( q ) , x ) , q e v e n , a p a i r of non-conc u r r e n t l i n e s (L,M) i s r e g u l a r i f f one o f t h e f o l l o w i n g c a s e s o c c u r s : ( i ) L a n d M a r e l i n e s o f t y p e ( b ) , ( i i ) i n W(q) some l i n e o f {L,M)' i s i n c i d e n t w i t h x.

.

1 7 . R-REGULARITY OF POINTS AND LINES 1. DEFINITIONS

Consider a generalized quadrangle S

(P,B,I)

of o r d e r (s,s+2),

s > 1. S i n c e 1 :. s .< t r e g u l a r p o i n t s c a n n o t o c c u r [ 9 1 . M o r e o v e r , i n t h e known e x a r z p l e s a l s o r e g u l a r l i n e s d o n o t o c c u r . T h e r e f o r e we i n t r o d u c e t h e c o n c e p t o f R - r e g u l a r i t y .

M. de Soete and J.A. Thas

174

I n what f o l l o w s w e a l w a y s assume t h a t t h e g e n e r a l zed q u a d r a n g l e 2 o f o r d e r (s,s+2) c o n t a i n s a s p r e a d R ( I R - s t l ) ) .

S = (P,B,I)

1.

= {z E P 1 I z -. x , z # x , zx 4 R u { X I I* For a p a i r o f d i s t i n c t p o i n t s x,y we d e n o t e t h e set x n yl* as

F o r x E P, we d e f i n e x

-y

.

I stl. b u t xy 4 R , t h e r e h o l d s I { x , y ) 1. II x E A } . So f o r a More g e n e r a l l y , f o r A C P we d e f i n e A'* 9Ix l*l* p a i r o f n o n - c o l l i n e a r p o i n t s x , y we have { x , y l - I u E P II u z, uz 4 R , Vz E ( x , y I 1 * } So we o b t a i n I I x , y ) ' * l * I < s t l . I f x y 1.1. = xy and s o ~ { ~ , y } ' *= ~s*t l~. b u t xy 4 R , t h e n c l e a r l y I x , y l Ix,y}'*.

If x f y or x

1.

-

.

-

A p a i r of d i s t i n c t p o i n t s x,y i s c a l l e d R-regular p r o v i d e d x -. y 1.1, and xy 4 R , o r x f y and I I x , y } I = s t l . A p o i n t x i s R-regular

p r o v i d e d ( x , y ) i s R - r e g u l a r for a l l y E P , y f x. A R-grid i n S i s a s u b s t r u c t u r e S ' = ( P f y B 1 , I 1 )o f S d e f i n e d as follows :

P'

I x i j E P II i = l Y . . . , s t 2 , j = l, . . . , s t 2 ,

B' = I L 1 , . . . , L s t 2 , I'

I n Li

((PI

17

Xijxji

Mj

M1,...

,Mst2}

X B ' ) U (€3'

X

Rji

c B \ R , and

P I ) ) , w i t h Li f L

x i j i f i f j , Li = Rij

and i # j } ,

E R for 1 6

i,

J

Mi -f- M j y

- xjj iyy and

x !j

j . Mi,

G st2.

11.

We d e n o t e t h e set { L 1y...,Lst2! ( r e s p . !MI,.. * ,Mst2)) by { L i y L j 1 o r {Mi,rCl1* (resp. (MiyM.Ill o r { L i , L . l l * ) f o r any i # j . J J J I f L1,L2 E B \ R, L1 j . L 2 , t h e n by d e f i n i t i o n t h e p a i r ( L 1 , L 2 ) R-regular i f f ( L l a L 2 ) b e l o n g s t o a R-grid. I n such a c a s e t h e r e e x i s t s a unique R E R for which L1 R L2. A l i n e L E B \ R i s

is

- -

weak R-reguZar i f f ( L,M) i s R - r e g u l a r f o r a l l M E B \ R w i t h L .f- M 1 and IIL,M} RI 1. A l i n e L E B \ R i s R-reguZar i f f L i s weak R - r e g u l a r and f o r a l l M E B \ p a i r (L,M) i s r e g u l a r .

R , L -f- M y w i t h I{L,Mll

R i f 1, t h e

F i n a l l y , n o t i c e t h a t R-regularity f o r l i n e s i s not t h e dual of R-regularity f o r points. 2 . EXAMPLES

2.1.

Theorem. C o n s i d e r P ( W ( q ) , x )

(P',B1,I')

and l e t R be t h e s e t

.

of a22 l i n e s of t y p e ( b ) i n B ' ( s e e 1.2.(d)) Then e ac h p o i n t Each l i n e of B ' \ R i s R-regular i f f q is e v e n . P r o o f . L e t W(q) = ( P , B , I ) . Choose a p o i n t x i n W(q). It i s o b v i o u s i s R-reguZar.

t h a t t h e set ( ( x , y l

11

I1 y E P , x f y l d e f i n e s a s p r e a d R i n P(W(q),x).

175

R-Regularit). of the Generalized Quadrangle P(Wls1. 1-11 L e t y,z E I", y % ' z . T h e n i t f o l l o w s t h a t y i s re&ular

'd(g). Let

ill

'-I x1

i [y,z)"

= {?<,!,I. 1

4 z. We know t h a t (y,z) 1

x1 b e d e f i n e d b y { y , z } l n x = { z g } 0 Prom 11.1. a n d 1;he d e f i n i t i o n o f R we

zO,

'*

2'

E

11.1

1

I *

I{y,z,} 1 = I{y,z) \ {zojl q a n d l{y,z} I 11 g. Helice e a c h p o i n t o f F ( W ( q ) , x ) i s R - m g u l a r . \ [ziti :{y,zI

s ~ t 3 i i ! !t h a t

=

-

D' \ R , L1 % ' L-,. if L1 L2, (L1,L2) i s a r e g u l a r 1 '< I- i n F ( W ( ~ l ) , x )a n d {L1,L21 c R . I f L1 4 L, but L1,L2 a r e b o t h k , q n c u r r , c r t t w i t h a same l i n e or W ( q ) through x , t h e n (L1,L2) i s L t 1 m ( b o t , k i n W(?) a n d i n P(W(q) ,x)) i f f q i s even. IIere we h a v e I' 8) R = 0. P i n a l l y , s u p p o s e t h a t L1 # L2 a n d L1,L2 a r e n o t {I,l,Lql coricuri7.ent w i t h a same l i n e t h r o u g h x i n l d ( q ) . T h e n I {L1,L2}" i? R I Let Ll,L2

6

i

= 1. If' q is e v e n , (L1,L2) i s r e g u l a r i n k l ( q ) . I n t h i s c a s e , l e t 1 11 - [ L1, . . ,Lq+ 1 1 and l e t tL,;,Lql = tM1,... Y M(-lt 1 1 Y L1 Y L2 1 L . I xi I !
-

L

i a R - g r i d i.n ? ( W ( q ) , x ) . I n d e e d , Li

- I

1

-

M . i f f i # j, a n d s i n c e f o r J xi, x.. x. (with x J1 J 11 ij -

a n y i # j t x i , x .J ) c t x i , x i j l , x J. .1 , w e have L i (1 X. and x j i = I,. 9 M . ) a n d h e n c e x.. E I x , x . . I J J 1 J 1 1J x . . - ' x.. w i t h x . . x . 6 R . We c o n c l u d e t h a t e a c h l i n e of B' \ 1J J1 1~ ~i R - r e g u l a r iff' q i s e v e n . n 2.2.

'-

R is

ii p o i n t x of 0 is R - r e g u l a r if a n d

Ttieoi~eni. L e t R b e t h e s e t oj- a 7 1 l i n e s t h r o u g h i n 1 1 ' T ( 3 ) . A point y, r e s p . a 7ine L @ R , L

;,+' O \ { X I < s n coizi,-. It i s i m m e d i a t e t h a t R i s ,JP~?:. IJ

c'onic

h o l d s T:(3)

tilerc'

2

3

? ? r e a d i n T;(O).

If 0 \

{XI is a

P(W(\q),y), q even. Applying t h e foregoing

L

f o l l o w s t h e R-regulcrity. Ti"0 p r o v e tkie c o n v e r s e , w? remark t h a t T;(O)

theorem therc.

P(T2(0'),(m)),

t x } ( 1 . 2 , ( d ) ) , Assume t h a t T;(O) = ( P , B , I ) a n d T2(0') = ( P 1 , B 1 , I 1 ) .C o n s i d e r a p o i n t y 1 E P s u c h t h a t yl i s R - r e g u l a r . 1'

0 \

T h e n yl i s

I

p o i n t o f type ( i ) i n T,(O').

We p r o v e t h a t y1 i s

r e g u l a r i n T2(O').

y2 f' y l Y w i t h y 2 a p o i n t of t y p e (i). If y1 'i y2, i s R - r e g u l a r i n T;(O). L e t [y1,y2) 181' -I" C tyl,y211' t h e r e r e s u l t s y i - - I Z {y,,y ? , . . . , yp). S i n c e {y1,y21 j' I' ll*l* (see 1.3.4. i n 1 9 1 ) . Consequently E {y,,y,] VYi E [YpY,] I U s i n g a g a i n 1.3.4. i n [ 9 I w e o b t a i n t h a t iY 1 ,Y?} l * l * C {y1,y21 L e t y2

E

P',

(yl,y,j

tlie p a i r

""J', .

is r e g u l a r i n T,(O').

(v,,y,) lint- y y

1 2

J1,...,N

meet 0 \

E

R.

D e n o t e by

-

Now r u p p o s e t h a t y 2 yly i.e. the e p o i n t s o f 0 \ { X I , a n d by

X ~ , . . . , X ~ +t h ~

i = 1, . . . , q + 1, w h i c h { x i = 0' i n a u n i q u e p o i n t . T h e n {ylYy21" = t W 1 , Wqtl}

q+1

t h e p l a n e s d e f i n e d by y1,y2,xi,

...,

M.de Soete and J.A. Thas

116

1 'I ' = { z II z 1 y,y,l u I ( - ) ) s i n c e y 1y 2 'i' a n d {Y,,Y21 V i l,...,qtl. A g a i n ( y l , y 2 ) i s a r e g u l a r p a i r i n T2(0'). N e x t ,

I' l e t X E P' be a p o i n t of t y p e ( i i ) w i t h y1 -f.' X . The t r a c e I y , , X l c o n t a i n s a t l e a s t t w o p o i n t s x1,x2 o f t y p e ( i ) w i t h x1 7L' x2. On t h e

contains, b e s i d e s yl and X, at least one p o i n t o t h e r hand y 2 o f t y p e ( i ) . C l e a r l y y2 7L' y l . From t h e f o r e g o i n g i t f o l l o w s t h a t (yl,y,) Iyl,Xl

i s r e g u l a r i n T 2 ( 0 1 ) . S i n c e {xl,x21 C I y 1 , y 2 I 1 ' a n d C Ixl,x2li', t h e p a i r ( y l , X ) i s a l s o r e g u l a r i n T2(O').

F i n a l l y , from t h e r e g u l a r i t y of (-) i n T 2 ( 0 ' ) , q e v e n , t h e r e r e s u l t s t h a t ( y l y ( - ) ) i s r e g u l a r . We c o n c l u d e t h a t y1 i s a r e g u l a r p o i n t i n T 2 ( 0 ' ) and hence 0' i s a c o n i c 1 9 1 .

Next we s u p p o s e t h a t L1 E B \ R i s R - r e g u l a r a n d t h a t ( L 1 , L 2 ) i s a R - r e g u l a r p a i r i n T;(O). W e u s e t h e n o t a t i o n s of 11.1. w i t h q s+l. F o r n a c h p a i r (LiyMi), 1 Q i G qtl, t h e r e holds {Li,Mill = I R i l y . . . y R i ,st1 1 C R . E a c h p a i r o f l i n e s o f R i s r e g u l a r ( t h e y d e f i n e t h e same p o i n t x o f 0 ( 1 . 2 . ( c ) ) . Hence ( L . , M . ) i s a r e g u l a r 1 1 p a i r o f T;(O), a n d s o d e f i n e s a p o i n t xi o f 0 , i = 1, ...,q t l 1'1' ( 1 . 2 . ( ~ ) ) . T h e r e f o l l o w s t h a t t h e s e t s {L1,L2} I - IL1'".'Lqtl I' a n d {L1,L21 = I M ly...,Mqtll a r e t h e t w o sets of g e n e r a t o r s o f a h y p e r b o l i c q u a d r i c Q i n PG(3,q). The o v a l 0 ' i s a p l a n e i n t e r s e c t i o n of Q. Hence 0 ' i s a c o n i c . fl 3 . R-REGULARITY AND AFFINE PLANES

3.1. Theorem. L e t x b e a R - r e g u l a r p o i n t of t h e g e n e r a l i z e d q u a d r a n g l e S o f o r d e r ( s , s + 2 ) w i t h s p r e a d R. T h e n t h e i n c i d e n c e I* s t r u c t u r e ( P ' , B * , I * ) where P* = x , B' {L E B II x I L and L 4 9 l U ( ~ y , z l ' * l * II y,z E X I * , z 7~ y l and I* t h e n a t u r a l i n c i d e n c e , i s a 2 - ( ( ~ + 1 ) ~ , s + l , ld)e s i g n i . e . an a f f i n e p l a n e of o r d e r s t l .

Proof. Immediate. 0 3.2,

Theorem. L e t S b e a g e n e r a l i z e d q u a d r a n g l e of o r d e r (s,st2)

w h i c h c o n t a i n s a s p r e a d R and a R - r e g u l a r p o i n t x w i t h x I L E R . T h e n e a c h p a i r (L,M), M E R, is r e g u l a r and IL,M}'l C R . Proof. L e t x E L E R a n d M E R \ { L l . C h o o s e a p o i n t y E M s u c h t h a t x 7L y . Then (x,y) i s a R - r e g u l a r p a i r . L e t { x , y }l * l * { z i y l G i G stll w i t h x zl, y = z stl. Each p o i n t zi i s i n c i d e n t w i t h a u n i q u e l i n e Ri E R w i t h R1 = L , R s + l = M , a n d w i t h a u n i q u e l i n e Li which d o e s n o t c o n t a i n a p o i n t o f I x , y l * * , f o r a l l 1 i G s+l. S i n c e z i 1 Rj, i # j , 1 < i , j 4 stl, w e o b t a i n ,

177

R-Regularity of the Generalized Quadrangle P( W(s), (mil

-

a p p l y i n g axiom ( i i i ) o f 1.1. w . r . t . z and R t h a t Li R for all i j' j y1 j # i , 1 G i , j Q s + l . So (L,M) i s r e g u l a r a n d m o r e o v e r {L,MI = ILiy 1 d i G s+ll a n d { L , M } l ' = I R i , 1 < i Q s+l). 0

3.3. C o r o l l a r y . L e t 1 b e a s e t o f n o n - c o n c u r r e n t l i n e s o f a g e n e r a l i z e d q u a d r a n g l e o f o r d e r ( s , t ) . T h e n 1 i s normal ( s e e I 7 l ) provided each p a i r o f l i n e s o f L i s r e g u l a r and t h e i r span i s a s u b s e t of 1. T h e f o r e g o i n g theorem shows t h a t i n a g e n e r a l i z e d q u a d r a n g l e S o f o r d e r (s,s+2) w i t h a s p r e a d R i n w h i c h a l l p o i n t s a r e R - r e g u l a r , t h e s p r e a d R i s a n o r m a l s e t . Then a n a f f i n e p l a n e o f o r d e r s + l c a n b e d e f i n e d i n t h e f o l l o w i n g way. L e t P' = R , B' = { { M , N l l l y M , N E R l a n d I* t h e n a t u r a l i n c i d e n c e . Then t h e s t r u c t u r e (P' , B ' , I * ) i s a n a f f i n e plane AR of order stl.

3.4. Theorem. L e t S b e a g e n e r a l i z e d q u a d r a n g l e of o r d e r ( s , s + 2 ) w i t h a spread R.

Let R, R'

E

R , R # R ' and a s s u m e t h a t ( R , R ' )

.

i s r e g u l a r w i t h {R,R'I1' {R1 R, R2,.. , R s + l = R ' I C R . If a l l p a i r s of p o i n t s i n c i d e n t w i t h d i f f e r e n t l i n e s of { R , R ' I L 1 a r e Rr e g u l a r , we c a n d e f i n e a n a f f i n e p l a n e (P',B',I') of o r d e r s+l

P I = t z E P II z I R ~ R~ , E ( R , R ~ I ~ ' , 1 < i Q s+i), 1.1' 11 tR,R'}' U tR,R'l II z,z' f PI, z U {{z,z'l

B'

+

2')

with

and 1'

t h e natural incidence. E P', z f z ' , t h e set 11 { z , ~ ' l ' * ~ *C P ' . L e t z I R i y z ' I R j y R i , R . E {R,R') From t h e J l*l* p r o o f o f 11.3.2. i t f o l l o w s t h a t for a n y zk E { z , z ' l the l i n e 11 Hence zk E P I . Now i t i s Rk E R , zk E R k , b e l o n g s t o { R i , R . ) J i m m e d i a t e t h a t ( P ' , B ' , I ' ) i s a 2-((~+1)~,stl,I) d e s i g n . Cl

Proof. F i r s t we p r o v e t h a t for z,z'

.

.

3.5. T h e o r e m . L e t S b e a g e n e r a l i z e d q u a d r a n g l e of o r d e r (s,s+2) w i t h a s p r e a d R and a R - r e g u l a r l i n e L. I f R , R ' E L1 R , R f R', 11 t h e n ( R , R ' ) i s r e g u l a r and { R , R ' I L1 fi R . Proof. L e t R , R '

E

L

1

n R,

R # R'. C o n s i d e r a n a r b i t r a r y p o i n t x I R ,

x 1 L . The p o i n t x i s i n c i d e n t w i t h s + 2 l i n e s o f B \ R . S i n c e IL1 n R ( = s+l t h e r e e x i s t s a t l e a s t o n e l i n e L ' E B \ R , x I L', such t h a t {L,L'll n R = I R I I n view o f t h e R - r e g u l a r i t y o f L, ( L , L ' ) b e l o n g s t o a R - g r i d . U s i n g t h e n o t a t i o n s of 11.1. w e p u t IRli, i 2 s+2) L1 n R w i t h L = L1, L' = L2. T h e n {L1,M1ll R R12, R f = Rlk, 3 G k Q s t 2 . A g a i n b y t h e R - r e g u l a r i t y o f L 11 = (L1,M1) i s a r e g u l a r p a i r . Hence ( R , R ' ) i s r e g u l a r a n d { R , R ' ) L'l nR. 0

.

,...,

hl. de Soete and J . A . Thas

178

3 . 6 . T h e o r e m . L e t S b e a g e n e r a l i z e d q u a d r a n g l e of o r d e r (s,s+2) w i t h a s p r e a d R . If S c o n t a i n s a R - r e g u l a r l i n e L, t h e n t h e i n c i d e n c e s t r u c t u r e (P',B~,E)w i t h P I = L* \ ( { L I u R ) , B' = { ~ K , K ' I ~ ' * n P I II K , K ' E P I , K Y K ' and I K , K ' I b e l o n g s t o a 11 R-grid} U I I K , K ' I II K , K ' E PI, K Z K ' and { K , K ' ) d o e s n o t b e L o n g t o a R - g r i d ; U { x B II x I L l w i t h xB t h e s e t of a l l l i n e s J f PI i n c i d e n t w i t h x, Proof. D e n o t e L

i s ax a f f i n e pLune of o r d e r s + l .

1 i-

R by {R1,

...,R s t l I .

Each p a i r of d i f f e r e n t l i n e s

o f t h i s s e t i s r e g u l a r (11.5.5.). L e t K , K ' E PI, K Y K ' a n d l e t I L1 E I K , K ' I , L f L1. Prom t h e R - r e g u l a r i t y o f L t h e r e r e s u l t s if I { L , L ~ ~n' R I

+

( L , L ~ )i s a r e g u l a r p a i r . I I e n c e ( K , K ' )

I that

is

a l s o r e g u l a r w i t h l I K , K 1 l L 1 I = s+l. I t i s o b v i o u s t h a t i n t h i s c a s e

i s t h e u n i q u e e l e m e n t of B' t h r o u g h K a n d K ' .

{K,K'l'' that

r- R / = 1 .

l{L1,Lll

Iri

d e f i n e s a R-grid.

(L,L1)

Now s u p p o s e

v i e w o f t h e R - r e g u l a r i t y o f L t h e pair

T h i s R-grid c o n t a i n s K and K ' .

Hence

l t K , K ' I l l * n P'l = s+l. C l e a r l y { K , K ' I l l * i s t h e u n i q u e e l e m e n t o f B' t i i r o u g h K a n d K ' . F i n a l l y , i f K , K ' E P' w i t h K I x I K ' t h e n xu is t h e u n i q u e l i n e o f B' t h r o u g h K a n d K f . We h a v e a g a i n l x B l = s+l. H e n c e , we c o n c l u d e t h a t (P',D',E) i s a 2 - ( ( s t 1 ) 2 , s t 1 , 1 ) d e s i g n . 0

3 . 7 . T h e o r e m . L e t S b e a g e n e r a l i z e d q u a d r a n g l e o f o r d e r (s,s+2) w h i c h c o n t a i n s a s p r e a d R . If x i s n R - r e g u l a r

p o i n t and

(L1,L2)

a R - r e g u l a r p a i r of l i n e s s u c h t h a t x i s i n c i d e n t w i t h no

line x

o f R ( w i t h t h e n o t a t i o n s o f II.l.), t h e n s is o d d .

ijxji

Proof. C l e a r l y x i s i n c i d e n t w i t h n o line o f [ L l , L 2 1 1 * . L e t {L1,L2I1'* = {L1 ,..., L s + 2 1 a n d l e t y i b e d e f i n e d b y x yi, , Y , + ~ a r e s t 3 p o i n t s of yi I Li, 1 G i 4 s + 2 . The p o i n t s x , y l ,

-

...

t h e a f f i n e p l a n e o f o r d e r s t l d e f i n e d by x (11.3.1.). E a c h l i n e t h r o u g h x i n t h a t p l a n e h a s a u n i q u e p o i n t i n common w i t h t h e s e t

,...,Y , + ~ I . . . . ,s t 2 1 , a r e

{yl

Suppose t h a t yi,yj,yk,

J

...

c o l l i n e a r i n t h a t a f f i n e p l a n e . If t h i s i s t h e c a s e , 1'

I

= s+l. S u p p o s e t h a t o n e of t h e p o i n t s I' which i s c o n c u r r e n t is i n c i d e n t w i t h a l i n e M E {L1,L21

t h e r e holds IIyi,yj,ykI yi,y.,yk

i f j # k f i, i , j , k E { l ,

w i t h Li,Lj,Lk,

e . g . yi I M.

o f M w i t h L j ( r e s p . L,).

Let u

Then ' I ,

j

E

( r e s p . u,)

be t h e i n t e r s e c t i o n

{yi,yk}".

Hence y

-

j arises, a c o n t r a d i c t i o n .

u k y j @ R , a n d a t r i a n g l e u.u y J k j T h e r e f o l l o w s t h a t t h e r e i s a l i n e o f {L1,L211*

uk, w i t h

which i s i n c i d e n t

o r y, a n d c o n c u r r e n t w i t h e a c t l y t w o of t h e l i n e s L i , L j or L,. S u p p o s e e . g . t h a t yi I M j , M . E { L l , L 2 1 1 * , a n d M I uk I Lk. J j From t h e d e f i n i t i o n o f a R - g r i d i t f o l l o w s t h a t L I x I Mk a n d

with yi,yj

j

jk

R-Regularity of the Generalized Quadrangle P(W(s), (-)I

uk

E R .

- yj,

j ' k'k

179

I* Hence y = x On t h e o t h e r h a n d u k E I y i , y k j and hence j jk' u k y j 4 R , a c o n t r a d i c t i o n . So t h e p o i n t s y i , y . , y k a r e n o t J

c o l l i n e a r i n t h e a f f i n e p l a n e of o r d e r stl. Hence I y l , . . . , y s t 2 } d e f i n e s a n o v a l i n t h e c o r r e s p o n d i n g p r o j e c t i v e p l a n e . The st2 tangent l i n e s

o f t h e o v a l a r e c o n c u r r e n t a t x . So we c o n c l u d e t h a t

t h e o r d e r s + l of t h e p l a n e i s e v e n .

3.8. C o r o l l a r y . L e t S b e a g e n e r a l i z e d q u a d r a n g l e of o r d e r ( s , s t 2 ) w h i c h c o n t a i n s a s p r e a d R . If x i s a R - r e g u l a r p o i n t and L a R-regular

l i n e such t h a t x

Z R,

n R , then s i s odd.

VR E L1

,... J s t d

P r o o f . L e t x I R x , Rx E R . T h e n Rx 3 L . P u t I L , R x j l = I K 1 1 and L n R (R1 , . . . , R s t l j . I f M E {Rl,R2)13 M # L , t h e n M

-

Ri,

for a l l 1 G i G s+l ( 1 1 . 3 . 5 . ) . Hence t h e r e e x i s t s a t l e a s t o n e l i n e L ' , L' # M , i n c i d e n t w i t h y , w h e r e y = P4 3 R j E {l, ..., stl}, 1 j' s u c h t h a t L' f K i , 1 i G s + l . Then { I , , L ' I n R {R.) a n d so 11' J (L,L') i s a R - r e g u l a r p a i r . L e t I L . L ' ) = IL1, ,Ls+2j. If R x i s

...

of t h e form x i j x j i

(w.r.t.

t h e R - g r i d d e f i n e d by L a n d L ' ) , t h e n

-

-

-

-

I L i , L .J) { L , L ' ) = $ and s o M1 . L , Mi L ' , Mj L , Mj L'. o n e o f t h e l i n e s Ki i s c o n c u r r e n t w i t h L ' , a c o n t r a d i c t i o n .

Hence 0

111. CHARACTERIZATIONS OF P ( W ( . b + l ) , ( m ) )

1. THEOREM Let S

(P,B,I)

be a g e n e r a l i z e d q u a d r a n g l e o f o r d e r ( s , s + 2 )

w h i c h c o n t a i n s a s p r e a d R . If aLL p o i n t s a r e R - r e g u l a r , isomorphic t o P(W(s+l)),(m)) w i t h

(m)

then S i s

a n a r b i t p a r y p o i n t of W ( s t 1 ) .

P r o o f . W e d e f i n e an i n c i d e n c e s t r u c t u r e S' = (P',B',I') as f o l l o w s .

P' c o n t a i n s t h r e e t y p e s of p o i n t s : ( i ) the points x

E

P; 1

( i i ) t h e s e t s IRl,R2} , R1,R2 E R ; (iii)a unique p o i n t (-). B' c o n t a i n s t w o t y p e s o f l i n e s : (a) the lines L E B \ R ; (b) s e t s , e a c h b e i n g t h e union of a l l t r a c e s o f p a i r s o f e l e m e n t s cf a same p a r a l l e l c l a s s i n A R ( c f . 1 1 . 3 . 3 . ) . I ' i s d e f i n e d as :

-

a p o i n t o f t y p e ( i )i s i n c i d e n t w i t h a l i n e o f t y p e ( a ) i f f t h e y are i n c i d e n t i n S; i t i s i n c i d e n t w i t h no l i n e o f t y p e (b);

M . de Soete and J.A. Thas

180

-

-

a point IR1,R2)

1

of type (ii) i s i n c i d e n t with each l i n e 1 and w i t h t h e u n i q u e o f t y p e ( a ) which b e l o n g s t o IR1,R2} l i n e of t y p e ( b ) from which it i s a s u b s e t . t h e symbol ( m ) i s i n c i d e n t w i t h a l l l i n e s o f t y p e ( b ) b u t w i t h n o l i n e of t y p e ( a ) .

It i s e a s y t o check t h a t S' s t l (see a l s o 1 7 I).

i s a g e n e r a l i z e d q u a d r a n g l e of order We p r o v e now t h a t m o r e o v e r S' W(s+l).

1st p r o o f . A p p l y i n g t h e t h e o r e m o f C.T. Benson 1 2 1 we o n l y h a v e I 'I ' IIx,y}

t o c h e c k t h a t e a c h p o i n t o f S' i s r e g u l a r , i . e . f o r a l l p a i r s ( x , y ) i n S ' w i t h x 7L' y .

S u p p o s e t h a t x a n d y a r e b o t h p o i n t s o f t y p e ( i )w i t h x Let x Y y . S i n c e a l l p o i n t s of S a r e R - r e g u l a r , we h a v e I{x I' 1 s t l . L e t x L Rx E R , y I R y E R . Then I x , y } = I x , y l l * U IRx,Ryl 1 .

From t h e p r o o f o f 11.3.4. it, f o l l o w s t h a t e a c h p o i n t o f I x , y } l " l ' is l*l* I 'I ' i n c i d e n t w i t h a l i n e of I R x , R I1. Hence { x , y l c Ix,y) Y 1'1 1 A p p l y i n g 1.3.4. o f [ 9 1 we o b t a i n I I x , y l I = s + 2 . If x I R I y , R E R, t h e n I x , y ) l ' c o n s i s t s o f t h e s t 2 p o i n t s {R,R'}', R' E R and 1 'I 1 R # R ' , o f t y p e ( i i ) .C l e a r l y I x , y l 3 I z II z I R I u { ( m ) } , a n d 1'1 ' so IIx,yl 1 = s t 2 . Hence ( x , y ) i s r e g u l a r i n S'.

.

Let x b e a p o i n t o f t y p e ( i ) and y IR,R'I1 a p o i n t o f t y p e ( i i ) , w i t h x 7 L 1 y . I f x1,x2 E { ~ , y } ~ w' i, t h x1,x2 p o i n t s o f t y p e ( i ) ,t h e

p a i r ( x 1 , x 2 ) i s r e g u l a r i n S'. Hence t h e r e f o l l o w s t h e r e g u l a r i t y o f ( x , y ) i n S'. Next we s u p p o s e t h a t x i s a p o i n t o f t y p e ( i ) a n d y = x I R E R , t h e n Ix,(-)}l'

(m).

c o n s i s t s of t h e s t 2 p o i n t s { R , R ' )

If

,

R a n d R # R ' , i n S ' . C l e a r l y I x , ( - ) } "" 3 I y II y I R I u I ( m ) } , 1 'I 1 and so / I x , ( - ) } I = s t 2 . Hence ( x , ( m ) ) i s r e g u l a r i n S'. 1 , y = IR , R ' I I , RxyR;, R , R t E R , b e F i n a l l y , l e t x = { R ,R;} X 1 y y Y Y I' t w o p o i n t s of t y p e ( i i ) , { R x y R i I 7L' { R R'}'. C h o o s e x1,x2 E I x , y )

R'

E

Y: Y

a r e two p o i n t s of t y p e ( 1 ) . From t h e r e g u l a r i t y o f (x,,x,) i n S ' , t h e r e f o l l o w s t h a t ( x , y ) is r e g u l a r in S ' . S i n c e t h e s e a r e t h e o n l y p o s s i b l e c o m b i n a t i o n s for a p a i r of d i s t i n c t p o i n t s ( x , y ) , x f t y , we c o n c l u d e t h a t S ' --L W(st1). Hence t h e q u a d r a n g l e S i s i s o m o r p h i c t o P ( W ( s t l ) , ( - ) ) . The l i n e s o f 1 'I ' t h e spread R i n S can be i d e n t i f i e d w i t h t h e sets { ( = ) , X I Y x Y ' (m). s u c h t h a t x1,x2

2nd p r o o f . I n [ l l ] C . S o m a d e f i n e s t h e S t e i n e r s y s t e m l*l*

P , B1 = B U I I x , y } 1I x , y E P , x f y I a n d I1 t h e n a t u r a l i n c i d e n c e . Now we p r o v e t h a t e a c h s u b s t r u c t u r e o f 0

D = ( P 1,B1,I1) w i t h P1

181

R-Regularit). of the Geiieralized Quadrangle Pl W(sl, (-I) g e n e r a t e d by a n a r b i t r a r y p o i n t x E P I , a n d a n a r b i t r a r y l i n e L

E

B

1'

x I, L,, i s ari a f f i n e p l a n e o f o r d e r s t l . w i t h L = { y , z J '*I* a n d x Z1 1,. ( i ) S u p p o s e x E Pi' L E U

wit11

(a) I f x E (y,zl

I*

1'

t h e n xw E

E \ R , Vw

E Iy,zl

1'1.

.

The

i s a s u b s t r u c t u r e of D which

s f f i n e p i a n e d e f i n e d by x ( c f . 1 1 . 3 . 1 . )

c o n t a i n s x a n d L. Hence x a n d I, g e n e r a t e an a f f i n e p l a n e o f o r d e r stl,

1'

( b ) I f X I * n {y,zj

d e f i n e d by r ( c f . 1 1 . 3 . 1 . )

= {ri, r # x, t h e n t h e n f f i n e p l a n e i s a s u b s t r u c t u r e o f 0 which c o n t a i n s x

a n d L . So i n t h i s c a s e , x a n d L d e r i n e a g a i n an a f f i n e p l a n e o f 3 r d e r stl. l*

l*l*

= $ a nd x ~- w w i t h w E { y , z ~ xl* a { y , z } , R E R , t h e n w e c o n s i d e r t h e a f f i n e xw 4 R . If j i I R z I RZ, R Y' y' z p l a n e o f o r d e r s+l d e f i n e d by {R ,RZ) ( s e e 1 1 . 3 . 4 . ) . T h e l i n e Y 1 ( s e e 11.5.2.). I i e n c e t h i s a f f i n e p l a n e i s a l s o g e n e X W E {Ry,HZI ( c ) Suppose

r a t e d by x and L .

(d) L e t x

I*

{ y , z l '*

= 8 and x

.-

w with w

E

{y,zl

l*l*

,

R , t h e n we c o n s i d e r t h e a f f i n e z I RZ, R ,R xw E A . I f y I R Y' Y Z p l a n e o f o r d e r s + l d e f i n e d by {R ,RZ1 ( s e e 3.4.). From 1 1 . 3 . 2 . i t E

11

f o l l o w s t h a t xw E {Zy,RZ}

.

Y

Again x and L g e n e r a t e a n a f f i n e p l a n e

of o r d e r s + l . 1

= 8 and xL* 5 {y,zl** = 9 . n { y , z l I*'* 1'1' C o u n t i n g t h e number o f p o i n t s u i n S s u c h t h a t u' n { y , z l f $ 1' 1' # 4 w e o b t a i n (stl)'(s-1)+2(~+1)~ = ( ~ + =l v). ~ p r u 17 t y , s l ( 2 )

Suppose t h a t x

IIence t h i s c a s e c a n n o t o c c u r . ( i i ) Suppcse x

E

PI., L

E

B w i t h L # R a n d x f, L . S i n c e S i s a

g e n e r a l i z e d q u a d r a n & l e r;here e x i s t , s a p a i r (y,M) E P

X

B such t h a t

x I M I y I L. (a) I f M E R we c o n s i d e r t h e a f f i n e p l a n e d e f i n e d by ( M , P ! ' l L , M' E R . T h i s p l a n e o f i ; r d e r s + l i s a l s o g e n e r a t e d by x a n d L . ( b ) I f P4 R we c o n s t r u c t t h e a f f i n e p l a n e of r o d e r s t l d e i ' i n e d b y y ( c f . 11.3.1.). T h i s p l a n e i s g e n e r a t e d by x a n d L . ( i i i ) L e t x E PI, L E B w i t h L E R a n d x I 1 L . S i n c e S i s a g e n e r a l ' z e d q u a d r a n g l e t h e r e e x i s t s a p a i r (y,.Nl) E P X B s u c h t h a t 1.1, L' E R a n d L' # L. T h e n we x I M I y I L w i t h :.I E R . L e t L' c o n s i d e r t h e a f ' f i n e p l a n e d e f i n e d by {L,L'l ( c f . 11.3.4.). A g a i n t h i s p l a n e i s g e n e r a t e d by x a n d L . (cf. 3.4.) with M

-

-

I n e a c h o f t h e c z s e s t h e p l a n e g e n e r a t e d by x a n d L i s a n a f f i n e p l a n e o f o r d e r s+l. I f s 2 3 w e c a n a p p l y t h e t h e o r e m o f F . Bueken-

M.de Soete arid J . A . Thas

182

h o u t 1 3 1 . Then D i s a t h r e e d i m e n s i o n a l a f f i n e s p a c e of o r d e r s + l .

By t h e e m b e d d i n g t h e o r e m of J . A . S

f

P(W(s+l),(-))y

or S

f

T;(O)

T h a s 1131 a n d s i n c e s 2 3, we h a v e w i t h 0 a c o m p l e t e o v a l of PG(2,s+l)

w i t h s o d d . S u p p o s e we a r e i n t h e s e c o n d c a s e . The s p r e a d R o f T;(O!

i s a n o r m a l s e t ( s e e 1 1 . 3 . 3 . ) . Hence a l l l i n e s o f R a r e c o n c u r r e n t i n a same p o i n t x o f 0 ( I I , l . ( c ) ) . S i n c e a l l p o i n t s a r e R - r e g u l a r , 0 \

{ X I i s a conic ( 1 1 . 2 . 2 . ) .

So i n t h i s c a s e we also h a v e T ; ( O )

f

P ( W ( s t l ) , ( m ) ) ( c f . I I . l , ( d ) ) . I f s = 2 ( r e s p . s = 1 ) t h e r e i s , up t o i s o m o r p h i s m , o n l y o n e g e n e r a l i z e d q u a d r a n g l e of o r d e r ( 2 , 4 ) (resp.

( 1 , 3 ) ) 1 9 I, and again t h e r e s u l t follows.

0

2 . THEOREM L e t S = (P,B,I) b e a g e n e r a z i z e d q u a d r a n g l e of o r d e r (s,st2) c o n -

.

If e a c h l i n e o f B \ R i s w e a k R - r e g u Z a r , ( m ) a n a r b i t r a r y p o i n t of w(st.1). Proof. A n a l o g o u s l y a s i n 111.1. w e c o n s t r u c t t h e g e n e r a l i z e d q u a d r a n g l e S' o f o r d e r s t l . Now w e p r o v e t h a t S' = W ( s + l ) , s o d d , u s i n g t h e d u a l of T h eo r em 5 . 2 . 6 . i n 1 9 1. Assume t h a t t h e l i n e s L 1 , L 2 o f S d e f i n e a R-grid o f S . From t h e d e f i n i t i o n of a R - g r i d i t f o l l o w s t h a t e a c h l i n e o f { L l , L 2 1 1 * , as w e l l as e a c h l i n e of IL1,L2)1L' is c o n c u r r e n t ( i n S') w i t h a u n i q u e l i n e o f t y p e ( b ) , a n d t h a t I* ~ a r e never concurrent d i f f e r e n t l i n e s f r o m I L ~ , L ~ I o r { L , ~L 1"' ( i n Sl) w i t h a common l i n e of t y p e ( b ) . Now we n o t i c e t h a t Li,Mi ll* w i t h Li E IL1,L21 , Mi E { L 1 , L 2 } ' * , 1 4 i < s + % , a r ec o n c u r r e n t w i t h a same l i n e of t y p e ( b ) a n d m o r e o v e r a r e i n c i d e n t w i t h a s a w p o i n t of t h a t l i n e of t y p e ( b ) . I n d e e d , for t h e p o i n t s xki I M i , k f i , xik I L i , k # i , t h e r e h o l d s xikxki E R . T h e r e e a s i l y f o l l o w s t h a t (L1,L2) i s a r e g u l a r p a i r i n S'. C o n s i d e r t h e p o i n t ( m ) . A n a l o g o u s l y a s i n t h e p r o o f o f 111.1. we show t h a t ( m ) i s r e g u l a r in 9 ' . Let x b e a n a r b i t r a r y p o i n t o f t y p e I' 1 ( i ) 2 n d p u t {x,(m)l = { ( R , R j . l , x I R E R , Ri E R } . L e t ( i l , L 2 , L , j b e a t r i a d o f l i n e s of S' w h i c h a r e i n c i d e n t w i t h p o i n t s J r of (x,(m) 1' . 'i'hen t h e following t wo c a s e s may occLir. The t h r e e l i n e s a r e of t y p e ( a ) , or t w o i i n e s a r e of t y p e ( a ) a n d o n e i s o f t y p o ( b ) . S u p p o s e t h a t L1,L2 a r c of t y p e ( a ) . I n S t h e l i n e s o f R w h i c h are c o n c u r r e n t w i t h L1 ( r e s p . L 2 ) f o r m a l i n e L1 ( I - e s p . L 2 ) sf t . h e a f f i n e p l a n e A R , S i n c e i n 2' t h e l i n e s L 1 , L 2 a r e c o n c u r r e n t w i t h d i f f e r e n t l i n e s o f t y p e ( t i ) , t h e l i n e s L1, l 2 o f A R a r e n o t p a r a l l e l . I f R' E R i s t h e C O ~ T I C e~ l~eIm e n t of L1 a n d L 2 , t h e n c l e a r l y 3' i s t h e u:iique e l e m e n t of R !;~hicil i s c o n c u r r e n t (in s ) w i t h L1 t a i n i n g a normal s p r e a d R then S

2

P(W(s+l),(m)), s odd, w i t h

I83

R-Regularit)' of the Generalized Quadrangle PI WIs), (-)I

a n d L 2 . H e n c e t h e p a i r f L L 1 b e l o n g s t o a R - g r i d i n S , a n d by t h e 1' 2 previous paragraoh it is a r e g u l a r p a i r i n S ' . By 1 . 3 . 6 . ( i i ) i n

1 9 1 t h e t r i a d (L1,L2'L ) h a s a t least one c e n t e r . Using t h e d u a l 3 o f 5 . 2 . 6 . i n I 9 1 , t h e r e r e s u l t s t h a t S' i s i s o m o r p h i c t o W ( s + l ) , s o d d . Hence t h e q u a d r a n g l e S i s i s o m o r p h i :

t o P ( W ( s + l ) , ( m ) ) . The

l i n e s of t h e spread R i n S can b e i d e n t i f i e d with t h e sets

{(-),xH1'l',

x Y'

( m ) .

0

3 . THEOREN Let

s

be

3

g e n e r a l i z e d q i l a d r a n g l e of o r d e r ( s , s + 2 ) c o n t a i n i n g a

spread R.

If e a c h Z i n e of B \ R i s R - r e g u l a r t h e n S P(W(stl),(=)) ( a ) an ar2bitrary poiqt of W(s+l). P r o o f . T h i s i s i m m e d i a t e f r o m 1121.5. a n d T h e o r e m 1 1 . 2 . 0 s ociLI, w i t h

T a k i n g a c c o u n t o f I I I . l . , 1 1 1 . 2 . a n d III.3., w e c a n f o r m u l a t e t h e next theorem.

4. THEOHEM L e t S be a g e n e r a l i z e d quadrangle of o r d e r ( s , s + 2 ) which c o n t a i n s a spread R .

Then t h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t

:

( i ) e a c h point i s R - r e g u l a r and s i s o d d ; ( i i ) e a c h line i s weak R - r e g u l a r and R i s a n o r m a l s e t ; ( i i i )each l i n e i s R-reguZur; (iv) S

2

P(W(s+l),

( m ) ) ,

s odd,

with

(m)

n n a r b i t m r y p o i n t of

W(s+l).

REFERENCES I11

Phi.en; , R . l u . a n d S z e k e r e s , G . , On a c o r n b i n a t o r i a l g e n e r a l i z a t i o n o f 27 l i n e s a s s o c i a t e d w i t h a c u b i c s u r f a c e , J . A u s t r . Math. S O C . 1 0 ( 1 9 6 9 ) 485-492.

12]

B ( , t l s r-,C.T., O n the s t r u c t u r e ot' g e n e r a l i z e d q u a d r a n g l e s , J . A l g e b r a 15 ( 1 9 7 0 ) 443-4511.

I31

B u e k e q h o u t , F . , Une c a r s c t ; r i s a t i o n d e s e s p a c e s a f f i n s b a s 6 e sur l a n o t i o n d e d r o i t e , I I P t h . 111 ( 1 9 6 9 ) 367-372.

I4 I

Dembowski,P.,

151

kIa11,N. J r . , A f f i n e g e n e r a l i z e d q u a d r i l a t e r a l s , S t u d i e s i n P u r e [ l a t h . ( e d . L. M i r s k y ) , Academic P r e s s ( 1 9 7 1 ) 1 1 3 - 1 1 6 .

161

-.

F i n i t e geometries (Springer-Verlag,l968).

P a y n e , S . E . , T h e equivalent? o f c e r t a i n g e n e r a l i z e d q u a d r a n g l e s , Col,ib. T h . 1 0 ( 1 9 7 1 ) 7'34-289.

J,

184

M . de Socte and J . A . Thas

I71 Payne,S.E., Quadrangles of order (s-l,s+l), J. Algebra 22 (1972) 97-110. 181

Payne,S.E. and Thas,J.A., Generalized quadrangles with symmetry, Part 11, Simon Stevin 49 (1976) 8 1 - 1 0 3 .

L 91

Payne,S.E. and Thas,J.A., Finite generalized quadrangles, Research Notes in Mathei:idtics # 1 1 0 ( P i t m a r ? Publ. Inc. 1 1 8 4 ) .

I 10

]

Segre,R., Lectures on modern geometry (Ed. Cremonc.;? Rorn? l ' h

1).

I111 Somma,C., Generalized quadrangles with parallelism, Annals of Discrete i""ath.14 ( 1 9 8 2 ) 265-282. 117 1

Thas,J.A., Combinatorics of partial geometries and generalized quadrangles, in : Higher combinatorics (ed. M. Aigner), Nato Advanced Study Institute Series, Reidel Publ. Comp. (1976) 183-199.

[ 131

Thas,J.A., Partial geometries in finitc affine spaces, Math. Z. 1 5 8 ( 1 9 7 8 ) 1 - 1 3 .

Annals of Discrete Mathematics 30 (1986) 185-202

185

0 Elsevier Science Publishers B.V. (North.Holland)

ON PERMUTATION ARRAYS, TRANSVERSAL SEMINETS AND RELATED STRUCTURES M i c h e l Deza and Thomas I h r i n g e r U n i v e r s i t i ! P a r i s V I I , U.E.R. de Math., P a r i s , France Technische Hochschule, Fachbereich Mathematik, Darmstadt, Federal R e p u b l i c o f Germany

E x p l o i t i n g some i d e a s o f C41, t h i s paper i s focused on t h e e q u i valence between s e t s o f m u t u a l l y o r t h o g o n a l p e r m u t a t i o n a r r a y s and a s p e c i a l c l a s s o f seminets ( t h e s o - c a l l e d t r a n s v e r s a l s e m i n e t s ) . Besides t h i s e q u i v a l e n c e , S e c t i o n 2 c o n t a i n s a c o n s t r u c t i o n method f o r t r a n s v e r s a l seminets u s i n g groups. Nonsolvable p e r m u t a t i o n groups o f p r i m e degree and t h e p r o j e c t i v e s p e c i a l l i n e a r groups PSL(2,Zm) y i e l d examples f o r t h i s method. I n S e c t i o n 3 some upper bounds a r e p r o v e d f o r t h e number o f m u t u a l l y o r t h o g o n a l p e r m u t a t i o n a r r a y s , depending on t h e i n t e r s e c t i o n s t r u c t u r e o f t h e s e a r r a y s . With t h e r e s u l t s o f S e c t i o n s 4 i t i s shown t h a t a l l examples o f Section 2 are row-extendible. Section 5 deals w i t h several i n c i dence s t r u c t u r e s a s s o c i a t e d t o t r a n s v e r s a l seminets. The consequences a r e i n v e s t i g a t e d when t h e s e i n c i d e n c e s t r u c t u r e s have s p e c i a l p r o p e r t i e s ( f o r i n s t a n c e , when t h e y a r e p a i r w i s e balanced d e s i g n s ) . S e c t i o n 6 d i s c u s s e s b r i e f l y t h e r e l a t i o n s o f t r a n s v e r s a l seminets w i t h o t h e r mathematical s t r u c t u r e s , e.g. w i t h t r a n s v e r s a l p a c k i n g s , g e n e r a l i z e d o r t h o g o n a l a r r a y s , and s e t s o f m u t u a l l y o r t h o g o n a l p a r t i a l quasigroups.

1. INTRODUCTION

A = ( a . . ) w i t h e n t r i e s a . . f r o m t h e s e t I1,2 ,...,r l i s 1J 1J c a l l e d a p e r m t a t i o n amuy i f each row o f A c o n t a i n s each o f t h e elements 1,2, A vxr matrix

. . . ,r

e x a c t l y once, i . e . i f t h e rows o f

r l . The i n t e r s e c t i o n s tr u ctu r e o f

A

A

represent permutations o f

i s d e f i n e d as t h e vxv m a t r i x

{l,Z

,...,

F(A) =

..=a. 1 . The v x r p e r m u t a t i o n a r r a y s (Fi i , ( A ) ) w i t h F 1. 1. , ( A ) : = t j I l s j s r , a IJ i'j A = ( a . . ) and B = ( b . . ) a r e c a l l e d e i r n . L Z a i * i f F ( A ) = F(B). C l e a r l y , A and 1J 1J a r e s i m i l a r i f and o n l y i f , f o r a l l i n d i c e s i , i ' , j ,

The p e r m u t a t i o n a r r a y s if, f o r a l l

aij

A

and

and

bij

B

B

a r e c a l l e d orthogonu2 i f t h e y a r e s i m i l a r and

i,i',j,j', = ai,j,

= bi ,j, ==3 j = j ' .

T h i s concept o f o r t h o g o n a l i t y g e n e r a l i z e s t h e well-known i d e a o f o r t h o g o n a l l a t i n r e c t a n g l e s . I t has been d e f i n e d and i n v e s t i g a t e d by B o n i s o l i and Deza i n C41.

M. Deza arzd T. lhringer

186

(See a l s o 1 7 3 f o r c l o s e l y r e l a t e d c o n s i d e r a t i o n s . ) S i m i l a r l y as f o r l a t i n r e c t a n g l e s and l a t i n squares, one w i l l be i n t e r e s t e d i n sets

4=

{A1,A 2,...,At}

of

mutually orthogonal permutation arrays, w i t h

t

p o s s i b l y g r e a t e r t h a n 2 . I n t h i s case t h e i n t e r s e c t i o n s tr uc tur e

t

F(A) =

(Fi ,i(f)i ) o f A i s d e f i n e d as t h e common i n t e r s e c t i o n s t r u c t u r e o f t h e A k' i . e . F(&):= F(A1) = F(A2) = = F(At). A p e r m u t a t i o n a r r a y A = ( a . .) i s c a l l e d standardized i f alj = j f o r a l l j 1J

...

.

Without loss of generality, i t w i l l be assumed i n t h i s paper t h a t each permut a t i o n array is standardized, and t h a t i t s a t i s f i e s the following n o n t r i v i a l i t y conditions (CII and ( C 2 ) . (C,)

The p e r m u t a t i o n a r r a y

A

A

has no c o n s t a n t column, i . e . each column o f

c o n t a i n > a t l e a s t two d i s t i n c t values, any two rows o f

(C,)

X

Let

A

are d i s t i n c t .

be a nonempty s e t , and l e t

d i s j o i n t s e t s o f nonempty subsets o f

Lo,L1,

...,L t

(with

X. The elements o f

tzl)

3 :=(X;Lo,L1 ,...,L t )

poi n t s and t h e elements o f ,..,, Lk Zines. Then i s c a l l e d a seminet o r (more p r e c i s e l y ) a ( t t 1 ) - s e m i n e t i f (S1)

any two d i s t i n c t l i n e s i n t e r s e c t i n a t most one p o i n t ,

(S,)

each c l a s s

Li

partitions the point set

w i l l be c a l l e d

X

u,,,,,

be m u t u a l l y

X.

C o n d i t i o n (S2) j u s t i f i e s t h e t e r m p a r a l l e l c l a s s f o r each o f t h e l i n e s

Li.

The

n o t i o n o f a seminet g e n e r a l i z e s such well-known s t r u c t u r e s l i k e a f f i n e p l a n e s , n e t s and (more g e n e r a l l y ) t h e p a r a l l e l s t r u c t u r e s o f Andre 121. A subset o f

X

3

in

c a l l e d a transi.arsa2 o f t h e seminet

5

e x a c t l y one p o i n t . I f p a r a l l e l class

Li

5

r:=

then

r

has a t r a n s v e r s a l c o n s i s t i n g o f

contains e x a c t l y

r

o f t h e seminet c o n s i s t s a l s o o f e x a c t l y versals o f

i f i t i n t e r s e c t s each l i n e o f

(X;Lo,L1

3

is

p o i n t s , t h e n each

l i n e s ( a n d hence each f u r t h e r t r a n s v e r s a l r

p o i n t s ) . I f T1,T

,...,Lt;T1,T2 ,..., Tv)

*,...,TV

are trans-

i s c a l l e d a transversa2

seminet ( o r transversal ( t t 1 , r ) - s e m i n e t i f each t r a n s v e r s a l c o n s i s t s o f

r

p o i n t s ) . B o n i s o l i and Deza r41 p o i n t e d o u t t h a t t h e r e i s a c l o s e r e l a t i o n s h i p b e t ween set.s o f m u t u a l l y o r t h o g o n a l p e r m u t a t i o n a r r a y s and o t h e r mathematical s t r u c t u r e s . F o r i n s t a n c e , t h e y proved t h a t each s e t o f p e r m u t a t i o n a r r a y s i s e q u i v a l e n t t o a 1-design w i t h number

r

and

ttl

mutually orthogonal v x r

t

v

treatments, r e p l i c a t i o n

m u t u a l l y o r t h o g o n a l r e s o l u t i o n s (see S e c t i o n 5 ) . Moreover,

i t was shown t h a t any o f these a r e e q u i v a l e n t t o a t r a n s v e r s a l ( t t 1 , r ) - s c m i n e t

with

v

t r a n s v e r s a l s . T h e r e f o r e many o f t h e examples and r e s u l t s i n t h i s paper

can be t r a n s l a t e d i n t o analogous statements on c o m b i n a t o r i a l designs w i t h m u t u a l l y ortogonal resolutions.

Oil

Pennutatioii Arrays

187

2. AN EQUIVALENCE AND A CONSTRUCTION METHOD

J :=

Let 1 2 {lo,l

o,...,lL).

(X;Lo,L1,..

. ,Lt;T1,T2,.

.. ,Tv)

Lo:=

be a t r a n s v e r s a l seminet w i t h

( I n f a c t , t h r o u g h o u t t h i s paper t h e s e t o f p a r a l l e l c l a s s e s , t h e Lo o f any t r a n s v e r s a l seminet a r e

s e t o f t r a n s v e r s a l s and t h e s e t o f l i n e s of

assumed t o be l i n e a r l y ordered, by t h e numbering o f t h e i r elements.) One can now k d e f i n e t m u t u a l l y o r t h o g o n a l v x r p e r m u t a t i o n a r r a y s Ak = ( a . . ) , k=1,2, ...,t, 1J i n t h e f o l l o w i n g way: F o r i E I 1 , 2 ,... ? v 1 , j E I 1 , 2 ,...,r l , k < I1,2 ,...,t ) l e t Ti n l;, and l e t

be t h e unique p o i n t . c o n t a i n e d i n

x

x. L e t

be t h e u n i q u e p o i n t w i t h

y

t h r o u g h y. F i n a l l y , d e f i n e one can conclude t h a t

a:j:=

1

y c T 1 n 1, and l e t

be t h e 1;

L k - l i n e through

be t h e

Lo-line

c . From t h e p r o p e r t i e s o f t r a n s v e r s a l seminets

&J):=IA1,A2, . . . ,A t )

i s , i n fact, a s e t o f mutually

orthogonal permutation arrays. The c o n d i t i o n s (C,)

o f S e c t i o n 1 can be paraphrased i n terms o f

and (C,)

t r a n s v e r s a l seminets as f o l l o w s : There i s no p o i n t o f t h e seminet w h i c h i s c o n t a i n e d i n a l l t r a n s v e r s a l s ,

(D1)

...,

. = ni=1,2, v Ti 1 , a n d v.2.)

1.‘.

line (D2)

0.

Any two t r a n s v e r s a l s a r e d i s t i n c t , i . e .

I n t h e c o n s t r u c t i o n procedure f o r

Ti

i

J?-(r) only

T

for

j

X have been T h e r e f o r e one can always

those p o i n t s o f Ti.

r e s t r i c t o n e s e l f t o t h e reduced t r a n s v e r s a l seminet

( X ’ ;LA,Li,.

d e f i n e d by

XI:=

ui=1,2,...,v

f o r each

iz j .

used which a r e c o n t a i n e d i n one o f t h e t r a n s v e r s a l s Tv)

I1 I t 2

(This implies, i n particular,

,

. ,LC;T1,T2,. . . ,

L k’ : = ( 1 n X ’ j l c L k I .

Ti’

I n t h e r e s t of this paper a l l transoersa2 seminets are assumed t o be reduced

arid t o s u t i s f g the ron,Zitions (0,) m d (DJ. Y

The process of c o n s t r u c t i n g m u t u a l l y o r t h o g o n a l p e r m u t a t i o n a r r a y s f r o m t r a n s v e r s a l seminets can be r e v e r s e d : L e t k p e r m u t a t i o n a r r a y s Ak = ( a . . ) , k=1,2, 1J

+

r ; , and l e t

& be ...,t .

a set of Define

be t h e e q u i v a l e n c e r e l a t i o n on

Y

t

Y:=

with

mutually orthogonal v x r {1,2. v) x {l,Z,

...,

...,

( i , j ) $ ( i ’ , j ’ ) i f and

j i Fi i ,(A). D e f i n e t h e p o i n t s e t X o f t h e seminet as t h e set o f equivalence classes o f 9 , i . e . X:= Y / $ = “ ( i , j ) l l b 1 (i,j)t:YI. For c = C C 1 , 2 ,.... r and k = 1,2,, . .,t l e t lo:= (I ( i , j ) l $ I j = c , i=1,2,...,v) and l k : = 1 2 k 1 2 ,...,1 L I . { I ( i , j ) l d j a . . = c l , and d e f i n e L o : = {lo,lo ,..., lor}, L k : = Ilk,lk

o n l y if j = j ’

and

~

1J

Finally, l e t (X;Lo,L1,..

T.:= 1

.,Lt;Tl,T2,,

= A .Summarizing,

’,r

(i,j)l$

..

1

j = 1 , 2 , . ..,rI, i = 1 , 2

,...,v .

Then

J(R):=

,Tv) i s a (reduced) t r a n s v e r s a l seminet w i t h one o b t a i n s

&(T(R))

M. Deza arid T. Ihririger

188

The existence of a s e t

2.1. PROPOSITION.

01

t

mutually orthogonal v x r perm-

tation arrays i s equivaZent t o the existence of a transversal (t+l,r)-seminet with

v

transversals. T h i s e q u i v a l e n c e was a l r e a d y observed i n 141. One can show even a l i t t l e more.

Let

. . ,Lt;T1,T2,..

= (X;Lo,L1..

.,Tv)

be reduced t r a n s v e r s a l seminets w i t h

.

and ‘U = (Y;Mo,M1,. ..,Mt;U1,U2,. .,Uv) 1 2 ,...,1 L I and Mo 1 2 ,..., Lo = ilo,lo Imo,mo

m i l , and assume R(’3’)= A(%). D e f i n e a mapping $ : X 4 Y as f o l l o w s . F o r X E Ti n 1: l e t $ ( x ) be t h e unique element c o n t a i n e d i n Ui nm:. Then 0 i s an and U , i . e .

isomorphism o f

0 satisfies

onto p a r a l l e l l i n e s ( i n f a c t , for all 2.2.

PROPOSITION.

If

y

&(J) =

T. nT. = 0

= Ui

$Ti

and

i i $lo = mo

u

are reduced transversa2 seminets with are isomorphic.

.

.

= ( X;Lo,Ll,. .,Lt;T1,T2,. .,Tv) corresponds t o a o f m u t u a l l y o r t h o g o n a l Zatin rectanglos e x a c t l y i f

IA1,A 2,...,Atl

for a l l

J

and

and

The t r a n s v e r s a l seminet

1

$Li = Mi,

i ) . This y i e l d s

J , ( J )= L ( U ) then

set

0 i s a b i j e c t i o n which maps p a r a l l e l l i n e s

i,j, i z j . The a r r a y s

A1,A 2,...,At

form a s e t o f m u t u a l l y

o r t h o g o n a l Zatin squares i f and o n l y i f tT1,T2,,..,Tv\,

(X;Lo,L1,, ..,Lt,Lt+l), w i t h Lt+l:= I n o t h e r words, t h e e q u i v a l e n c e o f P r o p o s i t i o n 2.1

i s a net.

s p e c i a l i z e s t o t h e c l a s s i c a l correspondence o f m u t u a l l y o r t h o g o n a l l a t i n squares w i t h nets. The f o l l o w i n g theorem p r o v i d e s a c o n s t r u c t i o n method o f s e t s o f m u t u a l l y o r t h o gonal p e r m u t a t i o n a r r a y s v i a seminets, u s i n g groups. 2.3.

THEOREM.

G he a f i n i t e group with neutral eZement e . Let

Let

t and

be p o s i t i v e integers, and Zet So,S1,.. .,St and F1,F2,.. .,F be nontrivial S subgroups of G such that the foZZowing conditions are s a t i s f i e d f c r aZl i , j E S

10,1,

..., tl,

k,l

E

11,2

,..., ~

4 Si n S j

(2)

i j

(2)

S. O F = {el,

(3)

k * 1 =$ Fk z F,,

(41

l F k I = CG:Sil.

i

= (el,

k

Then there e x i s t s a s e t o f S r : = I F I and v:= -./GI . 1 r

Proof. i.e.

1 :

F o r each

Li = {Sig

1

t

i E (O,l,

geG1. Then

mutualZy orthogonal v x r p e r m t a t i o n arrays, with

...,t l

let

(G;Lo,L l,...,Lt)

Li

c o n s i s t o f the r i g h t cosets o f i s a (tt1)-seminet:

Si,

C o n d i t i o n (S1)

On Permu tutiori Arru-vs

189

i s t r i v i a l l y s a t i s f i e d w h i l e ( S 2 ) i s a consequence o f ( 1 ) . Each r i g h t c o s e t

Fkh i s a t r a n s v e r s a l o f (G;Lo,L1, ..., L t ) : I t has t o be Fk n F k h l = 1 f o r a l l g,hsG. As a consequence o f ( 2 ) one o b t a i n s

o f one o f t h e subgroups show that

lSig

1S.g n F k h l i 1. Assumption ( 4 ) then i m p l i e s u f C F k S i f = G, and hence 1

lSig n F k h l

2

1. Each t r a n s v e r s a l has

d i s t i n c t r i g h t c o s e t s . Thus t h e r e a r e form

Fkh, w i t h

(D1) and

k t t1,2,

...,s l

and

r = lFll

elements, and each

v = :*IGI

Fk

IGI

has

d i s t i n c t transversals o f the

h e G. Finally, the n o n t r i v i a l i t y conditions

( D p ) a r e a consequence o f t h e n o n t r i v i a l i t y o f t h e subGroups Fk and o f

(3), r e s p e c t i v e l y . By P r o p o s i t i o n 2.1, t h e p r o o f i s complete. The seminet

seminet, i . e .

n

,,...,

L t ) o f t h e above p r o o f i s , i n f a c t , a transZation (G;Lo.L i t has a t r a n s l a t i o n group o p e r a t i n g r e g u l a r l y on i t s p o i n t s : I n t h e

r i g h t r e g u l a r represeritation o f

G

each maoping

XH xg,

g r G , maps e v e r y l i n e

o n t o a p a r a l l e l l i n e . On t h e o t h e r hand, each t r a n s l a t i o n seminet can be o b t a i n e d i n t h i s way f r o m a group

G

and subgroups

So,S l....,St

satisfying condition

(1). Analogous group t h e o r e t i c c h a r a c t e r i z a t i o n s have been given, f o r i n s t a n c e ,

f o r t r a n s l a t i o n planes, t r a n s l a t i o n n e t s , t r a n s l a t i o n s t r u c t u r e s and t r a n s l a t i o n group d i v i s i b l e designs ( s e e e.g.

rll,

1151. C221, [31 and 1201). Marchi r181 uses

s i m i l a r i d e a s f o r his c h a r a c t e r i z a t i o n o f r e g u l a r a f f i n e p a r a l l e l s t r u c t u r e s by p a r t i t i o n l o o p s . P r o b a b l y one can f o r m u l a t e an analogue o f Theorem 2.3 u s i n g l o o p s i n s t a e d of groups. The problem would be t o f i n d examples f o r such a g e n e r a l i z a t i o n . The r e s t o f t h i s s e c t i o n y i e l d s two c l a s s e s o f examples f o r Theorem 2.3. C f . Huppert 1141 and W i e l a n d t 1241 f o r t h e group t h e o r e t i c n o t a t i o n s .

2.4. EXAMPLE.

Let

G

be a n o n s o l v a b l e t r a n s i t i v e p e r m u t a t i o n group o f p r i m e

v:= p 2 , r:=-I,G I and l e t d be the p o s i t i v e i n t e g e r w i t h d < p - 1 P d = r (mod p ) . Then one can c o n s t r u c t a s e t o f t : = 1 mutually orthogonal

degree p . L e t and

i-

v * r p e r m u t a t i o n a r r a y s : Assume tl,Z, . . . , p 1

define

FkzF,

k z

for

So,S l,...,St,

1

Fk

since

G

t o o p e r a t e on

t o be t h e s t a b i l i z e r o f

6

*,.... a P I .

ial,a ak

in

G, i . e .

F o r each

k

Fk:= Ga

6

Then

k' i s d o u b l y t r a n s i t i v e ( c f . Theorem 11.7 o f r 2 4 1 ) . L e t

be t h e Sylow p-subgroups o f

G

(with

t ' 1 1 because

G

i s non-

s o l v a b l e ) . O b v i o u s l y , these subgroups s a t i s f y t h e assumptions o f Theorem 2.3. Hence t h e r e a r e show P

of

t'

m u t u a l l y o r t h o g o n a l v x r p e r m u t a t i o n a r r a y s . I t remains t o r G has e x a c t l y Sylow p-subgroups. L e t

t ' = t or, equivalently, t h a t

be a Sylow p-siibgroup o f

P

is

P

i t s e l f . Hence

a

G. The o n l y Sylow p-subgroup o f t h e n o r m a l i s e r

NG(P)

i s s o l v a b l e and t h u s o f o r d e r

pad'

NG(P)

with

( c f . r141, Satz 1 1 . 3 . 6 ) . T h e r e f o r e t h e number n o f Sylow p-subgroups G r satisfies n=[G:N ( P ) l = p . d ' = a T . From n = l (mod p ) one o b t a i n s d ' = r (mod p ) , G r i . e . d = d ' and n =

d"p-1

a.

The n o n s o l v a b l e t r a n s i t i v e p e r m u t a t i o n groups o f p r i m e degree have been comp l e t e l y determined, due t o t h e c l a s s i f i c a t i o n o f f i n i t e s i m p l e qroups ( s e e

M. Deza Q

190

I I T. ~ Ihringer

C o r o l l a r y 4.2 o f F e i t C111). N o t i c e t h a t solvabZe t r a n s i t i v e p e r m u t a t i o n groups o f p r i m e degree o cannot be used i n t h e above c o n s t r u c t i o n : These groups have e x a c t l y one Sylow p-subgroup, which would i m p l y 2.5. EXAMPLE.

t = 0.

F o r each i n t e g e r

mr2

one can c o n s t r u c t a s e t o f t m u t u a l l y m-1 rn ( 2 - 1 ) - 1 9 v : = (2m+1)2 and r : = m 2m(2m-l): Regard t h e p r o j e c t i v e s p e c i a l l i n e a r group G = PSL(2,q), w i t h q = 2 ,

orthogonal v x r permutation arrays, w i t h

t:= 2

as a p e r m u t a t i o n group o p e r a t i n g c a n o n i c a l l y on t h e

q+l

o f t h e p r o j e c t i v e l i n e o v e r t h e q-element f i e l d . F o r each define

Fk

t o be t h e s t a b i l i z e r o f

ak

in

G, i . e .

be t h e ( m u t u a l l y c o n j u g a t e ) c y c l i c subgroups o f

=

=-,

t'

o f conjugates o f

ttl, i.e.

k

t

..,aq+l) . . ,q+11

{al,a2,.

{1,2,.

Fk:= G L e t So,S l,...,Stl ak' q + l . By t h e r e s u l t s

o f order

G

t h e s e subgroups s a t i s f y t h e assumptions o f Theorem 2.3.

i n 1141, pp. 191-193, t h e number

points

one o b t a i n s

So

For

t ' + l= C G : N G ( S o ) l = w

t ' = t.

i n o r d e r t o conPSL(2,q) s t r u c t designs w i t h m u t u a l l y o r t h o g o n a l r e s o l u t i o n s . F o r i n s t a n c e , f o r each q E N o t i c e t h a t Hartman [121 used some o f t h e groups

{19,31,431 t i o n s and

there e x i s t s a design w i t h

v=qtl

t r e a t m e n t s , r =?

replica-

t+l=q mutually orthogonal resolutions.

3. BOUNDS FOR THE NUMBER

OF MUTUALLY ORTHOGONAL PERMUTATION ARRAYS

.

o f m u t u a l l y orthogonal p e r m u t a t i o n a r r a y s i s c a l l e d {A1,A2,. . ,At) maxima2 i f t h e r e e x i s t s no p e r m u t a t i o n a r r a y which i s o r t h o g o n a l t o a l l Attl Ak, k = 1,2 ,..., t. A t r a n s v e r s a l seminet (X;L0,L1, Lt;T1,T2 ,Tv) i s c a l l e d L-mo.ximaZ i f t h e r e e x i s t s no a d d i t i o n a l p a r a l l e l c l a s s Lt+l such t h a t (X;Lo,L1,

A set

...,

..

.,Lt,Lttl;T1,T2,. obvious. 3.1.

LEMMA.

. .,Tv)

i s a g a i n a t r a n s v e r s a l seminet. The f o l l o w i n g lemma i s

of mutually orthogond permutation arraps is maximal

A set

i f mid only if the associated transversal seminet

3.2. PROPOSITIOM.

,...

J (A)is

L-maximal.

of rnintualZy orthogoxu2 permutation arrays of

Eaeh s e t

Example 2 . 5 is maximal.

Proof.

Let

G = PSL(2,2m)

a s s o c i a t e d t r a n s v e r s a l seminet

be t h e group used f o r t h e c o n s t r u c t i o n of

7

has

G

as p o i n t s e t . The subgroups

...,St

a r e e x a c t l y t h e l i n e s t h r o u g h t h e n e u t r a l element

groups

F1,F2,.

. .,Fq+l

e

are e x a c t l y the transversals through

of

e

A.The SO'S1,

G, and t h e sub( c f . the proof

191

Oir Permu tation Arrays

G. Hence

o f Theorem 2 . 3 ) . By Satz 11.8.5 o f Huppert L141, t h e s e subgroups c o v e r t h e r e cannot be any a d d i t i o n a l l i n e through

e, and

i s t h e r e f o r e L-maximal.

J 2 J(&).

By Lemma 3 . 1 t h e p r o o f i s complete, s i n c e P r o p o s i t i o n 2.2 y i e l d s

0

A c t u a l l y , t h e a s s e r t i o n o f Propos t i o n 3 . 2 depends o n l y on t h e i n t e r s e c t i o n

F(&)

structure

of

a:L e t

gonal p e r m u t a t i o n a r r a y s w i t h

,..., B t , I

3

B1,B2

F(JJ )

= F(&).

S e c t i o n 2 shows t h a t t h e t r a n s v e r s a l seminets p o i n t s e t s and t h e same t r a n s v e r s a l s

n

y @ ) As .

number

Y

lines o f

through

e

7113)

7

G. T h e r e f o r e each l i n e o f

T(&)t h r o u g h

x

have t h e same o f the proof

contains e x a c t l y

J(&)

t h a t t h e same i s t r u e f o r

a consequence, t h e r e i s a p o i n t

t'+l of lines o f

and

pairwise d i s j o i n t transversals

0

=

7 7(A)i m p l i e s

p o i n t s , and

for

u s e s oF1s

S E S ~ w, i t h

')'(a)

The t r a n s v e r s a l seminet

n:= I S I = CG:F1l

o f Proposition 3.2 contains F1s,

be a s e t o f m u t u a l l y o r t h o -

The c o n s t r u c t i o n procedure o f

x

of

J ( 3 ) such

and a l s o

t h a t the

c a n n o t exceed t h e number

( i n f a c t , t h i s i s t r u e f o r each p o i n t

x

t+l o f

y(2)) .

of

T h e r e f o r e P r o p o s i t i o n 3.2 can be improved as f o l l o w s .

3.3. PROPOSITION.

Get

a ={A1,A

2,...,Atl

be one of t h e s e t s ofmutuaZZy

3

orthogonal permutation arrays of Exnmple 2 . 5 . Let

= IB1,B2,.

of mutually orthogonal permutation arrays with F ( B ) = F(&).

. . ,Bt, Then

be a s e t

1 t'

5

C..

The n e x t lemma g i v e s an upper bound f o r t h e number o f m u t u a l l y o r t h o g o n a l p e r m u t a t i o n a r r a y s depending on t h e i n t e r s e c t i o n s t r u c t u r e of t h e a r r a y s . The p r o o f o f t h i s lemma i s a u n i f i e d v e r s i o n o f t h e p r o o f s o f s e v e r a l r e l a t e d r e s u l t s i n

'41 and 171. 3.4.

LEMMA.

Let

A =IA1,A2 ,..., At] I

pemutatioil arrays, and l e t

..., r l

j o <{ l , Z ,

{ l , Z ,..., v},

i o c

1 1 , 2 ,..., vl,

J 5 {1,2,...,rI,

satisfg

II,

al

I

bi

t"il,i2t

I:

ai

'JiE I :

j o d F .1 i,

(A),

dl

V j c J

3i.I:

j + F i i

f

be a s e t of mutualZy orthogonu2 v x r

j O c Fil12'

(A),

0

(a.

I"he1i t ' r - ! J I - 1 .

-

P r o o f . L e t A = ( a . . ) be a v r p e r m u t a t i o n a r r a y w i t h F(A) = F ( & . 'J t h e r e e x i s t s an element i l k I . Define c:= a i . From b ) one o b t a i n s 1" 0 f o r a l l i I , and c ) i m p l i e s aio,of c. F o r each j c J t h e r e e x i s t s an witn

ai

OJ

= a i j , by d ) . Thus

jLj0

and

aijza..

'JO

= c . Hence

aioj

By a )

"jd

ic I

c . There-

192

M . Deza arid T. Iliringer

fore

aioj = c

f o r e x a c t l y one element

. . . ,rl \

(J

A,

with

c

kt+

j,

t J?-

mapping

o f the (r-iJl-l)-element set

j

{1,2,

I n p a r t i c u l a r , t h i s i s t r u e f o r each o f t h e p e r m u t a t i o n a r r a y s

{j,}).

IJ

and

j

r e p l a c e d by

ck

and

j,

.

By o r t h o g o n a l i t y ,

t s r - IJI

i s i n j e c t i v e . This implies

- 1.

the

0

The f o l l o w i n g c o r o l l a r y t r a n s l a t e s Lemma 3.4 i n t o t h e language o f t r a n s v e r s a l seminets. N o t i c e t h a t t h i s c o r o l l a r y c o u l d have been used i n o r d e r t o prove t h e P r o p o s i t i o n s 3.2 and 3.3. COROLLARY.

3.5.

seminet, and 7.et

I

al

. . ,Lt;T1,T2,. io c {1,2¶, . . , v l

= (X;Lo,L1,.

..,Tv)

. ., v l ,

and

I c {1,2,.

b e a transversa2

x

X

E

satisfy

6,

mi,, T ~ ,

b)

Ti

x

el Then

f

Lat

0

t sr-6

-

. 1, w i t h

r : = ITl]

and

ITi n (uicI

6::

Ti) I

0

3.6. COROLLARY.

Let

A={A1,A21...,A

uerrnutation arrays, and l e t

p: =

max { I Fi

t

.

1 be a s e t of rnutuaZZy orthogonal v x r

, (A) I I i ,i'=1,2,.

. . ,v,

i*i' I . Then

t i r - U - 1 .

~

Proof.

Choose

i,iot.il,2

,..., v l

and

j o t {112]

..., r I 1 w i t h

.

j o & F i i (A)D e f i n e I : = t i 1 and J : = F i l o ( & ) 0 s a t i s f y t h e assumptions of Lemma 3.4. 1-1 and

3.7. COROLLARY.

Let

pervnitution a r r a y s . Let T'hen t s r Proof.

-A-2 . Choose

A=IAl,A2, ...,A t 1 h:=

min { I F . #(.+?-)I li

. Then

IFi io(sZ)I =

IJ

I , io, J , jo

be a s e t of mutzia1Zy orthogonal v x r

1

i,i'=1,2,..,,v),

io,i1,iz8 i 1 , 2 ,...,v l , il*i2$ and

joF11,2

and assume

,..., r l

X z l .

with

jo'

(A)

(A).

Fili2(&), j o ~ F i l i o ( ~ ) and j o & F '2'0 . . D e f i n e I : = {ilyi21, J : = F.'1'0. IJ F . (f-). T h e n I , io, J , j o s a t i s f y t h e assumptions o f Lemma 3.4. Hence t h e 12iO proof i s complete i n t h e case IJI -Xtl. Assume now I J I = A . Then X = IF. (&)I = IF. .

A

(&)I,

1 l i O

IFi i (&)I = IJ u tjo)l = X t l . T h e r e f o r e t h e case 1210 1 2 i s s e t t l e d by C o r o l l a r y 3.6. U and t h u s

The C o r o l l a r i e s 3.5,

IJI

=

3.6 and 3.7 y i e l d s l i g h t g e n e r a l i z a t i o n s for some o f t h e

r e s u l t s i n L41, 171 and r 1 7 1 (which a r e f o r m u l a t e d i n terms o f designs w i t h mutually orthogonal r e s o l u t i o n s ) .

A v x r p e r m u t a t i o n a r r a y A i s c a l l e d row-transitive i f t h e rows o f s e t o f p e r m u t a t i o n s o p e r a t i n g t r a n s i t i v e l y on t h e s e t { l Y 2 , . ..,rl.

A

form a

193

Oil Permutatioii Arraj's

3.8. PROPOSITION.

A =IA1,A2, ...,At]

Let

be a s e t of mutuully orthogonal

v' r p e r m t a t i o n arrays. Assume one of these arrays (and hence a l l of them) t o be

row-transitive.

t c min Ir-1 , m t l }

Then

orti?ogonal Latin squares of order

Proof. line

, with m t h e largest number of rm*tualZy

.

R o w - t r a n s i t i v i t y i m p l i e s each l i n e o f t h e a s s o c i a t e d t r a n s v e r s a l semi-

T(&) =

net

r

1 cLi,

(X;Lo,L1

,..., Lt;T1,T2 ,...,T v )

i > l , i n t e r s e c t s each l i n e o f

i s a (t-1)-net

Lt) existence o f

t o have e x a c t l y Lo

r

p o i n t s , s i n c e any

e x a c t l y once. Thus

(X;Lo,L

l,...,

o f o r d e r r . The e x i s t e n c e o f such a n e t i s e q u i v a l e n t t o t h e

t - 1 m u t u a l l y o r t h o g o n a l l a t i n squares o f o r d e r r

A complete ( t t 1 ) - n e t o f o r d e r r has e x a c t l y

rtl

w i t h a t r a n s v e r s a l cannot be complete. T h e r e f o r e

. Hence

t-lsm.

p a r a l l e l classes. But a n e t

t + l c

rtl.

U

4. EXTENSION BY ROWS

A set

J1= IA1,A2,.

. .,At}

o f mutually orthogonal v x r permutation arrays i s

c a l l e d row-ertercdible i f i t i s p o s s i b l e t o a d j o i n a new row t o each o f t h e a r r a y s such t h a t t h e r e s u l t i n g ( v t 1 ) x r p e r m u t a t i o n a r r a y s a r e a g a i n m u t u a l l y o r t h o g o n a l . A t r a n s v e r s a l seminet

(X;Lo,L1,

...,Lt;TI,T2,

...,T V )

i s c a l l e d zrwsversal-

estendibZe i f t h e r e e x i s t s a t r a n s v e r s a l seminet (Y;Mo,M 1,...,Mt;T1,T2,...,TV, Tvtl) w i t h Y 2 X and Li = I m n X 1 m c M i l . As transversal-extendibility i s t h e obvious t r a n s l a t i o n o f r o w - e x t e n d a b i l i t y i n t o t h e language o f t r a n s v e r s a l semin e t s , one o b t a i n s

4.1. LEMMA.

o f inutun1L;j orthogonaz permutation arrays i s

A set

extendible ij' and only i f t h e associated transversal seminet

J (&)

TOW-

i s transver-

sa 1-e.-cteadib l e . The above d e f i n i t i o n o f r o w - e x t e n d a b i l i t y i s s t r i c t l y s t r o n g e r t h a n t h e one g i v e n i n r 4 1 where t h e r e s u l t i n g a r r a y s were o n l y assumed t o be s i m i l a r . Both def i n i t i o n s coincide i f

Y =

X i n t h e t r a n s v e r s a l seminets used i n t h e d e f i n i t i o n o f

transversal-extendibility. I n p a r t i c u l a r , t h e d e f i n i t i o n s c o i n c i d e i f

...,L t )

(X;Lo,L1, i s a n e t ( c f . P r o p o s i t i o n 4.4 o f r 4 1 ) . I n t h e case o f m u t u a l l y o r t h o g o n a l

l a t i n squares, r o w - e x t e n d i b i l i t y i s e q u i v a l e n t t o t h e e x i s t e n c e o f a common

colLumz-tj.ans?~ersaZ ( i .e. a u s u a l t r a n s v e r s a l o f l a t i n squares w i t h t h e c o n d i t i o n "no two c e l l s a r e on t h e same row" r e p l a c e d by t h e weaker c o n d i t i o n " n o t a l l c e l l s a r e on t h e same r o w " ) ; see 141, P r o p o s i t i o n 4.2. Tv)

Let

,...,

(X;Lo,L1,...,Lt;T1,T2

be t h e t r a n s v e r s a l seminet a s s o c i a t e d t o t h e l a t i n squares

A1,A2,.

..,At.

194

M . Dezu urid T. Ilrringer

C l e a r l y , t h e l a t i n squares have a common column-transversal e x a c t l y i f t h e r e

Tvtl

e x i s t s a transversal

T v I . There i s no such

.. ,Lt)

(X;Lo,L1,.

i f the net

Tv+l

A1,A 2,...,At

if

(i.e.

of

with

a Lt+l:=

Tv+l

(X;Lo,L1,...,Lt,Lt+l)

i s an a f f i n e p l a n e

f o r m a complete s e t o f m u t u a l l y o r t h o g o n a l l a t i n s q u a r e s ) .

I n g e n e r a l , i t i s an open q u e s t i o n whether such a t r a n s v e r s a l e x i s t s i f

...,Lttl)

..,

tT1,T2,.

(X;Lo,L1,

i s a transversal-free n e t i n t h e sense o f Dow C101.

A l t h o u g h t h e n e x t p r o p o s i t i o n i s easy t o prove, t h e r e s u l t i s q u i t e s u r p r i s i n q .

a

4.2. PROPOSITION. 411 s e t s of muti*aZZy o?thogonaZ permutation arrays 0.7 the Examples 2.4 and 2.5 are row-extendible.

Proof.

G

Let

JL, and

of

let

associated t o

be t h e group used i n Example 2.4 o r 2.5 f o r t h e c o n s t r u c t i o n

r = (X;Lo,L1 ,...,Lt;TI,T2

G and t h e subgroups

,...,T v )

,..., S t

So,S1

X = G,

R e c a l l f r o m t h e p r o o f o f Theorem 2.3 t h a t

i E {O,l,

for all

...,tl,

4 . 1 and because o f

IT1,T2

,...,T v }

and

,...,F,

F1,F2

of

G.

I gcG1 = Isif( f t F 1 } k=1,2 ,...,s } . By Lemma

Li = ISig

= IFkg

1

gcG,

i s s u f f i c i e n t t o show t h a t t h e t r a n s v e r s a l

i s transversal-extendible.

seminet with

and

r ( R ) ,i t

J

be t h e t r a n s v e r s a l seminet

Let

Fl

=

{flfcF1)

b e a copy o f

F1

F1OX=O, and d e f i n e Y : = X u F 1 . A s a l l subgroups Si,

c o n j u g a t e t o each o t h e r , t h e r e e x i s t Mi:=

{Siaifut-f}

and

Li 2 tmnX

iaifl=Siaif2. S o n F1

I feF1}, i = O , l , ...,t, and d e f i n e I mcMi}. I n o r d e r t o show Li = ImnX I mcMi}, Then

le}

(a.S 1 0ay1)aifl 1

one o b t a i n s

Li = I m n X i mcMil.

fore

i=O,l,...,t, are -1 w i t h S. = a . S a , L e t 1 i o i Tv+l:= F1. T r i v i a l l y t h e n Y g X

...,at E G

ao,al,

= (a.S 1 0ar1)aif2 1

fl = f2. T h i s i m p l i e s

assume

and t h u s lLi I = ltmnX

The same argument i m p l i e s c o n d i t i o n

...,Mt).

I t remains t o show (S1).

satisfy

iz j, x r y

and

x,y

6

Let (Siaifl

i,j u

E

{O,l,

Ifl})

...,t } ,

n (S.a.f

3 5 2

fl,f2

E

I mcMi}

F 1 and From

flf;'cSo.

I, and t h e r e -

(S2) f o r (Y;Mo,M1,

fl,f2

IJ IT2}).

E

F1 and

As

x , y ~X

(X;Lo,Ll,~..,

X. Assume y k X. Then y=Tfl=f2 and x c S . a . f n S . a . f = (a.S aT1)a.f n (a.S aT1)a.f = aiSoflnajSofl. 1 1 1 J J 2 1 0 1 1 1 J O J J 1 and thus Si = S contraT h i s i m p l i e s xf;' E a.S n a.S Hence aiSo = a.S 1 0 JO' J O j' d i c t i n g i* j . 17 Lt)

i s a seminet, e i t h e r

x

or

y

cannot be c o n t a i n e d i n

The a l t e r n a t i n g group

f o r some o f t h e r e s u l t s o f t h e A 5 y i e l d s an example 2 p r e c e d i n g s e c t i o n s . Since A 5 and PSL(2,2 ) a r e isomorphic as p e r m u t a t i o n groups, b o t h o f t h e Examples 2.4 and 2.5 i m p l y t h e e x i s t e n c e o f a s e t

of

5

m u t u a l l y o r t h o g o n a l 25x12 p e r m u t a t i o n a r r a y s . By P r o p o s i t i o n 3.2, t h e s e t

59,

is

maximal, and by P r o p o s i t i o n 4.2 -9:

of

5

i s row-extendible, i . e . there e x i s t s a s e t

m u t u a l l y o r t h o g o n a l 2 6 x 1 2 p e r m u t a t i o n a r r a y s . I t i s unknown whether

i s row-extendible o r not.

195

Oit Pcrniutatiori Arra),s

5. TRANSVERSAL SEMINETS CARRYING DESIGNS

The concept o f t r a n s v e r s a l seminets comprises a l a r g e v a r i e t y o f d i f f e r e n t mathematical s t r u c t u r e s . F o r d e t a i l e d i n v e s t i g a t i o n s one has t h e r e f o r e t o add f u r t h e r r e s t r i c t i o n s . S e c t i o n s 2 and 4 and p a r t s o f S e c t i o n 3 t r e a t e d t r a n s v e r s a l seminets w i t h a group o f t r a n s l a t i o n s o p e r a t i n g r e g u l a r l y on t h e p o i n t s . I n t h i s s e c t i o n a d d i t i o n a l assumptions w i l l be imposed on t h e f o l l o w i n g i n c i d e n c e s t r u c t u r e s which a r e a s s o c i a t e d t o each t r a n s v e r s a l seminet T2,...,TV)

(let

o f l i n e s and

I(T,X)

-

T:= IT1,T2,

i.e.

L

J ):

t h e t r e a t m e n t s a r e t h e t r a n s v e r s a l s and t h e b l o c k s a r e t h e p o i n t s

J,

t h e t r e a t m e n t s a r e t h e p o i n t s and t h e b l o c k s a r e t h e l i n e s arid

The i n c i d e n c e s t r u c t u r e

J.

I(T,X)

exactly i f

J.I n

i s e s s e n t i a l l y t h e seminet o f

I(X,L

cases t h e in c i dence i s d e f ined n a t u r a l l y . For example,

I(T,X)

x r T.. 1

x

E

and

X

x,yt X

x,y

E

or

1

.

are i n -

a r e d e f i n e d t o be

p a r a l l e l i n t h e i t h p a r a l l e l c l a s s i f and o n l y i f t h e r e e x i s t s a l i n e I(T,X)

a l l three

Ti ,- T

i s the design w i t h mutually ortho-

gonal r e s o l u t i o n s mentioned i n S e c t i o n 1: The b l o c k s with

. . ,L t '- T 1' i s the set

'j',

blocks a r e the l i n e s o f

the transversa s o f

cident i n

= (X;Lo,L1,.

. . . ,T V l ,

t h e t r e a t m e n t s o f t h i s i n c i d e n c e s t r u c t u r e a r e t h e p o i n t s and t h e

of I(X,LUT)

and

the s e t o f transversals o f

T

-

I(X,L)

LouL1u ... uLt

L:=

1 t Li

Two i n t e r e s t i n g cases d i s c u s s e d l a t e r i n t h i s s e c t i o n o c c u r when

I(X,LuT)

a r e PBD's ( p a i r w i s e balanced d e s i g n s ) . F o r i n s t a n c e ,

has t h i s p r o p e r t y i f t h e a s s o c i a t e d s e t o f p e r m u t a t i o n a r r a y s i s a com-

I(X,LuT)

p l e t e s e t of m u t u a l l y o r t h o g o n a l l a t i n r e c t a f i g l e s ( t h e examples a f t e r P r o p o s i t i o n

5 . 3 show t h a t t h e converse o f t h i s s t a t e m e n t i s n o t t r u e ) . B e f o r e g o i n g i n t o det a i l , some d e f i n i t i o n s a r e necessary. The seminet

3=

exactly

n

ILi I = r

f o r a l l i ), and

has

..., L t )

(X;Lo,L1,

's

p o i n t s . I n t h i s case

5

i s c a l l e d n - r e p Z a r i f each l i n e c o n t a i n s i s a (tt1,r)-seniinet

i s a l s o c a l l e d an ( r , n ) - h n o

n s r, w i t h e q u a l i t y i f and o n l y i f

3

with

r = IXl/n

(i.e.

c o z f i p i m t i o z . One

i s a net or, equivalently, i f the

a s s o c i a t e d p e r m u t a t i o n a r r a y s a r e r o w - t r a n s i t i v e (see S e c t i o n 3 ) . An n - r e g u l a r t r a n s v e r s a l seminet s a t i s f i e s

n

5

v

w i t h e q u a l i t y i f and o n l y i f t h e T i ' s

rv = C. ITiI 2 1x1 = r n ) , 1=1,2, . . . ,v are pairwise d i s j o i n t . I n t h i s s i t u a t i o n

(because

t h e s e t o f t r a n s v e r s a l s can be c o n s i d e r e d as a new p a r a l l e l c l a s s

Lt+l,

associated permutation arrays are l a t i n rectangles. Therefore, i f

n = r = v, then

I(X,LIJT)

i s a (t+2,r)-net,

and t h e a s s o c i a t e d p e r m u t a t i o n a r r a y s a r e l a t i n

squares, The t r a n s v e r s a l seminet

and t h e

r= (X;Lo,L1

,..., Lt;T1,T *,.. .,Tv)

i s called

M . Deza and T, lhririger

196

q- uniform i f each p o i n t i s c o n t a i n e d i n e x a c t l y q t r a n s v e r s a l s . A q - u n i f o r m t r a n s v e r s a l seminet i s n - r e g u l a r w i t h n = v/q. Each column o f each o f t h e assoc i a t e d p e r m u t a t i o n a r r a y s c o n t a i n s each element

i e

..., r l

{1,2,

either

q

or

0

t i m e s . An example o f an n - r e g u l a r ( w i t h n=4) b u t not q - u n i f o r m t r a n s v e r s a l seminet i s p r o v i d e d by t h e o r t h o g o n a l p e r m u t a t i o n a r r a y s

IA1,A21 i s a row-extension o f two o r t h o g o n a l l a t i n squares. I t i s easy t o check t h a t A1,A2 have no o r t h o g o n a l mate.) (The s e t

For each p o i n t

x

t r a n s v e r s a l s through

o f a t r a n s v e r s a l seminet, l e t x, and d e f i n e

the incidence s t r u c t u r e treatments

I(T,X)

i : ={: I

incident with

E ;

The b l o c k s o f

I(T,X)

r

treatments. Notice t h a t have

ttl

denote t h e number o f

X E X I . F o r each t r a n s v e r s a l seminet

i s a 1-design

i s incident w i t h exactly

TicT

Y

Sr(l,i,v),

i . e . each o f t h e

b l o c k s , and each b l o c k I(T,X)

x

v is

may have repeated b l o c k s .

m u t u a l l y o r t h o g o n a l r e s o l u t i o n s ( i . e . any two

p a r a l l e l c l a s s e s o f d i s t i n c t r e s o l u t i o n s c o i n c i d e i n a t most one b l o c k ) . The r e s o Lo,L1 ,...,Lt. I f t h e r e i s a nonnegative i n t e g e r

l u t i o n s a r e g i v e n by

i j

A

with

ITinTT.l

J

( i . e . i f t h e t r a n s v e r s a l s f o r m an ( r , h ) - e q u i d i s t a n t

f

= A

for all

code), t h e n

PBD w i t h any two d i s t i n c t t r e a t m e n t s c o n t a i n e d i n e x a c t l y

becomes a

i,j

with

I(T,X) A

blocks.

E q u i v a l e n t l y , t h e a s s o c i a t e d p e r m u t a t i o n a r r a y s a r e equidistant w i t h Hammingdistance

r-A, 1 . e . any two rows o f each o f t h e p e r m u t a t i o n a r r a y s c o i n c i d e i n

exactly I(T,X)

p o s i t i o n s . I f , moreover, t h e t r a n s v e r s a l seminet i s q-uniform,

A

i s a 2-design

if I ( T , X )

i s an

case, any A'

t'

S,(2,q,v).

SA,(t',q,v),

As a consequence, with

t ' 22, then

then

r ( q - 1 ) = A ( V - 1 ) . Analogously,

(:,--\)

r (?,--ll) = h'

.

I n this

d i s t i n c t rows o f each o f t h e p e r m u t a t i o n a r r a y s c o i n c i d e i n e x a c t l y

positions.

The t r a n s v e r s a l seminets c o n s t r u c t e d i n t h e p r o o f o f Theorem 2.3 a r e n - r e g u l a r and q-uniform, w i t h n = -I G I and q = s . F o r none o f these examples I ( T , X ) i s a r PBD. Many examples o f t r a n s v e r s a l seminets a r e g i v e n i n r171, w i t h

I(T,X)

an

ITi n T . 1 < 1 f o r a l l d i s J t i n c t i , j ) . For i n s t a n c e , by Theorem 1 . 5 o f r171 t h e r e i s a p o s i t i v e i n t e g e r v1 such t h a t , f o r v z vl, t h e c o n d i t i o n v = 3 (mod 12) i s e q u i v a l e n t t o t h e e x i s SA,(t',q,v)

o r a g r o u p - d i v i s i b l e d e s i g n ( i n t h i s case

tence o f a t r a n s v e r s a l seminet S(2,3,v).

with

t

2

and t h e p r o p e r t y t h a t

I(T,X)

i s an

There a r e s i m i l a r r e s u l t s o f o t h e r a u t h o r s ; some o f them a r e l i s t e d here:

I97

On Pwmictation Arrays

1) D i n i t z l81.

F o r each p r i m e power o f t h e f o r m

a positive

c > l an odd i n t e g e r , t h e r e e x i s t s a t r a n s v e r s a l seminet such t h a t

t=c-1

I(T,X)

and

I(T,X) 3)

k

i n t e g e r and

i s an

i s an

S(2,2,qtl).

There e x i s t s a t r a n s v e r s a l semient such t h a t

Kramer e t a l . C161.

2)

k q = 2 c t 1, w i t h

t = 12

and

5(5,8,24).

Hartman C121.

There e x i s t t r a n s v e r s a l seminets such t h a t

S(3,4,v)

( v , t ) = (20,3),(32,5),(44,7).

for

I(T,X)

i s an

T r a n s v e r s a l seminets can be r e p r e s e n t e d by t h e f o l l o w i n g ( t t 1 ) - d i m e n s i o n a l array.of side r : the c e l l

(io,il,.

,,,it)

contains the s e t

1

ITicT

xcTi}

if

l ~ o n l ~ l n . . . n l ~ t= { X Iand , i t i s empty o t h e r w i s e ( t h e l i n e s o f each p a r a l l e l 1 2 r c l a s s a r e assumed t o be l i n e a r l y ordered, i . e . Lk = {lk,l ky...,lk}). I f I(T,X) i s an

S(2,2,v),

t h e n t h i s a r r a y i s c a l l e d a Room ( t t 1 ) - c u b e o f s i d e

such a r r a y i s e q u i v a l e n t t o size

(v-l)x(v-l)

(see 191). I f

(t+l)-dimensional S(5,8,24)

1-design

i s an

S,,(t',q,v),

t h e n t h e above

y i e l d s a 13-dimensional Room d e s i g n o f s i d e 253, see [161.

I 1I

Sttl(l,;

I(X,L)

I

I(X,L)

o f t h e seminet

I(X,L)

l c L } ,I X I )

becomes an

purullel structure) i f ments o f

I(T,X)

a r r a y i s a ( t + l ) - d i m e n s i o n a l Room design; f o r i n s t a n c e ,

The i n c i d e n c e s t r u c t u r e then

r = v - 1 . Each

p a i r w i s e o r t h o g o n a l symmetric l a t i n squares o f

ttl

(X;Lo,L l,...,Lt)

w i t h o u t repeated blocks. I f

5

Sttl(l,n,rn). I(X,L)

3=

3

is a

i s n-regular,

i s c a l l e d an An&& seminet ( o r a f f i n e

i s a l i n e a r space, i . e . i f any two d i s t i n c t t r e a t -

a r e i n c i d e n t w i t h e x a c t l y one b l o c k . O b v i o u s l y , AndrG seminets

do n o t a d m i t t r a n s v e r s a l s . A t r a n s v e r s a l seminet i s c a l l e d aZmost-Andr6

( o r com-

p l e t e ) i f any two d i s t i n c t p o i n t s a r e e i t h e r connected by a l i n e o r by a t r a n s v e r s a l . As each almost-Andre t r a n s v e r s a l seminet i s L-maximal, Lemma 3.1 y i e l d s 5.1, REMARK.

raiatecl s e t

Let t h e warisverstil seniinet

r)

fL(

0.fm : i t i t a

be aZmost-An&&.

Then. the asso-

2Zy mtho
A c t u a l l y , P r o p o s i t i o n 3.2 was proved e s s e n t i a l l y hy showing t h a t t h e i n v o l v e d t r a n s v e r s a l seminets a r e almost-AndrG. Recall t h a t a s e t o f c a l l e d .?ompZete i f 5.2. PROPOSITISN. sols. Then

t .. r-1,

t

mutually orthogonal l a t i n rectangles o f s i z e

vxr

is

t = r-1.

Let

be a trwi?suersiiZ ( t t 1 , r ) - s e m i n s t

wit;z e p n l i t y if

U ~ L oxLy !

with

if t h e associated s e t

v

trcrnsver-

A(3')

of

pemnutation arrags is a z o ! p i e t e se% of ni!tunllLg orthogonal L i t i n r e c t a n g l e s . I n

this i?me

I(X,LuT)

is dr:

S(P,jr,v},rv)

.

M. Dezu and T. Iliringiv

198

Proof. The inequality t 5 r-1 i s a t r i v i a l consequence of Corollary 3.5. Assume now t =r-I. By Corollary 3.6 then u = O f o r Jt(T) o r , equivalently, & ( T ) i s a s e t o f l a t i n rectangles. The completeness of fL(J) implies 7 t o be complete, i . e . any two elements of X are connected by a b l o c k from LuT. As T i r i T . = O for a l l i , j with i t j , one obtains t h a t I ( X , L u T ) i s a n J S(Z,{r,v),rv). n

Notice that complete sets of mutually orthogonal l a t i n rectangles have been constructed in Quattrocchi, Pellegrino C191 f o r a l l v n o t exceeding the smallest prime divisor of r For a transversal seminet (X;Lo,L1,.. .,Lt;T1,T2 ,...,T v ) and M c {1,2, ...,v l , M Z ~ ,l e t tM:= T ~ I, and define d : = E ~ ~ ~ , ~ ,Mz0 - ~ ,. .. ,v} Then

.

inicM

with equality if the transversal seminet i s almost-Andr6. 5 . 3 . PROPOSITION. Let s a k . Let Thev t =

w7

J

be a transversaZ (ttl,r)-semiMet i i ) i t h v

transuer-

be n-regular, and asswne I(X,LuT) to bc an S ( 2 , { r , n l , r n ) . V 1 , anti there are eractly transversals through. each x

n n-

n

t

X,

Proof. As the transversal seminet i s almost-Andr6, ( 1 ) i s valid with equality. Since I T i nT.1 5 1 for i z j , one o b t a i n s o = El:. tli',) J '-1 I V for a l l 1 - L and 1x1 = rn, ( 1 ) turns into

=

v(;).

With

lll=n

This can easily be transformed into the claimed equality for t . Let Q. be the number o f transversals through some x c X. Then v - l = r n - l = a ( r - 1 ) +( t + l ) ( n - 1 ) together with the equality j u s t proved yields Q. .=: 3 A class of examples satisfying the assumptions of Proposition 5 . 3 can be obtained as follows. Let (X;Lo.L1, ...,L r ) be an affine plane of order r . For kz2, consider the transversal seminet J = (X;Lo,L1,. . . ,Lr-k;T1,T2,. .. ,Trk) L i . Then satiswhere T1 ,T2,. . . ,Trk are the lines contained in Ur-k
'

On Permurariorr A F T U ~

199

v = n r . Together w i t h P r o p o s i t i o n 5.3, t h e

w i t h the additional property that

r e s u l t s o f [131 and [51 on t h e s e designs i m p l y 5.4.

PROPOSITION.

S (2, I r,nl, r n )

A n n - r e g u l a r transversa2 ( t + l , r ) - s e m i n e t

I ( X,LuT)

with

an

ezists

a)

i n t h e case

n = 2, r ?3

if a n d onZy if e i h t e r t = 0, r = 3 o r t = r-1,

bi

in the case

n = 3, r = 2

if and onZy if

t = 0,

ci

in t h e case

n=3, r = 4

if and onZy if

t = O

or

t=3.

N o t i c e t h a t a l l seminets i n t h e above p r o p o s i t i o n a r e e i t h e r complete s e t s o f m u t u a l l y o r t h o g o n a l ? a t i n r e c t a n g l e s ( t = r-1) or t r a n s v e r s a l d e s i g n s (t=O).

6. SOME STRUCTURES RELATED TO TRANSVERSAL SEMINETS I t i s well-known t h a t n e t s a r e e q u i v a l e n t t o s e t s o f m u t u a l l y o r t h o g o n a l l a t i n

squares, t o t r a n s v e r s a l d e s i g n s ( v i a d u a l i t y ) , t o o r t h o g o n a l a r r a y s , t o o p t i m a l codes, e t c .

P r o p o s i t i o n 2.1 i s a g e n e r a l i z a t i o n o f t h e f i r s t o f t h e s e e q u i v a l e n -

ces. Next, t h e second e q u i v a l e n c e i s g e n e r a l i z e d t o t r a n s v e r s a l seminets.

.

Each t r a n s v e r s a l s e v i n e t = (X;Lo,L1,. . ,Lt;T1,T2,. . . ,Tv) i s e q u i v a l e n t t o a ~ P ~ ~ ~ ~ J C Ip aW cJk Li n g , v i a t h e i n c i d e n c e s t r u c t u r e I ( L , X ) . Each b l o c k X C X h i t s each o f t h e gyi"7u:s t i n c t groups

Li,L.

L.

1

J

i n e x a c t l y one t r e a t m e n t

lcLi,

two t r e a t m e n t s f r o m d i s -

a r e j o i n e d b y a t most one b l o c k , and t h e r e i s no such b l o c k

if

L . =L.. Moreover, t h e r e a r e a d d i t i o n a l mzin tr.eatmeurts Ti; 1 $1 t i o n s t h e treatments o f I(L,X) i n t o d i s j o i n t blocks.

each

Ti

parti-

...

and

Supoose now X and each Li t o be l i n e a r l y o r d e r e d , i . e . X = Ix1,x2, I 1 2 r Li = { l i 7 1 .,..., l j 1 . L e t t h e m a t r i x B = ( b . . ) o f s i z e ( t t 1 ) X I X I be d e f i n e d 1J

1

b . . = k :0 x. i . T h i s m a t r i x i s an OA (orthogomZ a r r a y ) i f t h e seminet 1J J i s a n e t ( s e e e.g. 191). 11.1 t h e general case, t h e s e t o f a l l 1x1 columns o f B

by

forms a code o f ZeilgLh

t+l o v e r t h e a l p h a b e t

{1,2,

...,r l

with

1x1

w r d s and

distance t , s i n c e any two d i s t i n c t columns c o i n c i d e i n a t most one p o s i t i o n . Each t r a n s v e r s a l T. corresponds t o a f a m i l y o f codewords (columns o f 6 ) J such t h a t , f o r a l l k ' 11,2 ~ ,...,r I and a l l i c [0,1, . . . ,t l , t h e r e i s e x a c t l y one codeword i n t h i s f a m i l y w i t h v a l u e k i n row i , The t r a n s v e r s a l T . can a l s o

miriimaZ

be regarded as an ii!je,:tioe

, i i a g o n r l subset o f

r words o f l e n g t h t + l o v e r

{l,Z,

..., r ?

{1,2,

. . . ,rIttl

J ( i . e . as a s e t o f

which d i f f e r i n an-y c o o r d i n a t e ; see

161). L e t the associated permutation arrays or, equivalently, l e t Doints

xcX

Ti i . , T . = 0

J

and o f t h e l i n e s

li.Li

for a l l (i21)

A1,A2,

...,At

now b e l a t i n r e c t a n g l e s

i , j , i z j . Assume t h e numbering o f t h e

now be more s p e c i a l t h a n above: L e t

M . Deza and T. Ihririger

2 00

k l = l k if lnT1zlo, andlet x=x i j these assumptions t h e f i r s t row o f B

m.1,

for

t h e m t h row o f

i.e.

'm-19

(bm,(i-l)r+l

9

(i :;)

of

A

b

U

=k

if

k xcTinlo.

Under

j = k (mod r )

and,

bm,(i-1)r+2

3..

3

i s t h e i t h row o f

bm,ir)

corresponds now t o t h e s e t o f columns

Ti

Am-l.

(i-l)r+l

,

o f B . F o r example, c o n s i d e r i n g t h e s i m p l e complete s e t A1 = 1 2 3 2) o f m u t u a l l y o r t h o g o n a l l a t i n r e c t a n g l e s , one o b t a i n s A2 = (3

,

The p e r m u t a t i o n a r r a y s of

satisfies

if

.. . , i r

(i-l)rt2,

j

j=(i-1)rtk

becomes a " l i n e a r i z a t i o n " o f t h e l a t i n r e c t a n g l e

B

Notice t h a t the transversal

Q(A,A')

with

A

and

A'

a r e s i m i l a r e x a c t l y i f , f o r a l l j , column

can be o b t a i n e d f r o m column

A'

be t h e become

Q(

r x r

k's

matrix

(q..)

i n column

[; :;)' (;

'J

J

:23)

j

of

d e f i n e d by of

=

A ' , and

A

by renaming t h e symbols. L e t

qij= k

i f the

qij=*

o t h e r w i s e . For example,

i's

i n column

j

(; ;:). * 1 3

As a l l p e r m u t a t i o n a r r a y s a r e assumed t o be s t a n d a r d i z e d ,

qii

=i

for all

Q ( A , A ' ) . I n no column o f t h i s m a t r i x a symbol a p e a r s t w i c e . T h e r e f o r e

i in

Q(A,A')

can be c o n s i d e r e d as ( t h e m u l t i p l i c a t i o n t a b l e o f ) a p a r t i a l l e f t - c a n c e l l a t i v e g r o u p o i d d e f i n e d on

{1,2,

..., r l .

The p e r m u t a t i o n a r r a y s

A,A'

are orthogonal i f

Q ( A , A ' ) t h e n becomes i s a complete quasigroup i f and o n l y i f

and o n l y i f t h i s g r o u p o i d d l s o i s r i g h t - c a n c e l l a t i v e , and a p a r t i a l quasigroup. Moreover,

Q(A,A')

the permutation arrays are row-transitive.

Let

A1,A2¶.

. . ,At

be s i m i l a r permuta-

t i o n a r r a y s . By P r o p o s i t i o n 1.2 o f 141, these a r r a y s a r e m u t u a l l y o r t h o g o n a l exactly i f

Q(A1,A2j,

Q(A1,A3),

. .. , Q(A1,At)

are mutually orthogonal p a r t i a l

quasigroups. N o t i c e t h a t t h e a s s o c i a t e d t r a n s v e r s a l seminet i s n - r e g u l a r i f and Q(Ai,A.), i z j , has t h e f o l l o w i n g p r o p e r t i e s : Each i E (1,2, ...,r } J appears e x a c t l y n times, and t h e r e a r e e x a c t l y n d i s t i n c t symbols i n each rcw

o n l y i f each

and i n each column. There a r e many i n c i d e n c e s t r u c t u r e s b u i l t f r o m n e t s . Examples a r e t r a n s v e r s a l geometries 161, d-dimensional n e t s 1211, and e x t e n s i o n s o f d u a l a f f i n e planes

C231. I t w i l l be i n t e r e s t i n g t o s t u d y s i m i l a r s t r u c t u r e s w i t h n e t s r e p l a c e d by t r a n s v e r s a l seminets.

On Permutation Arrays

101

REFERENCES

[I1

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r21

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131

J. Andr'e, u b e r P a r a l l e l s t r u k t u r e n . T e i l 11: T r a n s l a t i o n s s t r u k t u r e n . Math. 2. 76 (1961) 155-163.

141

A. B o n i s o l i and M. Deza, Orthogonal p e r m u t a t i o n a r r a y s and r e l a t e d s t r u c t u r e s . Acta U ~ i u .CuroZirzue 24 (1983) 23-38.

151

A.E. Brouwer, A. S c h r i j v e r and 1-1. Hanani, Group d i v i s i b l e d e s i g n s w i t h b l o c k s i z e few. Discrete ?.k&zerriatks 20 (1977) 1-10.

I61

M. Deza and P. F r a n k l , I n j e c t i o n geometries. J . Comb. Theory (B) 37 (1984) 31-40.

I71

M. Deza, R.C. M u l l i n and S.A. (1978) 322-330.

181

J . D i n i t z , New l o w e r bounds f o r t h e number o f p a i r w i s e o r t h o g o n a l symmetric l a t i n squares. Cong~essus1"hmercmtium 22 (1979) 393-398.

191

J . D i n i t z and D.R. S t i n s o n , The spectrum o f Room cubes. E d r o p . J . Combinatorics 2 (1981) 221-230.

r 101

S . Dow, T r a n s v e r s a l - f r e e n e t s o f s m a l l d e f i c i e n c y . A w h . Math. 41 (1983) 472-474.

Vanstone, Orthogonal systems. ileq. r b t h . 17

I 1 1 1 W . F e i t , Some consequences o f t h e c l a s s i f i c a t i o n of f i n i t e s i m p l e groups. Pruiteedin2s of i'ymposic; ix Fur-e kfu.athematics 37 ( 1980) 175-181.

r 122

A. Hartman, Doubly and o r t h o g o n a l l y r e s o l v a b l e q u a d r u p l e systems. I n : R.W. Robinson e t a l . ( e d s . ) , i,'~~!t,i,zat.jr?:uZ M~ithe.oiaticsVII. L e c t u r e Notes i n Mathematics 829 ( S p r i n g e r , B e r l i n H e i d e l b e r g New York, 1980).

r 131

Ch. Huang, E. Mendelsohn and A. Rosa, On p a r t i a l l y r e s o l v a b l e L - p a r t i t i o n s . AvnuZs of iiiscrele k t h e m c t i c s 12 (1982) 169-183.

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CY14pp?7!

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r 151 D.

J u n g n i c k e l , E x i s t e n c e r e s u l t s f o r t r a n s l a t i o n n e t s . I n : P.J. Cameron e t a1 , (eds . ) , Ipinitc geonetries uid d e s i g i i s . L e c t u r e Notes London Math. SOC. 49 (Cambridge U n i v e r s i t i y Press, Cambridge New York, 1981).

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E.S.

Kramer, S.S.

M a g l i v e r a s and D.M.

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J . Comb, Theory ( A ) 29 (1980) 166-173.

l17J

E.R. Lamken and S.A. Vanstone, Designs w i t h m u t u a l l y o r t h o g o n a l r e s o l u t i o n s . Ezirop. J . Curnbimtorics ( t o a p p e a r ) .

r 181

M. Marchi, P a r t i t i o n loops and a f f i n e geometries. I n : P.J. Cameron e t a l . (eds ) , Finite 2eometr.ies m i designs. L e c t u r e Notes London Math. SOC , 49 (Cambridge U n i v e r s i t y Press, Cambridge New Y G r k , 1981).

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Sprague, T r a n s l a t i o n n e t s . ~ i t c .,..,th.

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This Page Intentionally Left Blank

Annals of Discrete Matliematics 30 (1986) 203-216 0 Elsevier Science Publishers B.V. (Nortlr.Hollatid)

PASCALIAN CONFIGURATIONS I N PROJECTIVE PLANES

Giorgio Faina D i p a r t i m e n t o d i Matematica Universita d i Perugia 06100 P e r u g i a ITALIA

INTRODUCTION Following T i t s [19],

l e t P he a p r o j e c t i v e Oval of a p r o j e c t i v e

p l a n e TI and l e t P ( d ) be t h e f i g u r e formed by a i l o f t h e s e c a n t s o r t a n g e n t s t o 5: which a r e p a s c a l i a n l i n e s w i t h r e s p e c t t o R (see [4,p.

3701).

The f i g u r e s P ( 2 ) a r e c a l l e d ? - p a s C a l i a n

configurations

of 2. The p r o b l e m o f d e t e r m i n i n g t h e c o n f i g u r a t i o n P(G) was i n t r o d u c e d by F. E u e k e n h o u t i n [ 4 ] a n d , by u s i n g t h e s e c o n f i g u r a t i o n s , i t i s

p o s s i b l e t o produce an i n t e r e s t i n g c l a s s i f i c a t i o n f o r t h e p r o j e c t i v e o v a l s (see, a l s o , [ 7 ] ) . The p u r p o s e of t h i s p a p e r i s t c p r o v e t h e following r e s u l t :

I t seems e x t r e m e l y i n t e r e s t i n g t o m e n t i o n t h a t i n a G a l o i s p l a n e

h P G ( 2 , q ) , q = p , t h e p r o b l e m of d e t e r m i n i n g t h e non-empty R - p a s c a l i a n configurations i s completely resolved. I f o f S e g r e [18],

q

i s o d d , by t h e t h e o r e m

t h e o v a l s a r e c o n i c s , f o r which e a c h l i n e i s p a s c a l i a n

(see [4, p. 3 7 2 1 ) .

I f instead

q

i s e v e n t h e n t h e r e is, o t h e r t h a n

G. Fairia

204

the class of conics, another class of ovals (called L k a n . & U v n ow& [lo]) which have a single non-exterior pascalian line and such a line is a tangent (see [lo]). We also note that there are projective ovals having non-exterior pascalian lines in non-desarguesian planes, but in this setting the problem is very far from being resolved (see [4, p. 3801 , [21] 1 . At present two other R-pascalian configurations were found: a) If G is the Wagner's oval [21]

then the R-pascalian configuration

contains a unique line (see [7] : Buekenhout also discovered that this line is a tangent to S7 (see 1 4 1 ) b ) If R is the Tits ovoide 2i

.

&abt&ldon [19],

set of tangents to R (see [4, p. 3821 and

then P(Q) coincides with the

171).

So we see that in 2 ) and 3) of our Theoran we have exhibited

new types of

2-pascalian configurations.

1. DEFINITIONS

PRELlMINARy RESllLTs

For definitions of the terms projectibe plane, aollineation, translation plare, desarguesian plane and near-field see, for exanple, Segre [MI. A phojective u u d is defined (see [19]) as a set of pints R of a projective

plane I such that

M)

three are collinear and through each there passes one and

only ore line (the tangent) that contains IY) other pints of R.

is a 6-ple (a a a b b b ) of p i n t s of R , m t recessarily 1' 2' 3' 1' 2' 3 distinct, such that : ai,bj # aj,bi, ai#aj and b #b for i#j and ai#bi i j (i,j=lr2,3), where a , h is a w e n t of R when a.=b Hexagons have three 11i j' (ifj) and are called pcrlc&aii when distinct hiagorule p o i a t h a.,b,n

A hexagon of

c)

_ I

1

_ I .

3

c,% J i

these three points are collinear. The €amus theorem of Euekenhout

143 m y

be

stated as: Id each i a c h i b e d lwxagot1 in a pm jcc..Citive v u d R 0 p a c d i a n -then R in a t u n i c i n a p o jeotiwe pappian ptane.

Let R be a projective oval ii1 a projective plane TI. A line R of

R

205

Pusculiatr Corij~gurutiorzs111 Projective Planes

i s c a l l e d R-puncae-Lun i f e a c h hexagon i n s c r i b e d i n R which h a s t w o d i a g o n a l p o i n t s on L a l s o h a s t h e t h i r d d i a g o n a l p o i n t on 1. The

II which a r e

f i g u r e P ( Q ) formed by a l l o f t h e p a s c a l i a n l i n e s o f

s e c a n t o r t a n g e n t t o R i s c a l l e d R - p a d c a L h n c u n d i g u f i a t i o n o f II.

Oval d o u b l e l o o p s

1.1.

I t h a s b e e n o b s e r v e d by many a u t h o r s ( [4]

, [ 6 ], [7] 1 ,

that a projecti-

ve o v a l may be u s e d t o d e f i n e o p e r a t i o n s o f a d d i t i o n @ and m u l t i on i t s p o i n t s . W e w i l l d e s c r i b e a method f o r d o i n g t h i s

plication

which i s a s l i g h t m o d i f i c a t i o n o f B u e k e n h o u t ' s p r o c e d u r e [ 4 , p.

3731

w i l l c a l l e d a n o v a l dou6Le

The r e s u l t i n g a l g e b r a i c s y s t e m (Q ; @ , O ) T

Loop. Let

R be a p r o j e c t i v e oval i n a p r o j e c t i v e p l a n e ll. A r b i t r a r i l y

s e l e c t t h r e e p o i n t s on R and l a b e l them

ylo,i

p o i n t of i n t e r s e c t i o n of t h e t a n g e n t s a t p o i n t s of R, o t h e r than

y

and t h e n l a b e l t h e

o

and

i s a s s i g n e d a and

o

is assigned 1 .

i

I f a and b a r e t h e symbols a s s i g n e d t o t h e p o i n t s

if 0;

in

},

a#o if

x . The

y , a r e t h e n a r b i t r a r i l y a s s i g n e d symbols

with the r e s t r i c t i o n t h a t

Q\&

with

a

and

b

of

we d e f i n e t h e sum a@6 i n t h e f o l l o w i n g way:

-

then the l i n e a=o

then l e t

a point

z

w i l l m e e t R i n a point

a,x

a'=o. If

o t h e r than

bfo

x; i f

then t h e l i n e b=o

then l e t

a'

o,b

other than

meets

z=x. L e t

x,y

C be t h e

~

p o i n t o f i n t e r s e c t i o n of t h e l i n e

a ' ,z n R } \ { a ? =pl

then

c=a'

a',z

. Letting

w i t h QN{a'j; i f

c be t h e symbol a s s i g n e d t o

t h i s p o i n t we d e f i n e u616=c. If

a r b E R \ { o l y ] , w e d e f i n e t h e p r o d u c t a o b as f o l l o w s :

the tangent a t then the l i n e

i

b,j

__

meets t h e s e c a n t

o,y

i n a point

w i l l m e e t R i n another point

j; i f

b'#i;

if

a,i

b#i b=i

with t h e then l e t b ' = i . L e t t h e i n t e r s e c t i o n of t h e l i n e l i n e o , y be h. L e t c be t h e p o i n t o f i n t e r s e c t i o n of t h e l i n e

-

h,b'

-

with t h e s e t R\{b'] ; i f { h,b'nR}

N

jb'} =$

then

c=b'. Letting

c be t h e symbol a s s i g n e d t o t h i s p o i n t w e d e f i n e a 0 6 = c . I f w e c a l l t h e s e t of symbols u s e d

Q , then i t i s e a s i l y seen t h a t

.

G.Fuiria

206

0

together with the operations defined above is a double loop which

is denoted by where

which is also called an o v a l double l o o p ,

(Q,;@,')

{o,y,i}. This leads us to

S:=

LEMMA 1.[6]- The loop (Q

S

-

the line

,@)

is an abelian group if, and only if, i-

is a R-pascalian line; the loop (Q,,')

x,y

group if, and only if, the line

+ Qs:=Qx{o}.

cy

is a R-pascalian line, where

It is not difficult to verify that every point fied with a involutorial permutation

is an abelian

I(p)

p€II\R can be identi-

of the points of R as

follows: if

p ~ u \ R ,two points of R are a pair in the involutorial permutation if they lie on the same line through

I(p)

LEMMA 2 . [ 4 ] -

p.

For a line l of a projective plane containing a pro-

jective oval R , the following are equivalent: 1)

.i? is

R-pascalian, and

2) for each triple of involutions I(p) , I ( q ) , I (r) with centers

on l, the composition

I(p)I(q)I(r)

is also an involution

with center on 1. An a u t o m o 4 p h i n m of a projective oval R is a permutation $ of the points of R which preserves the involutorial permutations I(p), where

v

p~ll\R,that is to say:

PErI\R,

3!

q&n\R

: $ I(p)lp=I(q).

The automorphisms of R form a group. Denote this group by

AutR.The

following is easily proven: each c u L l k n e a t i o n mirkph44tn 0 6

06

ll t h a t pekmuteh R i n t o i t b e t 6 i n d u c e s an a u t o -

0.

We also have the following result.

LEMMA 3.[4]-

Let 52 be a projective oval of a projective plane Il and

let a be a collineation of TI that permutes R into itself. The line

207

Pascaliaii Coiifigirrarioiisiir Projwtive PIaiies _.

x,y

,

where

x,y~:R,is a R-pascalian line if, and only if, the line

(x),w.(y) is a (1-pascalian line.

1.2. The near-field of order nine (Andrb [l]) Let

2

x =-1

an irriducible quadratic over GF(3). Let

of all elements of the form where we assume

on K

a+bi

as

a

and

b

K

be the set

vary over GF(3),

2

i =-1. We wish to define an addition and a product

in such a way that, using the field GF(3) addition, K will

be a near-field. We define the addition as follows (a+bi)+ (c+di):= (a+c)+ (b+d)i f for all

a,b,c,deGF (3).

We define the product in the following way: ai=ia, for all

a€GF(3)

a(btc)=ab+ac, for all ab+ba=O, for all

a,b,c~K

a,bEK\GF(3), where

afb

and

a+b#O.

It is evident that (K,+) is an abelian group and that K\{O} is a group. G’ven

(a+b#O), there is a unique

a,b,cEK

XEK

such that

ax+bx+c=O. 2

Finally, a =-1

for all

acK\GF(3).

1.3. The non-desarguesian translation plane of order 9 (Andr6 [l] From the Rear-field of order 9 K

)

we may now construct a transla0

tion affine plane of order nine, denoted by T

,

as follows (see,

for example, [1] :

-

points are the pairs (x,y) for all

x,y~K;

lines are defined as sets of points (x,y) whose coordinates x,y satisfy an equation of one of the forms (aEK),

(1) x=a

(2)

y=ax+b

(a,bcK).

There is, up to isomorphism, a unique projective plane that

0

T =T\{d}

infinity of same

TO.

To

for a line d of

T

such

T , where d is called the line at

and its points are called points at infinity of the

C.Faitia

208

If

p

is the point at infinity of

y=ax+b

is the point at infinity of

by (a). If

p

denoted by

(m).

then it will be denoted x=a

then it will be

It has been shown by Denniston [5] and Nizette [14] that in the translation non-desarguesian plane of order nine dual

T

(and in its

T I ) the ovals fall into a single transitivity class under the

collineation group. The self-duality property make it unnecessary to study

and T'

T

jective oval in

Rodriguez )6] the group oval of

T

separately; so the following example of pro-

T will suffices:

discovered the oval

AutR

of

32

collineations that leaves invariant an

and proved that AutR

I J

x' =ix-iy y' =ix+iy

and Nizette [14] has studied

R

have generators

x ' =-x

x'=x

Y"Y

with

io=-ir a € Aut K.

1.4.

The Hughes plane of order nine (Zappa [22])

From the near-field of order plane, denoted by

H

9 K

we may now construct a projective

as follows:

- the points of H are the triplets

( x ,x ,x 1 , where

x.EK, other 1 2 3 than ( O , O , O ) with the identification (x1,x2,x3)=(kxl,kx rkx3) for 2 all non-zero k in K;

- the lines of H will now be the sets of points satisfy an equation of the form any automorphism of

K

(x,y,z) which

x+yt+z=O, teK, such that if cf is

then the mapping

x'=a xa+b yo+c zu 1 1 1

z'=a xO+b yO+c z(J 3 3 3 with

(VaitbircicGF3), i=1,2,3)

det(a,,b,,c,)#O, is a collineation of

H.

209

Pascaliun Configurations in Projective Planes

Denniston [ 5 ] and Nizette [14] have discovered that in the Hughes plane

H, ovals fall into two transitivity classes under the colli-

neation group of 48

riant under

H. An oval,

D , in one of these classes is inva-

collineations, as against

16

collineations for

the other class. So the following examples of non-isomorphic projective ovals of

H

will suffice:

N={(l,i,O), (1,-i-l,O),(l,-l,i+l),(l,-l,-i-l), (O,i,l),(O,i,-l), (l,~,-i-l),(l,~,i+l), (l,l,i),(l,l,-i)l. In [14], Nizette proved also that

AutN have generators

a

a

x'=x-y

x =-x

x'=x +y

y 1 =-y

y'=x+y

a a y'=x - y

Z'=Z

Z'=Z

1

0

(with i'=-i-l).

2. PASCALIAN CONFIGURATIONS JJ Let

R

T

AND IN

T'

be the Rodriguez oval of the non-desarguesian translation

plane of order nine T. First we show that each non-exterior line (to R ) through the point ( 0 , O ) is a R-pascalian line. Let the tangent

y=ix

(O,l)=i, (0,-i)=-i,

(i,-i)=2i+l, (-i,i)=2+i, (i,i)=2+2i, (-i,-i)=l+i Q

denote

and label the points of R in the following way:

(-i)=O, (i)=-, (-l,O)=l, ( 1 , 0 ) = 2 ,

Letting

.t

be the symbol assigned to the set Rx{(i)l

coinciding with the set of elements of the near-field the following triplet of points on R :

. (i.e. Q K),

is

we select

G. Fabra

310

By

1.1, the algebraic system (QS;@,O) is an oval double loop. Now, with a straightforward proof which we omit for shortness, we can to check that a@b=a+b, for all

a,bEQ (1.e. for all

a,bEK).

Since (K,+) is an abelian group, we have that (QS ,@)

is an abelian

group. Therefore, by Lemma 1, we have that the tangent at

(i)=-

is a R-pascalian line. Also, s h c e AutR {

is a transitive permutation group on the set

,

(i), (-i)I C R (see [ 4 ] )

is

by Lemma 3 , we have that the tangent y=-ix

R-pascalian.

NOW, in order to show that the secant

x=O

it is only necessary to prove that, for + o

loop ( Q s ,

)

, where

+ QS=R\{ (0,l), (0,-1)1 ,

is a

S=I (0,l), (O,-l), (i)1 , the is an abelian group. A very

long, but straightforward computation, shows it. that (see [ 1 4 ] )

AutR

fixes the point

tive on the points of R*{(i),(-i)}. all non-exterior lines through

R-pascalian line,

It is well known

( 0 , O ) and that it is transi-

Thus, from Lemma 3, we have that

( 0 , O ) are R-pascalian.

Now we will prove the non-existence of non-exterior R-pascalian (0,O).The points of R may, for

lines not passing through the point shortness, be denoted by digits from

0

to

9

as follows:

!i)=o, (-i)=1, (-1,0)=2, (1,O)= 3 , (i,-i)=4 ,(-i ,i)=5 ,(1,i)=6 ,(-i,-I)=7 , (0,1)=8 and (0,-1)=9. By [4, p. 3831 and [20, table 32/34],

it follows that, if we denote

by G(8) the group of all elements in AutR

then

IG(8)

1=4

and that

G(8)={f

f

f

which fix the point

8,

f 1 , where

1’ 2 ’ 3 ’ 4

Finally, since AutR acts transitively on R\{ (i), (-i)1 , by Lemma 3, the only thing remaining to be shown is that the lines

- - - - 8r8

I

5,8

r

O,8

x e not R-pascalian.

i

218

r

0,l

21 I

Pascalinn Configurations in Projective Planes

First of all, consider the following points of ll\R: __-

-~

-~

--

-__

p =1,9r18,8, p =1,6n8,8, p =5,8nO,O, p =5,8nOI2, p =0,8fll,l, 1 2 3 4 5

--

--

--

~~

____

p =0,8ni,2, p =2,8no,o, p =2,8noI3,p =0,1n2,2, p =0,1n2,4, 6 7 8 9 10 --

pl1=0,1fl2,6. Now, without giving the proofs (which are straightforward but time-

consuming) we remark that

p1,p2,p3~8,8 but

I(p1)I(p2)I(p3)

is

-

not a involutory permutation of R with center on

8,8; thus, by

__

Lemma 2, it follows that the line

8,8

is not R-pascalian.

Repeating this process, replacing

I (p,) I (p,) I (p,)

by I (p,) I (p4)I(p,)

,

~

gives that the line

5'8

Similarly: I (p5)I (P,) I (P,)

is not R-pascalian.

,

I (P,) I (P,) I (p7)I I (P,) I (pl0)I (pll) are -

not involutory permutations of R with center in

~

0,8, 2'8, 0,1,

respectively. Hence these lines are not R-pascalian.

The R'-pascalian configuration of the dual

T'

of

T

is again of

the same type and we omit the analogous proof.

3. PASCALIAN CONFIGURATIONS

IN

H

In [8], Hughes reproduces the plane

H

in the useful following way:

- the points are the symbols A i , B i ,C.,D ,E.,FifGifi=O,1,..., 12; i i i -seven of the lines are the following sets of points 1) IAO,A1 'A3 I Ag,Bo tCo I Do I Eo I Fo 'Go 1 2) IAotB1rB8tD3 'Dll

tE2 tE5 ,E6,G7,Gg1

-

217

G.

G){A

0'

C

7'

C,D 9

D

2'

FUiflO

D E E F,F) 5' 6' 3' 11' 1 8

7) {AOiB3rB11 r C 2 rC5rC6 r D7 i D g rG1 rG8);

- the remaining lines are found by successively adding one to the sub-scripts, reducing modulo 13. In this notation, we remark that (see [5] and [ 1 4 ] )

the ovals D und

N of section 1.4 are the following sets of symbols:

B0rE0rC 6 r D 6 r C 7 rD 7 I

P'{A4'A5'All'A12'

N={B

0'

c0' c 4 ,G4' c 6 ' D 6' B 7' F 7' B ll'E1l'

'

We first show that the D-pascalian configuration is the empty set. The suggestive term JLeaL is used for the points A

of H I then in D i there are four real points and six i m a g i n a h y points. It is well

known that (see [5]

, [14])

IAutPj=48 and we note further important

properties: 1) Auto is generated by: (A11A12)(BoEo)( C 6 C 7 ) ( D 6 D 7 )

2) AutD is transitive on the set of real points of 0; 3 ) AutD is transitive on the set of imaginary points of

D;

4)

I (AutD)xl=12

5)

(AutD)x is transitive on the set of imaginary points of D for

for all real point

XED;

all real point xrD; 6 ) if

is a real point of D , then

x

set of real points of 7 ) if

(AutD)x is transitive on the

D.{xl;

is a imaginary point of D then

y

I (Autp)

Y

1=8;

8) (AutD) =(AutD) : BO EO 9) (AutD) BO

is transitive on the set {C

D C D I. 6' 6 ' 7' 7

We omit the proof which is very long, but not difficult. By the above properties of AutP and Lemma 3 , the only thing remaining

213

Pascalian Configurations in Projective Planes

to be shown is that no one of the lines

- - - A4rA4r A12iA12

I

A4iA5

r

BOrEO

I

BOrC6

I

A4rA12

is D-pascalian. As in the proof of section 2, it is sufficient to exhibit some appropriate involutorial permutation of the points of

D

. First of

all, consider the following points of i l \ P

:

Now we remark that the following permutations

are not involutory permutations of type in

-

I(p) with the centers p

~

-

-

A4,A4, A12,A12, A4,A5, A4,A12, B O I E O ~BO'C6

respective1.y. Hence, by Lemma 2, these lines are not D-pascalian. Finally, we must show that the N-pascalian configuration is the empty set too. Also in this case, it is well known that (see [5] and [14]) 1AutNI=16 and it is not difficult to check that:

1) AutN is generated by

X=(C C F E G D B B

4 6 7 1 1 4 6 7 11

and

2) AutN fixes the set {Bo,Co}; 3 ) AutN is transitive on the sets {B

0'

C

0

1

and I=N\$ 0 ,C0 1 respecti-

vely; 4) (AutN) is transitive on 1; BO 5 ) (AutN) =(AutN) : BO cO =(AutN) =(Id,ul G4

.

Hence, in order to prove that P(N)=gf, it is sufficient to show that no one of the following lines is N-pascalian:

G. Faina

214

A repetitionofthe arguments used in the earlier proof of this CtiOn

98-

shows that the permutations

are not involutory permutations of the points of N

with center in

the above mentioned lines, respectively, wile we have that:

--

C41G4, E 8' E 0EC 4 ' C 6' C 5' G0EC 4' D 6' E 2' A 1EC 4IB7' Hence, by Lemma 2, no one of these lines is a N-pascalian line.

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ad un ,teahema di B. Sqhe

3. U. BARTOCCI, Condidekazioni d d h Universita di Rcana (1967).

&O&a

4.

F. BUEKENHOUT, 333-393.

5. R.H.F.

d&e

ova&

6u.i

q-mcki,

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06

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6. G. FAINA, Sut d o p a cappio ahdociato ad un ova&, (5) 15-A (19781, 440-443.

eOll. Un. Mat. Ital.

7. G. FAINA, Un m66inamento deRea o e a 6 b i 6 i c a z i o f l e di BuetzenhoLLt peh g k 5 o v d i a b w , Boll. Un. Mat. Ital. (5) 16-B (19791, 813-825. 8. D.R. HUGHES, A ceadd 9 (1957), 378-388.

06

n o n - P t b a q u u h n p o j e c f i v e p h n e ~ ,Canad. J . Math.

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Od

215

Pascalian Configurations in Projective Planes

10. G. K O R C W R O S , SuRee ow& di .Ouu&zhne i n un piano di G d v D di ohdine p&, Rend. Accad. Naz. XL (5) 3 (1977-78),55-65. 11. G. KORCHMAROS, U n a geneA&zzaLane d d teahema di Buehenhout A&e ow& p a s c a l h n e , Boll. Un. Mat. ItdL. (5) 18-B (19811, 673-687. 12. R. MAGARI, Le c o n d i g m a z i o n i p a h z i ~ckiube contevwte net piano P 4ut quabicohpo a ~ n o c i a t i w odi ohdine 9 , Boll. Un. Mat. Ital. (3) 13 (1958), 128-140. 13. G. M E N I C H E T T I , Sapha i k-UJLCki [email protected] n& piano g h a d i c ~di a h h ~ a z i o ne di ohdine 9, Le Matematiche 21 (1966), 150-156. 14. N. N I Z E T T E , P e t e h m i n a L b n den o w d e n du p h n de .i%anS.b.x%on nun ahguesien e,t du p&n de Hughen d'ahche m u d , Bull. Soc. Math. Belg. 23 (1971), 436-446. 15. T.G. OSTROM, 1-18.

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20. A.D. THOMAS, G.V. WOOD, Ghoup T a b t e a , S h i v a P u b . , O r p i n g t o n 1980. 21. A. WAGNER, On pmpec..tiui,tien 71 (19591, 113-123.

06

&&Lte phojecfiwe p&UwA,

ZAPPA, SLLi g m p p i di c o U n e a L o n i d e i pian; M a t . Ital. (3) 12 (1957), 507-516.

22. G.

Math. Zeit.

di Hughen, Boll. Un.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 217-224 0 Elsevier Science Publishers B.V. (North-Holland)

217

MONOMIAL CODE

- ISOMORPHISMS

Pave1 F i l i p and W e r n e r H e i s e (*) Mathematisches I n s t i t u t der Technischen U n i v e r s i t a t Munchen 20 2420, D -8000 Munchen 2

P.0.Box

Germany

Let

D be two l i n e a r subspaces of GF(q)" and a Hamming weight preserving l i n e a r b i j e c t i o n . It

and

C

v : C+D

w i l l be proved t h a t cp i s t h e r e s t r i c t i o n o f a monmial transformation o f GF(q)" , i.e. an n x n - m a t r i x over GF(q) , which i s t h e product o f a diagonal and a permutation m a t r i x . An a p p l i c a t i o n shows t h a t t h e group o f a l l Hamming weight preserving l i n e a r b i j e c t i o n s o f t h e

, r 2 3 , of

code HAM(r,q)

length

isomorphic t o t h e general l i n e a r group Let

F be a f i n i t e s e t c o n s i s t i n g o f

For

i EZ":= { 1 , 2

. .. ,n l

F"

Cartesian product

by

associates t o every p a i r t h e words

?

3

and

2

we denote t h e

n z l

be an integer.

ithpr oje c tion of t h e

-

n fold

. With respect t o t h e

; ( ~ , $ ) j I { i € ~ " ; ~ ~ ( ~ ,) which *~~(3)}t

(?$) E F"

x

F"

t h e number

p(?,g)

d i f f e r , t h e Cartesian product

A non empty m e t r i c subspace I t s minimal d i st a n ce d(C)

is

t

elements and

q22

mi : F"+F ; ( X ~ , X ~ ~ . . . ,)+xi X

p:F"xF"+lN0

H m i n g d i st a n c e

q - a r y Hamming

... t qr-1 GLr(q) .

1t q t q

CcF"

F"

i s c a l l e d a (bloc k ) code o f length n over F .

i s defined as d(C) : = m i n i

I C I 2 2 and as d(C) := n + 1 o r

o f p o s i t i o n s i n which

becanes a m e t r i c space.

d(C)

:=a

if

?+;I

p(?$);?,;€C,

if

C c o n s i s t s o f only one codeword.

.

be two codes o f length n over F In c o n t r a s t t o sane more pragmatic d e f i n i t i o n s i n t h e coding t h e o r e t i c a l l i t e r a t u r e we take a sanewhat mathematically p u r i s t i c a t t i t u d e . We define a code -isomorphism cp: C - r D as a Let

C

and

D

b i j e c t i o n cp frm p(cp(?),cp($))

=p(sf,$)

C

onto for a l l

D which preserves t h e Hamming distance, i.e. +-D

x,y~C

. Extending t h e range o f

cp

frm

0 t o F"

i s a code -rnonaorpMsm o f

we a l s o say t h a t

c p : C+F"

r e s t r i c t i o n OIc

o f a code-autmorphism

fonns a t r i v i a l code) t o a code

CcF"

C , Clearly, t h e 0 from t h e whole space F" (which

i s always a code-monanorphism of

C

.

(*) The authors g r a t e f u l l y thank prof. L u c a - M a r i a A b a t a n g e l o f o r h e r e x c e l l e n t organisation o f t h e congress and S e r g i o P o v i a f o r h i s extraordinary care during our sojourn a t t h e h o t e l Riva d e l Sole i n Giovinazzo.

P. Filip and W.Heise

218

Every permut a t io n $ o f t h e symmetric group Sn o f Zn induces on F" a c o d e automorphism y : F"+F". We s e t "$(?)) f o r a l l i E n n and a l l ZEF" Two codes C,DCF" a r e c a l l e d equivalent, i f t h e r e e x i s t s a permut at ion ~1 o f Zn w i t h y ( C ) = D Sometimes ( e . g . i n [ 2 1 f o r t h e b i n a r y case q = 2 )

IT$(^^(?)

.

.

i s c a l l e d t h e automorphism group o f t h e code C Othertimes (e . g. i n [l] ) i t s f a c t o r group := {y ; $ E G I i s c a l l e d t h e a u t m o r F r m o u r p u r i s t i c p o i n t o f view, we do n o t agree w i t h t h e s e phism group o f C d e f i n i t i o n s (*), because t h e r e a r e o t h e r t y p e s o f Hamming d i s t a n c e p r e s e r v i n g bijections t h e group G : = { $ c S n ; y ( C ) = C }

.

-

.

.

L e t x : = ( A ~ , A ~ , . . . , ~ , ) E S : b e a v e c t o r o f l e n g t h n , whose components i E Z n , a r e permutations o f t h e s e t F We d e f i n e a code -aut anorphism

.

by s e t t i n g

F"

ni($!(?)) :=Ai(ni(?))

for a l l

i EZn and a l l

hiesF , $! o f

2EF" ,

F r m now on l e t q be a power o f a prime number and l e t F be t h e G a l o i s f i e l d is o f order q , F=GF(q) The Hamming m e t r i c p on t h e v e c t o r space F"=V,(q) + induced by t h e k m i n g w e i g h t , i . e . t h e norm y : F " + N o ; x - , I ~ i E Z n ; ni(?)+0Il,

.

i n t h e u s ual way. F o r ?,?EF"

we have

P(?$)

=y(?-;)

. I n case

n= l

, the

Hamming w eight y : F + { O , l I i s t h e ordinary characteristic function o f t h e s e t F* : = F \ { O } Using t h e Kronecker symbol 6i,j we have y ( x ) = 1 for all xEF

. For

.

a subset

CcF"

we a b b r e v i a t e

y(C) : = F y ( t )

.

EC

By a Zinear (n,k) -code

C

F we understand a m e t r i c subspace C c F "

over

which i s a l s o a k - d i m e n s i o n a l

l i n e a r subspace o f

F"

. Let

CcF"

,

be a l i n e a r

i EZn t h e r e s t r i c t i o n nilc : C+F of t h e ithp r o j e c t i o n fran F" t o t h e l i n e a r code C i s a l i n e a r form. T h i s l i n e a r f orm

(n,k) -c o de. F or "i

: F"+F

, i.e. nilc i s non - t r i v i a l i f and o n l y i f t h e derivation Ai(C) := ker(nilc) Ai(C) = { ~ E C;mi(?) = 01 , of C i n the position i i s a ( k 1) - d i m e n s i o n a l l i n e a r subspace o f C , A code-monomorphism (p:C+F" w i l l be c a l l e d Zinear, i f

-

i t i s a l i n e a r mapping. A l i n e a r i n j e c t i v e mapping c p : C+F"

phism, i f and o n l y i f i t preserves t h e Hamming weight , i.e. for a l l ~ E C

i s a code-monanor iff

-

y(cp(-d))=y(?)

.

.._

Fo r i E Zn we denote by -#ei (6i,l,6i,z,...,6 ) t h e ithcanonical unit i,n aector of F" Now l e t $ € S n b e a g a i n a p ermut at ion o f Zn Then

.

~


i= 1

Hence

.

(i)(;).ti = e n i ( ? )

i s a l i n e a r transformation o f

k=1

F"

.t

(i)

for a l l ZEF"

, $EGLn(F) , which

.

can be represent ed w i t h r e s p e c t t o t h e canonical by t h e permut a t io n m a t r i x ( 6 i , $ - 1 ( j )1 1 S i , j 9 n b a s i s t z i ; i ~ Z n } o f F" Mow l e t h=(A1,A2,...,An)E(F*)" be a v e c t o r

.

(*) Remark by W. H e i s e : What do I c a r e about my r u b b i s h s a i d y e s t e r d a y !

Monornial Code-lsomorphisms

219

where a l l cmponents a r e non -zero. For i E Z n we i d e n t i f y t h e element A ~ E F w i t h t h e permutation XiES, defined by Xi(x) :=Xiax f o r a l l x € F , Then n

n

i = l

i =1

f ( > ~ ~ ( ? ) - z ~=) >ni(z).xi.ti

f(?) =

transformation o f

,

F"

( A i o 6 i , j ) l < i <, jn

for a l l

w i t h respect t o t h e canonical basis o f

X = (X1,X 2,...,A,)

i.e. i f

ni(O(z))

= h i a n

i s a linear

F"

i s called m o n a i a 2 , i f there exists a

i €Zn and a l l

for a l l

monomial l i n e a r transformations o f

.

F"

and a permutation $ E S n

E(F*)"

(?) $(i)

f

, which can be represented by t h e d i a g o n a l m a t r i x

'?EGLn(F)

A l i n e a r transformation @EGLn(F) o f

vector

. Thus,

?EF"

such t h a t

2EF"

0 =

by ,

. Obviously,

preserve t h e Hamming weight o f every v e c t o r

F"

.

.

? EF" L e t C be a l i n e a r (n,k) -code over F A l i n e a r code-monanorphism c p : C-F" w i l l be c a l l e d monomia2, i f i t can be extended t o a monanial l i n e a r

transformation a : F"+F"

; i.e. i f t h e r e e x i s t s a monanial transformation

, whose r e s t r i c t i o n t o

c p : C+F"

i s m o n m i a l , i f and only i f t h e r e e x i s t

and a permutation all F.

is

C

such t h a t

$ES,

cp

n

J. M a c W i l l i a m s and N . J. A . S l o a n e [2;p.2381

group o f a l i n e a r (n,k)-code l i n e a r transformations code

C

C

over

OEGLn(F)

o f the

h l y X 2y...,AnEF*

i E Z n and

for a l l

d e f i n e t h e autanorpnism

o f t h e whole vector space

nology, t h e group o f l i n e a r code-autcmorphisms o f cpEGLk(F)

elements

F as t h e group o f a l l those monanial

i n v a r i a n t , i.e. f o r which we have a(?) E C

transformations

code -moncmorphism

=Xi.n6ci.(t)

mi(&)

.

?EC

.A

, (oJc =cp

OEGLn(F)

for a l l C

, which

F"

leave t h e

. I n our

ZEC

termi

-

consists o f a l l l i n e a r

k -dimensional vector space

, which preserve

C

.

t h e Hamming weight, i.e. f o r which y(cp(2)) = y ( t ) f o r a l l ? E C So t h e group o f a l l l i n e a r code -automorphisms o f any l i n e a r equidistant (n,k) -code (i. e.

v(z)

for a l l

=d(C )

?€C\ldl

problem 33 o f [2;p.231fl t h i s paper l i n e a r code

- with

i s t r i v i a l l y t h e group GLJF)

from t h e binary,

q=2

. Extending

, t o t h e general case we prove i n

our d i f f e r e n t conception o f a code - isomorphism

- ismorphism i s monomial.

- that

every

This then gives evidence f o r t h e f a c t , t h a t

our p u r i s t i c a t t i t u d e i s n o t too f a r from

F. J . M a c W i l l i a m s ' and

N. J. A . S l o a n e ' s pragmatic p o s i t i o n . For t h e proof we make sane a u x i l i a r y

propos it i ons Let

.

C be a l i n e a r (n,k)-code over F=GF(q) , cp:C+F"

morphism and

D:=cp(C)

a l i n e a r code-mono-

,

The f i r s t proposition, as an easy consequence of t h e rank formula f o r matrices, whose l i n e s form a basis of t h e l i n e a r code s t r u c t u r e of t h e code

C

.

C

, does n o t make use o f t h e m e t r i c

220

P. Filip and W.Heise

PROPOSITION 1 . L e t t b e a n i n t e g e r w i t h O i t s k and i ( 1 ) , i ( 2 ) , . . . , i ( t ) € Z n p a i m i s e d i f f e r e n t i n d i c e s . The t p r o j e c t i o n s TI^(^)^^ , h = 1 , 2 , ..., t a r e l i n e a r l y independent i f and o n l y i f f o r e v e r y c h o i c e o f t ( n o t n e c e s s a r i l y a l y ~ 2 , . , . y a t E Ft h e r e a r e e x a c t l y qk-t codewords t € C

d i s t i n c t ) elements with

T T ~ ( ~ ) ( ?,) h== 1~, 2~,

...,t .

Note, t h a t t h e case t = k p r o v i d e s us w i t h a c h a r a c t e r i s a t i o n of t h e l i n e a r so c a l l e d MDS -codes, c f . [ l ; p . l 6 4 f .I o r [2;p.317ff .I.I n t h i s paper p r o p o s i t i o n 1

-

i s o n l y used i n t h e c a s e t = 1 I n t h i s case p r o p o s i t i o n 1 i s a mere r e f o n n u l a t i o n o f t h e "Satz u b e r d i e G l e i c h v e r t e i l u n g d e r Zeichen i n l i n e a r e n Codes" from r1;p.2103.

-

.

.

02s I n - k

.

If p r e c i s e l y s o f t h e n p r o j e c t i o n s TI.1Ic i EZn , a r e t r i v i a l , t h e n a l s o p r e c i s e l y s o f t h e n , j €Zn , a r e t r i v i a l . p r o j e c t i o n s nj PROPOSITION 2

Let

s

be an i n t e g e r w i t h

ID

Proof. F o r each non - t r i v i a l p r o j e c t i o n milc , iE Z n , b y p r o p o s i t i o n 1 we have F y ( n i ( z ) ) = qk-'- ( q 1) F o r each t r i v i a l p r o j e c t i o n nilc , i€Zn, we have

- .

EC

= 0

>y(ni(?))

. Th e r e f o r e

y(C) =

>Y>y(ni(z)) i = l ~

6€C

E

= (n

- s).qk-'.(q

c

- 1) .

E x a c t l y i n t h e same manner we prove y ( D ) = ( n-a). qk-'a(q -1) , where a i s t h e number o f t h e t r i v i a l p r o j e c t i o n s n j J D , j EZn The b i j e c t i o n c p : C + D p r e serves t h e Hamming weight, so we have y(C) = y ( D ) , whence a = s

.

.

The p r o o f o f p r o p o s i t i o n 2 i s e s s e n t i a l l y t h e o n l y p l a c e i n which we make use o f the fact, that

F= G F( q )

i s a f i n i t e f i e l d . However H a n s K e l l e r e r

(Math. I n s t .

d. TU Munchen, n o t Hans K e l l e r e r , Math. I n s t . d. LMU Munchen) showed t h a t p r o p o s i t i o n 2 h olds a l s o f o r codes o v e r a n i n f i n i t e f i e l d F : Denote by n ( h ) , h = 1,2,.. . , s , t h o s e s i n d i c e s from Zn f o r which t h e p r o j e c t i o n s TI

n(h) I C ,j

njID

a r e t r i v i a l . By a we denote t h e number o f t h e t r i v i a l p r o j e c t i o n s

€Zn . S i n c e F i s i n f i n i t e , t h e r e i s a codeword

?€C w i t h ni(?) * O f o r each i E Z n \ I n ( l ) , n ( ( 2 ) , n ( s ) 1 , hence y(cp(2)) = y ( t ) = n - s and t h u s -P n -s 5 n - u S i n c e t h e r e i s a l s o a codeword d E D w i t h ~ ( d=) n - U we g e t

...,

.

+

-

t

-

.

I ndependent ly y(cp-l(d))=y(d) = n - a and t h u s n - I S 5 n s , whence a = s L u d w i g S t a i g e r (ZKI d. Akad. d. Wiss. d. DDR) gave another p r o o f o f t h e same f a c t which w i l l be i n c l u d e d i n h i s f o r t h c o m i n g paper "On c o v e r i n g codewords" i n t h e A t t i d e l Seminario Matematico e F i s i c o d e l l ' U n i v e r s i t a ' d i Modena. As a consequence t h e theorem o f t h i s paper does n o t depend on t h e f i n i t e n e s s o f t h e underlying f i e l d F

.

N = I n(l),n(2),...,n(s) 1 t h e s e t o f those indices n ( h ) E Z n , .,s , f o r which t h e p r o j e c t i o n i s t r i v i a l . Then, b y p r o p o -

We denot e by h=l,2,..

nn(h)

Ic

22 I

Monomial Code-Isomorphisms

s i t i o n 2 there i s a set t h e projections j e c t i o n nilD

M=Im(l),m(2),

m ( h ) ID

n

(h)

...,m ( s )

...,s

lcZn

.. , r .3

denote by

R

IN1 U I K ( j )

;j E R

.

3

O f course, f o r a l l

i n d i c e s , such t h a t

j EZn\N

the pro-

we i n d i c a t e by

'

j E Zn\N

ah EF *

with 1 -< r . 5 n k t 1 . We I K ( j ) , j EZn\N Then

is a partition of

7Ln

-

we have

a system o f r e p r e s e n t a t i v e s of t h e s e t s

l

s

1 t h e s e t of a l l t h o s e r . i n d i c e s j ( h ) EZn\N ,

such t h a t t h e r e e x i s t s a n element

Ic = ah-n j Ic

of

,are t r i v i a l . For i E Z $ M

i s always non - t r i v i a l . F o r each index

K ( j ) := I j(l),j(2),...,.i(r.)

h = l,Z,.

, h=1,2,

n

and we have

.

n =c r ts ~ E jR

.

.

PROPOSITION 3 L e t j E R b e an i n d e x . Then t h e r e i s a s e t L ( j ) = { i ( l ) , i ( 2 ) ,..., i ( r j ) 1 cZn\M o f r j i n d i c e s and t h e r e a r e n e c e s s a r i l y d i s t i n c t ) elements the r . projections 1

ni(h)

ID

..

The d e r i v a t i o n

1

f o r h = 1 , 2 ,... ,r. 3 L(j) nL(j') = 0

t h a t ni(,,(cp(Z)) = hi(h,.nj(h,(?) jlER\{jl i s an o t h e r index, t h e n

Pro o f .

r

(not

1

E F * such t h a t any two of X i ( l ) , X i ( * ) ,... "i (r ) , h = 1,2,. ,r. , a r e l i n e a r l y dependent and such and a l l

S'EC

.

A:=A.(C)=ker(n.I 3

I C

( k -1) - d i m e n s i o n a l l i n e a r subspace o f t h e

)

of

i n the position

C

. If

j

k - d i m e n s i o n a l v e c t o r space

is a C

.

It

c o i n c i d e s w i t h t h e d e r i v a t i o n s A. ( C ) of C i n t h e p o s i t i o n s j ( h ) , 1(h) h = 1 , 2 ,..., r . . The r . p r o j e c t i o n s n j ( h ) ( A , h =1, 2 ,...,r a r e t r i v i a l as 7

3

IA

.

j

w e l l as t h e s p r o j e c t i o n s n n ( h ) , h = 1,2,...,s The l i n e a r subspace B :=cp(A) o f t h e k - d i m e n s i o n a l v e c t o r space D=(p(C) has dimension k - 1

-

. We

a p p l y p r o p o s i t i o n 2 t o t h e codes A and B and t o t h e l i n e a r code isanorphism There a r e r . t s t r i v i a l p r o j e c t i o n s nilB , i EZn among them t h e

cpIA : A + B

.

7

p r o j e c t i o n s nm(hl J B , h = 1,2,.

..,s . The o t h e r

r . t r i v i a l projections 3

a r e r e s t r i c t i o n s o f t h e non - t r i v i a l p r o j e c t i o n s n 1(1) . IB'ni(2) IBI...,~ i ( r j )IB t o t h e subcode B c D Thus t h e d e r i v a t i o n s ni ( 1 ) ID"i(2) i ( r . )ID (D) o f D i n t h e ' p o s i t i o n s i ( h ) , h=1,2y...yr. , a l l coincide with B A (h)i 3 Thus t h e p r o j e c t i o n s , h = 1 , 2 , ...,r.1 , a r e p a i r w i s e l i n e a r l y dependent. + S inc e C i s a l i n e a r code we can choose a codeword z E C \ A w i t h n . ( b ) = 1 By -b I s u p p o s i t i o n we have n j c h ) ( b ) = a h , h = l Y 2 , . . . , r , . It+i s cp(z)Ecp(C\A) = D \ B , 1 thu s f o r h = 1,2,. . . , r . t h e element B ~ ( := ~ ni ) ( h ) (cp(b)) E F i s always non - z e r o . L + + with Each codeword t E C = < b > @ A can u n i q u e l y be w r i t t e n as c = B - b t ; B E F and ~ E .A Fo r h = l y 2 y . . , y r we s e t Xi h ) : = B i ( h ) / ~ h E F * Then f o r j d)) = h=lY2,...,r. we have ~ ~ ~ ( ~ ) ( c p ( ~t ~) )~ ( ~ ) ( B . c ~ ( i ) t =v (E*nich)(cp(lf)) 1 -b + + = B.Bi(,)*ah/ah = B*Xi(h)*ah* nj ( b ) = j(h)(b) = Xi(h)'nj(h)(B*b) =

.

.

ID

.

.

+ +

= Xi(h).nj(h)(6'b+a)

= 'i(h)*"j(h)(')

The s e t L ( j ) c o n s i s t s o f t h e r j i n d i c e s i ( 1 ) y i ( 2 ) y . . . , i ( r . ) .Analoguosly t h e 3 Suppose t h e r e s e t L ( j ' ) c o n s i s t s o f rjl i n d i c e s i 1 ( l ) , i ' ( 2 ) , . . . , i ' ( r . , ) 3 e x i s t s a n index i E L ( j ) n L ( j ' ) Then, b y what we have seen above, t h e r e e x i s t

.

.

P. Filip and W.Heise

222

(t)

and n,(cp(t)) = n;.n 1' for two elements ni ,rl; E F* w i t h ni(w(t)) = ni-nj(?) a l l t E C Hence n ITj, are 1 i n e a r l y dependent, c o n t r a d i c t o r y t o our and j Ic supposition t h a t j and j ' a r e two d i s t i n c t indices from R

.

THEOREM

.

Ic

.

The l i n e a r code -monmorphism

Proof.

The system tation JIES, o f

i s monanial.

cp

I M 1 u { L ( j ) ;j ER 1 i s Zn by pasting together

a partition of

Zn .We define a pennu-

t h e b i j e c t i o n s I)N : M + N ;m(h)+n(h)

...,

.

and J, : L ( j ) + K ( j ) ; i ( h ) + j ( h ) , j E R For h=1,2, s w e s e t A m ( h ) : = I (we 1 could choose as w e l l any other element o f F* ) and get (2) f o r h=1,2,.. .,s and a l l t € C , by proposi nr' ( h ) ('(')) = = nr' (h) " J , (m (h)) + t i o n 2. By proposition 3 we get f o r a l l j ER , ( ~ ( c ) =) X i (h);nJ, ( i ( h ) ) f o r h=1,2 ,...,r and f o r a l l ?EC Thus @ = t o $

-

.

j

Note t h a t t h e permutation J, and i f s = O o r s = l t h e n projections nj

.

.,An)

if

jER

3

I n F"

.

i s uniquely determined, i f

r. = 1 f o r a l l

j ER

3

I n c a s e r . = l f o r a l l j E R and s = O ; i . e . i f a l l 7 , j E Z , , a r e pairwise l i n e a r l y independent then t h e

Ic

vector

A = (A1,AZ,.. r.=1 forall

(t)

E (F*)"

,s=l

i s uniquely determined, also. The same i s t r u e and q = 2 .

we use the usual s c a l a r product

F"

x

F"+F

;

(2,;)

n

+

:=>ni(x)*ni(y)

+

.

i=1

The dua2 (a more appropriate but unusual name would be "orthogonal")

C* := r?EF";?.t=O 1 of a l i n e a r (n,k) -code C over F i s a l i n e a r code over F , which i s not necessarily a canplement t o C i n F" . Now l e t jections

be a l i n e a r

C

nj

Ic

(n,k) -code over

, j EZn , are

F

, such

(n,n-k)-

t h a t any two o f t h e n pro

-

l i n e a r l y independent. By t h e "Untere Abschatzung des

Minimalabstands 1 inearer Codes" [1;p.2271 t h i s c o n d i t i o n i s equivalent t o t h e f a c t t h a t t h e minimal distance o f t h e dual CL o f C i s a t l e a s t three, d(C*) 2 3

.

(Other codes are i n many respects f a i r l y u n i n t e r e s t i n g . ) Then t h e r e i s f o r every l i n e a r code -autmorphism cp o f C p r e c i s e l y one monomial transformation

@Ic .

*,...,

=Q = 2 07 E GLn(F) o f F" w i t h X = ( A l , X An) E (F* ) n , J, € S n and Therefore i n case d(C*) 2 3 t h e r e i s no d i f f e r e n c e between MacW i11 iams I and S l o a n e ' s concept o f the l i n e a r autanorphism group o f the code C and t h e author authors' concept. These groups a r e isomorphic. The transformation a' = t-' 07 as

Q,

a monomial transformation o f F" dECl

.

preserves t h e Hamming weight o f every codeword From O(C) =C we deduce O'(C1) = C * Thus t h e map cp+Q' i s a hano

.

IcL

-

morphism fran t h e group o f a l l l i n e a r code-automorphisms o f C i n t o t h e group o f Ifany two o f t h e n projections nilcI , a l l l i n e a r code-autmorphisms o f C*

.

i EZ, , a r e a l s o l i n e a r l y independent, i.e. i f d(C) 23 , then t h e r a l e s o f C = (CL)l can be interchanged i n our argunentation and t h e groups o f a l l l i n e a r code-autanorphisms o f C and C* are isanorphic.

Monomial Code-Isomorphisms

For over

r=3,4,

...

F =GF(q)

t h e simplex code HAMl(r,q) o f length

223

i s d e f i n e d as a l i n e a r

(n,r) - c o d e

n = (qr - l)/(q - 1) , which as a l i n e a r subspace o f F"

i s generated by t h e rows o f a

r x n - m a t r i x over

F

, whose

columns f o r m a s y s t e n

o f r e p r e s e n t a t i v e s o f t h e o n e - d i m e n s i o n a l l i n e a r subspaces o f

Fr ( c f . e . g.

-

[ l ; p . 2 3 2 ] ) . O f course, t h e r e a r e many code i s a n o r p h i c v e r s i o n s o f HAMl(r,q) One can change t h e o r d e r o f t h e columns and g e t e q u i v a l e n t codes. One c a n a l s o choose o t h e r systems o f r e p r e s e n t a t i v e s . N o t e t h a t e v e r y l i n e a r code

C

.

with

d(CI) 2 3

c a n be o b t a i n e d by p u n c t u r i n g ( i . e. d e l e t i n g t h e components i n sane f i x e d p o s i t i o n s i n a l l codewords) a s u i t a b l e s i m p l e x code. The d u a l o f HAMl(r,q)

. The s i m p l e x code i s . Indeed, any l i n e

i s t h e q - a r y Hannning code HAM(r,q) m i n i m a l d i s t a n c e d(HAMl(r,q)) m a t r i x has (qr-'

- l ) / ( q - 1)

= qr-l>4

equidistant with of i t s g e n e r a t o r

z e r o e n t r i e s and any non - z e r o codeword o f

can be i n t e r p r e t e d as a l i n e i n a g e n e r a t o r m a t r i x o f

HAW(r,q)

HAMl(r,q)

which i s

o b t a i n a b l e from t h e o r i g i n a l g e n e r a t o r m a t r i x by a p p l y i n g o n l y elementary o p e r a t i o n s on t h e l i n e s . The group o f a l l l i n e a r code - a u t a n o r p h i s m s o f tIAtP(r,q)

. Since

i s t h e g e n e r a l l i n e a r g r o u p GLr(F)

b y d e f i n i t i o n any t w o o f t h e l i n e a r

, i cZn , a r e l i n e a r l y independent ( i n f a c t i t i s forms nilml(r,q) d(HAM(r,q)) = 3 ) t h e g r o u p o f a l l l i n e a r c o d e - a u t a n o r p h i s m s o f t h e Hamming code HAM(r,q)

i s isanorphic t o

GLr(F)

.

F i n a l l y , we make a bow t o p r o j e c t i v e g e a n e t r y and remark t h a t t h e theorem o f t h i s paper a p p l i e s m u t a t i s mutandis t o "semi

- linear

code

- ismorphisms".

REFERENCES

[11 Heise, W. and Q u a t t r o c c h i , P. , I n f o r m a t i o n s - und C o d i e r u n g s t h e o r i e ( S p r i n g e r , B e r l i n - H e i d e l b e r g - N e w York -Tokyo, 1983).

121 MacWilliams, F. J . and Sloane, N. J . A., The t h e o r y o f e r r o r - c o r r e c t i n g codes ( N o r t h - H o l l a n d , Amsterdam

- New

York - O x f o r d , 1977).

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 225-242

225

0 Elsevier Science Publishers B.V.(North-Holland)

ONTHE CROSSING NUMBER OF GENERALIZED PETERSEN GRAPHS S . Fiorini

Department of Mathematics, University of Malta

ABSTRACT la,, a*,

The Generalized Pcte:-scfiGmnn P ( n , k ) is defined to De the qraph on 2n vertices !abel led an,bl,b2 ,...,bn} and edges ta.b.,a.a. 1 1 1 i+1 'bibi+k:

...,

i = 1,2,.. .,n; subscripts modulo nl. The crossing numbers v(n,k)of P(n,k) are determined as follows: ~ ( 9 ~ 3= )2, v(3h,3) = h, v(3h+2,3) = h+2, h+l&v(3h+l ,j)Lh+3,v(bh,4)=2h; various conjectures are formulated. All graphs C I (V(G),E(G)) considered will be simple, i.e. contain no loops or multiple edges. 'be Generalized Petersen Graph P(n,k) is defined to be the graph of order 2n with vertices labelled ia,a2 ,...,an,bl,b2 bn} and edges (aibi,aiai+,,bibi+k:i=l,2,.. ,n; subscripts modulo n ,l6ki.n-I1 'The derived Generalized Petersen Graph denoted Pt(n,k) is obtained from P(n,k) by contracting all edges of form ai,bi, called spokes; edges of form bibi+k in P(n,k) are then called chords of the n-circuit al,a2, PRELIMINARlES

.

,...,

...,

an,a,. A drawing of a graph in a surface is a mapping of the graph into the surface in such a way tnat vertices are mapped to points of the surface and edges vw to arcs in the surface joining the image-points of v and w and the image of no edge ccntains that of any vertex. In our case, the only in the surface we consider is the plane and all our drawings will be sense that no two arcs which are images of adjacent edges have a common point other than the image of the c o m n vertex, no two arcs have more than one point in common, and no point other than the image of a vertex is c o m n to more than two arcs. A common point of two arcs other than the image of a c m o n vertex is called crossing. A drawing is said to be optimal if it minimizes the number of crossings; clearly, an optimal drawing is necessarily good. rhe number of crossings in an optimal drawing of a graph C is denoted by v(G); the number of crossings in a drawing U of C is denoted by vp(G).

S.Fiorini

226

TECHNIQUES The technique of proving that t h e crossing number of some graph C is some p o s i t i v e i n t e g e r k is q u i t e standard. Some g o d drawing is e x h i b i t e d whereby a n upper bound for k is e s t a b l i s h e d . By some ad hoe method it is then shown that t h i s number i s also a lower bound. Embodied i n t h e theorems of t h i s s e c t i o n we p r e s e n t some conclusions of a general n a t u r e which h o p e f u l l y could be used also i n determining the lower bounds of c r o s s i n g numbers of o t h e r graphs. If two g r a p h s C and H are homeomorphic, t h e n t h e i r crossing numbers are i d e n t i c a l . / I COROLLARY1 ('Ihe Monotone l'heorem) If u = ( k , n ) , t h e greatest comnon d i v i s o r of k and n , and i f 2 6 u 6 k < i n ,

THEORW 1

then

and where w

'n, k fvn-n f a , k-klo ' n,k

denotes v(P(n,k.) f

.

P B Let H be obtained from P ( n , k ) by d e l e t i n g k s u c c e s s i v e spokes and l e t i( be obtained from P(n,k) by d e l e t i n g every k ' t h spoke i n t h e c a s e u 4 2. Then H is homeomorphic t o P(n k,k) and i f u 5 2 , then K is homeomorphic t o P(n-n/o ,k-k/a). 'The r e s u l t f o l l o w s from 'Theorem 1. / /

-

If C is a graph and X 5 V(G)oE(G) t h e n the subgraph induced by X is denoted by u(>.

THEORM 2

If v3= PROOF

(The Decomposition Theorem) Let 0 be an optimal drawing of a 0. graph G and l e t E(C) = XWYJZ, XnY = YnZ = ZlrX 0 , then v ( G ) = vo(uY> + vaCd,z>

v(G)

E

vD(C) = v O U ~ Y >+ vyo[uz>

-

vVCD

+

k , where k

is t h e number of c r o s s i n g s of form Y x 2 5 VV<XUY>

+

= vyu(aY> +

w p z > v

- va*

0aCuZ>, s i n c e v 0(x>

0

The followiw c o r o l l a r y r e a d i l y follows by i n d u c t i o n on k:

COROLLARY

2

Let D be an optimal drawing of a graph G i n which some s u b s e t X of E(C) makes 0 c o n t r i b u t i o n t o vu(C).

, of E(C) t h e n

YinY. = 0 ( i d j ) is a d e c m p o s i t i o n J

v(G) f

&

v <XUYi>.

/I

The Crossing Number of Generalized Petersen Graphs

THEOREM 3 (The Deletion 'heorem) Let u be the least number of edges of a graph G whose deletion fran G results in a planar subgraph H of C. Then u (GI b a. PROOF Assuming on the contrary that w < a, then deleting the (at most)v edges being intersected results in a planar subgraph of G I contradicting the minimlity of a. / / We often make use of this simple conclusion in conjunction with Euler's polyhedral formula as in the following:

THEOREM 4

w(9,3)

I

2

PROOF The graph of Figure 1 (i) shows that 2 is an upper bound for v(9,3). M o s that it is also a lower bound we note that P(9,3), contains as subgraph a homeomorph of the graph C of Figure 1 (ii); (the subgraph is

obtained by deleting an edge from each of the three triangles of P(9,3). ) has 12 vertices, 18 edges and girth 5, so that if u edges are deleted to obtain a planar subgraph HI Euler's formula for H implies that

G

5(b

Thus, ~ ( 9 ~ 3 + )a

5

- a) 6 2(18 - a).

r4/31 = 2. / /

(ii) Fig.1.

227

S.Fiorini

228

THEOREM 5 (lhe Contraction Theorem) Let 0 be a grawing of a graph C and l e t e E E(G) make 0 contribution t o vi) ( G ) . Let G be t h e graph obtained from C by c o n t r a c t i g t h e edge e = uv to a single vertex u = v and let 0' be the drawing of G induced by 0. Then wv ( C ) p wDi

PROOF (i)

(ii)

Let wv

fEE(G) such that f is adjacent t o e

If f d E(Ge) ( t . g uw is missing i n 0' ;

E

uv,

E ( G ) ) , then any crossing involving f i n 1)

If f E E(Ge) and f i s crossed by some edge t u i n 0, then t h i s crossing is a l s o missing i n 0'.

Since a l l o t h e r crossings a r e unaffected, i n a l l cases

v

If is an g p t i m l drawing i n which e then v(G) 5 w(G 1.

COROLLARY 1

t o v,(G),

Pi3OOF

E

E ( G ) niakes 0 contribution

By t h e Contraction Theorem, v(G)

uD(C)

a u u , ( G e ) 9 "(Gel. / /

Repeated use of the Contraction Theorem y i e l d s t h e following:

...,

COROLLARY -2

Let <e e > be a sequence of edses of G each of whicb makes 0 contribution to vD!h) i n iome drawing U . I f w e d e f i n e recursively G = G,

Oo = 0, Gi = (G1-'Iei,

Vi 0

t h e drawinp; of Gi induced by Ui-',

uOO(G ) 2 <,,((?I

1

13

... 2 wyt(G t 1.

then

//

Let D be an optimal drawing of C and let H be a subsraph of CSROLLARY 3 C such that for Rach edge eHof H , e makes 0 contribution t o w U ( G ) . ' h e n w(G) 2 v ( G ), where C is obtained from G by contracting each edge e of' H.

-PROOF

W e order t h e edges o f :I and apply C o r o l l a r i e s 1 and 2.

//

RMAHKS

(i) In t h e Contraction meorem and its c o r o l l a r i e s , t h e condition t h a t "e makes 0 contribution t o w l l is v i t a l . Consider the graph G obtained from K by "expanding1' any vertex i n t o two adjacent v e r t i c e s u , v , of valency 7 4. It is not d i f f i c u l t t o s e e t h a t 7 2 w ( C ) ) U ( C ~ ~ I )v(K7) 9. ( i i ) 'The reverse inequality w(Ce) 2 w ( C ) cannot i n general be proved, even i f o t h e r conditions (e.g; i f G c o n t a i n s no t r i a n g l e s ) a r e imposed.

The Crossing Number of Generalized Petersen Graphs

6 If Ck d e n o t e s t h e d e r i v e d graph P 1 ( 3 k , 3 ) , then v(Gk) = k and there e x i s t s an optimal drawing i n which the ( 3 k ) - c i r c u i t C does n o t i n t e r s e c t itself.

-THEOREM

PROOF ~

That v(Gk) 4 k follows from t h e drawing of F i g u r e 2:

Fig. 2 To e s t a b l i s h the r e v e r s e inequali.ty we n o t e t h a t t h e d e l e t i o n of any t h r e e s u c c e s s i v e edyes of C y i e l d s a subgraph homeomorpnic t o GK-1' 'The s t a t e m e n t is now proved by i n d u c t i o n on k. To start t h e i n d u c t i o n we apply t h e D e l e t i o n Theoren t o G4, for which m = 24, n = 12 and g = 4 , s o that f = ( 2 4 - ~ ) - 1 2 + 2 =1 4 - a =>

=>

-

4(14 a ) & 2(24 v 2 u 2 4,

- a)

and t h e s t a t e m e n t is v a l i d i n t h i s case.

W e now c o n s i d e r an optimal drawing U of Gk and assume, for c o n t r a d i c t i o n , t h a t

v3 (C,)

k-I. If' C does n o t i n t e r s e c t i t s e l f i n 3 , t h e n by t h e Decomposition t h e i ' t h set of three s u c c e s s i v e chords [heorem w i t h CC, = C and Yi ( i = 1,2, ...,K), we conclude t h a t v(Gk) k , s i n c e ( X U Y i ) = 1. It follows t h a t i n t h i s c a s e v ( G x, 1 = k and t h e r e e x i s t s a drawinq i n which C does n o t i n t e r s e c t itself. I f , on t h e o t h e r hand, C i n t e r s e c t s i t s e l f i n some edge e , then by d e l e t i n g e and two s u c c e s s i v e edges of C, we o b t a i n Ck-l f o r which Lhe inductive hypothesis implies:

a contradiction.

/I

The same a r q c n e n t , o n l y s l i T h t l y modified, h o l d s for PV(3k+h,3) ( h = 1,2) and determines t h i s c r o s s i n g number as k + h. However, s i n c e the i n d u c t i v e argument f a i l s i n i t s i n i t i a l s t e p for h = 1 ( t h e g i r t h of P 1 ( 7 , 3 ) = 31, we

start w i t h k = 3 for t h i s case.

229

230

S.Fiorini

THEOREM 7 If Ck denotes t h e derived graph P1(3k+h,3), then f o r h = 1 , k 3 3 and for h = 2 , k & 2 , v(G,) = k + h. F u r t h e r , t h e r e exists a n optimal drawing i n which t h e ( j k + h ) - c i r c u i t C does n o t i n t e r s e c t itself. That v(Gk) c k + h follows from t h e drawings of Fig. 3.

To e s t a b l i s h t h e r e v e r s e i n e q u a l i t y we proceed by induction and n o t e t h a t for ( h , k ) = ( 1 , 3 ) or ( 2 , 2 ) t h e g i r t h is 4 and (n,m) = (10,201 and (8,16) r e s p e c t i v e l y . 'he Deletion Theorem, then y i e l d s : PROOF

f

= 12

a and f

10

L 2(20

- a)

- a)

4(12 respectively;

-

- a,

r e s p e c t i v e l y , so that

and 4(10

- a ) c 2(16 - a),

i n e i t h e r case v 2 a .r 4 = k

+

h.

Now suppose t h a t C makes 0 c o n t r i b u t i o n t o vI) i n some drawing 0 .

.men C

is p l a n a r l y embedded and a l l chords e i t h e r l i e i n I n t ( C ) or i n Ext(C).

Case ( i )

if a l l a d j a c e n t chords l i e i n d i f f e r e n t r e g i o n s , then two d i s t i n c t

sub-cases a r i s e none of which

i3

optimal;

Case (ii) If some p a i r of ad.jacent chords ai,3ai, aiai+3 both l i e i n t h e same r e g i o n , then two f u r t h e r sub-cases, according as ai-2ai+, l i e s i n the same or i n d i f f e r e n t r e g i o n s a s t h e s e , arise. In a l l cases that l o c a t e

ai,lai+2, a re-drawing is p o s s i b l e which both does not i n c r e a s e which some chord i n t e r s e c t s C.

v

and i n

i4e conclude t h a t i n all. cases t h e r e e x i s t s a n optimal drawing i n which C is i n t e r s e c t e d i n some edge e. Assuming for c o n t r a d i c t i o n t h a t v(Ck) < k + h , d e l e t i n g t h e edge e and two s u c c e s s i v e edges, we o b t a i n a homeomorph of Ck-l for which t h e i n d u c t i v e hypothesis implies: k + h

-

1

v(Ck-.,) f v(Gk)

-

1 5 k

+

h

-1-

1,

a contradiction. The drawings of Figure 3 are then seen t o be both optimal and i n which C does not i n t e r s e c t itself. / I

The Crossing Number of Generalized Petersen Graphs

Fig.3.

23 1

S. Fiorini

232 THEORkM 8

k + 3

w

(3k + 1,3) 2 k + 1

PROOF That

v(3k + 1,3) 5 k + 3 follows from t h e drawing of Figure 4. 'lo show that t h e lower bound a l s o holds, we consider two cases for a minimal

counterexample: Case ( i ) If t h e r e e x i s t s an optiml drawing i n which no spoke is i n t e r s e c t e d , then t h e Contraction Theorem implies that

v(3k + 1,3) 2 v'(3k + 1,3) = k + 1 (By Theorem 71, for k ? 3. That v(7,3) =

3 follows from

t h e work of Exoo, Harary and Kabell.

Case ( i i ) If some spoke is i n t e r s e c t e d , then d e l e t i n g t h a t spoke and two successive spokes, w e o b t a i n a homeormorph of P(3k 2,3) whose crossing number is k, by t h e rninimality of k. But then,

-

~ ( 3 +k 1,3) 2 v(3k a contradiction.

- 2, 3) + 1

//

Fig . 4 .

= k + 1,

233

The Czossing Number of Generalized Petersen Graphs

The remaining two cases: v(3k + h,3) = k + h (h exactly the same way once we prove that u(d,3)

= 0,2) are established in

4 = u(12,3).

4'

Fig .5 Proofs which are not case-by-case are elusive. 'To facilitate presentation we sketch the method of procedure. We assume, for contradiction, that the crossing number is at roost 3 and consider separately the cases where (i) no crossing is a spoke intersection, (ii) where a l l three crossings, (iii) two of the crossings, and (iv) exactly one crossing is a spoke intersection. 'The Contraction Theorem deals with (i) whereas 'Theorem 1 deals with (ii). 'Thereafterthe armwent takes the following sequence: A large (usually Hamiltonian) circuit H is chosen in the grapn. If H is planarly embedded in some optimal drawing of the 2-spoke-deleted graph, then a contradiction is obtained by virtue of the Decomposition 'heoremwith H = X. If not, then H must intersect itself in exactly two of its edges to yield a 2-looped drawing of itself. A contradiction is obtained for each pair of edges. 'To this end heavy use is made of the following remarks. We define the planarization induced by a drawing of a graph C to be theplanar grapn obtalnL4 DY replacins eat?n Lrossins oj a new vertex with U incident edges, in the obvious way. He aiso define a pair of parallel. ckol of a circuit C to be a pair of e&es (a,b),(c,d) in G\C such that s
c.

S. Fiorini

234

Remark 3: If C t o g e t h e r with chords ( a , b ) , ( c , d ) , ( e , f ) is homeomorphic t o K then a 2-looped p l a n a r i z a t i o n o f H t o g e t h e r w i t h t h e s e edges a l s o c o n t a i n s 3'3 i f t h e c r o s s e d edges of H b o t h l i e i n a segment of H a homeomorph of K c o n t a i n i n g a t rn0s3'~one of t h e vertices { a , b , c , d , e , f ]

That v(d,3) 5 4 foliows from t h e drawing of Fiqure 5(i). 'To establish t h e reverse i n e q u a l i t y we n o t e t h a t s i n c e n 16, m = 24 and t h e g i r t h g I6 , then t h e Deletion Theorem i m p l i e s t h a t

--PROOF:

6(10

so t h a t

v> u

-

CO =

2

2(24

- a)

3.

Lf t h e r e e x i s t s an optimal drawing i n which no spokes i n t e r s e c t , then t h e Contraction 'heorem t o g e t h e r w i t h Theorem 7 imply that w

(8,3)

2

v'(t1,3) = 4.

Thus we can assume t h e r e e x i s t s an optimal drawing i n which e x a c t l y t h r e e c r o s s i n g s occur one of wnich is a spoke i n t e r s e c t i o n : If a l l t h r e e i n t e r s e c t i o n s are spoke c r o s s i n g s , then d e l e t i n g t h e t h r e e crossed spokes should r e s u l t i n a p l a n a r graph. But d e l e t i n g t h e s e t h r e e spokes ( i n a l l p o s s i b l e ways) t o g e t h e r with a n a p p r o p r i a t e f o u r t h spoke we can always o b t a i n a honeormorph of t h e graph obtained from P ( 5 , 2 ) by d e l e t i n g a spoke, which is non-planar; a contradiction.

Case (i)

Case ( i i ) If e x a c t l y two spokes are i n t e r s e c t e d , tnen d e l e t i n g t h e s e two spokes should r e s u l t i n a grapn with c r o s s i n g number 1 and i n each optima.1 drawing of which no spoke is i n t e r s e c t e d . However, i f t h e two spokes are e i t h e r s u c c e s s i v e ( a t d i s t a n c e 1 on t h e rim) or alternate (at d i s t a n c e 2 ) , then d e l e t i n g an a p p r o p r i a t e t n i r d spoke r e s u l t s i n a homeomorph of P ( 5 , 2 ) wnose c r o s s i n g number is 2. If tne d i s t a n c e is 3 ( r e s p . 4 ) t h e n the r e s u l t i n g d e l e t e d graph c o n t a i n s a homeomorph of t h e graph of F i g u r e 6 (i) ( r e s p . ( i i )); both t h e s e cyaphs are seen t o p o s s e s s t h e Hamiltonian c i r c u i t H <1,2,3, 10,11,12,1>. If t h e r e e x i s t s s n o p t i r a l drawinq i n which H is p l a n a r l y embedded, t h e n by t h e Decomposition 'Theorern w i t h X = H , Y = t ( 1 , 5 ) , ( 2 , 7 ) , ( 4 , 9 ) 1 and Z = i(6,11),(8,12), (10,3)1, t h e f'irst graph is seen t o have crossing number 2 ; for t h e second graph we t a k e Y = ~ ( 1 , 5 ) , ( 2 , 1 0 ) , ( 3 , 1 2 ) }and z = t ( 4 , a ) , ( 6 , 9 ) , ( 7 , 1 1 ) } . I n each case ~ U Y >= K = GUD. Thus we m y assune that t h e only optimal drawings are t h o s e i J ' d h i c h H intersects i t s e l f e x a c t l y once i n non-spoke edges. For t h e f i r s t graph, i t S , , d e n o t e s t h e segment <6,7 12> and S2 = <1,2 7>, and i f we assume that ( 6 , " ) = S 1nS 2 is not c r o s s e d , then some edge i n S l \ ( 6 , 7 ) must cross some edge i n S2\ ( 6 , 7 ) , by Remark 3. Now, for sL1 p a i r s of edges ( e ,e2)cSlxS, t h e r e e x i s t s a p a i r of parallel edges s e p a r a t i n g tnem e x c e p t *or e ( 4 : 6 ) , which is a spoke, anyway. 'Thus ( 6 , 7 ) must i n t e r s e c t some edge i n & r j , S>, by Remark 2. But (6,7) is s e p a r a t e d from each edge i n <10,11,. by parallel edges (3,10),(4,9),so it cannot cross any of them, by Remark 1. O f t h e remaining edges, tne only non-spokes are ( d , g ) and ( 4 , 5 ) , both of which cases are d i s pensed w i t h by Remark 3. A s to t h e second graph, some edge i n segnent

...,

,...,

,...,

...,

..,>

235

The Crossing Number of Generalized Petersen Graphs

..

.

,11> must c r o s s some edge i n segment S2 = <12,1 ,.. ,5> by is K and t h e Remrk 3, s i n c e H together w i t h chords (6,i2),(7111),(8,4) 393 same holds for chords ( 2 , 1 0 ) , ( 3 , 1 2 ) , ( 1 , 5 ) . Now for each p a i r of edges

S, = <6,'T,.

(el,e2)e(S, x S2), there e x i s t s a pair of p a r a l l e l edges s e p a r a t i n g them. Thus by Remark 1, d cannot i n t e r s e c t itself.

2

B (ii)

Fig. 6 Case ( i i i ) : If' exactly one spoke is i n t e r s e c t e d , then d e l e t i n g t h i s spoke, 7 1 , l l ) s a y , should result i n a graph whose crossing number is 2 ; a s i n t h e previous case we show t n a t t h i s l e a d s t o a contradiction. 'The c i r c u i t (Fig 7 ( i ) ) C = <1,2,3,3~,6',11,4',4,5,6,7,'~',2~,5',8',8,1> is seen t o be a Hamiltonian c i r c u i t i n t h i s graph and h a s chords ( 2 , 2 ' ) , ( 5 , 5 ' ) , ( 7 , d ) which together w i t h 'Thus, i f C is planarly embedded i n some optimaL C a r e horneornorphic t o K 313' drawing, then one o r o t h e r of' che spokes ( 2 , 2 ' ) , ( 5 , 5 ' ) is n e c e s s a r i l y crossed, e conclude that C must i n t e r s e c t itse1f;if eyact.Ly contrary to assumptions. W ; otneronce, then t h i s i n t e r s e c t i o n m u s t occui* i n t h e segment <2,1,...,6,5> wise e i t h e r ( 2 , 2 l ) or (5,5') is crossed. T n i s i n t e r s e c t i o n must also occur i n t h e segment <5r,2',...,4',6'>, s i n c e as before, C together w i t h chords W e conclude that t h e c r o s s i n g ( 6 , 0 * ) , ( 5 , 5 ' ) , ( 4 ' , 7 ) is hmemiorpnic to K must occur i n t h e i n t e r s e c t i o n of these 3'3'seqnents, i e ; i n t h e segment < 5 8 , 2 1 , 7 f , 7 , 6 , 5 > . Since (7,7') is a spoke and a loop n u s t contain a t Least two v e r t i c e s four cases a r i s e according as t h e crossing occurs i n : ( a ) ( a ) (5,6) i( ( ' 7 f , 2 1 ) , ( b ) ( 5 , 6 ) s ( 2 , , S ' ) , ( c ) ( 6 , 7 ) x ( 7 ' , 2 ' ) o r ( d ) ( t i , ' / ) x (2',5').

i n t h e unique embeddin? of t h e p l a n a r i z a t i o n of' c spokes ( 2 , 2 ' ) , ( 5 , 5 ' ) , ( 6 , 6 ' ) can be drawn uncrossed i n only one way i n cases ( a ) , ( d ) , i n two ways i n case ( c ) and i n no way i n ( b ) . I n ( a ) ('7,d) and one (3,4) or ( 3 ' , 8 ! ) must each c o n t r i b u t e 1 t o v ; i n ( d j , ( 7 ' , 4 ' ) and one of (3,4) o r ( 3 ' , d l ) each cont r i b u t e 1 t o V ; and i n ( c ) each of ( 7 , 8 ) and ( 7 ' , 4 ' ) c o n t r i b u t e 1 t o v i n e a c h embedding, Since a l l cases yi-eld a c o n t r a d i c t i o n we conclude t h a t C must be twisted twice, i n three Loops, i n such a way t h a t a l l o t h e r edges can be drawn i n without f u r t h e r crossinqs.

S. Fiorini

236

(ii)

F ip .7. 'To i d e n t i f y t h e two p a i r s of' i-ntersecting edges OP C we inake use of t h e foliowing remarks: If ( a , b ) and ( c , d ) (a w i t h some edge i n segnent or between some edge i n w i t h an edge i n <j, b>, tnen edges ( a , b ) and ( c , d ) cannot be drawn without crossing each other. Furtnermor*e t h e planarization of C and tne paraLJ.el edges must be drawn a s i n Fip;ure / ( i i ) ie: with both end loops i n the e x t e r i o r (or equivalently, the i n t e r i o r ) of r;he

Henark 1:

m s i <j(b)

...,

middle loop. 2: Ln a good drawing h crossing riurnber V l each of the end-loops must t h e middle loop has a t most two vertices, contain a t ieast 2 v e r t i c e s . ocher than v e r t i c e s of chords of t h a t loop, then t h e r e e x i s t s another drawing w i t h a t most w crossings in which C does not i n t e r s e c t i t s e l f ' ; t h i s reduces t o 3 previous case.

-.Heinark

7

.

i f ' both twists occur i n the segnent <2' ,'/',. .6' ,3'> then the chords ( 7 , d ) , ( 2 , 2 ' ) , ( 3 ' , 8 ' ) necessarily i n t e r s e c t . so t h a t one twist involves one of' t h e edr;;es ( 2 l , 5 ' ) , ( 5 I ~ l ) , ( ~ , 2 ) , ( 2 , 3 ) We i n v e s t i g a t e each of these separate1.y. I'he f i r s t ( i n a counter-ciockwise sense) candidate t o cross ( 2 ' , 5 ' ) is ( d , 2 ) which is separated from it by tne p a r a l l e l chords (5,5'),(8'-,3'). Hence t h e second t w i s t must occur Detween one of (5',2'),(2','7'),('7,6),(6,5) and one of (d,2),(2,3) , by v i r t u e of Henark 1. O f t h e f i r s t s e t , ( 7 , 6 ) and ( 6 , 5 ) are excluded s i n c e they a r e also separated by tne paralle.1 chords ( 8 ' ,3' ,4% 'The reminino; cases are disposed of by Hemark 2. Arguing i n t h i s Way we conc l u d e t h a t the set of edges t h a t can c r o s s ( 2 ' , 5 ' ) is empty. For t h e otner edges, the only crossings t h a t need discussing are (5', d ' x ( ~ , 2,(5' ) ,a' x (2,3), (5',Yl) x ( ' / l , 2 l ) , (2,2) x ( 3 1 , 6 v ) . W e assume witnout loss of Senera l i t y chat v e r t i c e s 8 and 8' l i e on r;he l e f t loop and consider t n r e e possible l o c a t i o n s of' 3 ' : on che r i g h t loop, on t h e lower branch of t h e middle loop,

.

and on t h e upper branch. I n each cas ( 6 v , 3 c ) can be drawn i n uniquely. i n t h e l a s t case, 2 must l i e on t h e P hr; 1 . 0 0 ~s~o t h a t ( 2 ' 2 ' ) is necessariLy crossed. I n the o t h e r cases, 7 must e on t h e upper branch o f t h e middle loop, s o that one of ( 2 , 2 ' ) or ( 5 , 5 ' ) is crossed. The o t h e r cases are s i m i l a r l y d e a l t with. / I

237

The Crossing Number of Generalized Peterseii Graphs Theorein 10

w ( 1 2 , 3 ) = 4.

.

ro i'hat v ( 1 2 , 3 ) 5 4 follows from t h e drawing of F i g u r e 5 ( i i e s t a b l i s h t n e r e v e r s e i n e q u a i i t y we assume f o r c o n t r a d i c t i o n t h a t u 5 3 and n o t e t h a t i f none of t h e spokes i n t e r s e c t i n some o p t i m a l drawinq then a p p l y i n g t h e k l e t i o n rneorem t o t h e d e r i v e d Sraph for wnich m = 4 , n = 12 and g i r t h is 4 , we g e t

Proof:

4(14

-

CX)

5

2(24

-

a)

so t h a t v > a > 4 .

'Thus sone spoke is i n t e r s e c t e d and we proceed t o c o n s i d e r three cases a c c o r d i n 5 a s t h e number of spokes involved is e x a c t l y 3 , 2 or 1.

Case ( i ) : if^ a l l tnree i n t e r s e c t i o n s are spoke i n t e r s e c t i o n s , then deletin;< t n r e e spoKes should r e s u l t i n a p l a n a r graph. If' two of the d e l e t e d spokes w e e i t h e r c o n s e c u t i v e ( a t d i s c a n c e 1 on t h e 1 2 - c i r c u i t , C ) o r a l t e r n s t e ( a t d i s t a n c e 2 on C ) , then d e l e t i n 5 a f o u r t h a p p r o p r i a t e spoke r e s u l t s i n a homeomorph of P ( 9 , 3 ) less a spoke, which is non-planar. rhus we assume t h a t t h e d i s t a n c e on C between d e l e t e d spokes is a t least 3. If t h e y are e q u a l l y spaced, then t h e resuLcin.7 p l a n a r graph c o n t a i n s zi homenorph o f X a c o n t r a d i c t i o n . There r e n a i n s t h e r e f o r e two s u b c a s e s a c c o r d i n g 3' 3'as tne s u c c e s s i v e d i s c a n c e s on t h e r i m are ( 3 , 3 , 6 ) or ( 3 , 4 , 5 ) .

(ii)

Fig. 8. i n t h e first i n s t a n c e , t h e r e s u l t i n g p l a n a r graph is one or other of' t h e graphs i n d i c a t e d i n Figure d ( i ), whereas t h e second case g i v e s rise t o one of t h e two g r a p h s implied i n F i g u r e d ( i i ) . Ln each i n s t a n c e one of the d e l e t e d edges n e c e s s a r i l y i n t e r s e c t s more than one edge.

Case ( i i ) : if e x a c t l y tao spokes are i n t e r s e c t e d , then d e l e t i n g t h e s e two spoKes should r e s u l t i n a qraph whose c r o s s i n g number is one. As i n t h e p r e v i o u s case, i f t h e d i s t a n c e between the d e l e t e d spokes is e i t h e r 1 or 2 a l o n g t h e 1 2 - c i r c u i t C , tnen d e l e t i n g a f u r t h e r a d j a c e n t spoke a p p r o p r i a t e l y y i e l d s a hoioeomorph of P(9,3) whose c r o s s i n g number is 2. 'The remaining cases are d e a l t with s e p a r a t e l y . I n each case we d e l e t e t h e spoke ( 1 , l ' ) and one of ( 4 , 4 ' ) , (5,5'),(6,6'),(7,7'), r e s p e c t i v e l y . Each o f t h e F i r s t t h r e e graphs i s s e e n t o p o s s e s s a tlamiltonian c i r c u i t H as follows: <1,2,2',5',5,4,3,3',6',6,7,7',4',1', 10' , l o , 1 1 , l l ' ,a' , 6 , 9 , 9 ' , 1 2 ' , 1 2 , 1 > , < I ,2,2' ,5' , d ' , 1 1 ' , 1 1 , 1 0 , 1 0 ' , 1 ' , 4 ' ,7' ,7,8,9,

9',~',6,5,4,3,3',12',12,1>,< 1 , 2 , 2 ' , 1 1 ' , 1 1 , 1 0 , ~ , ~ , ~ ~ , 5 ' , 5 , 6 , ~ , 7 ' , 1 U ' , 1 ' , 4 ' , 4 , 3 , W e assume t h a t H is p l a n a r l y embedded is some o p t i m a l

3',b8,9',12',12,1>. drawing.

S.Fiorini

238

Then, i n t h e first case the p a i r of chord t r i p l e s ( ( L j 1 , 8 l ) , ( 7 , 8 ) , ( 9 , 1 0 ) ) and ((31,121),(11,12),(21,111)) each c o n t r i b u t e 1 t o t h e c r o s s i n g number so t h a t by t h e Decomposition Theorem t h e c r o s s i n g number is a t least 2. I n t h e second case, t h e t r i p l e of chords ( ( 4 , 4 1 ) , ( d l ~ 1 ) l ( 9 , 1 0 )n)e c e s s a r i l y y i e l d s a c r o s s i n g involving one or o t n e r o f t h e spokes ( 4 , 4 ' ) o r ( 8 , 8 ' ) , reducing t o Case ( i ) ; s i m i l a r l y , i n t h e t h i r d case, a c r o s s i n g must arise among t h e t r i p l e of chords ( ( ~ , ~ ~ ~ , ( 1 0 , 1 0 ~ ) , ( ~a~g a, i1n 1reducing ~~), t o Case ( i ) . Since a l l cases imply a c o n t r a d i c t i o n we conclude t h a t H must i n t e r s e c t i t s e l f , g i v i n g rise t o e x a c t l y two l o o p s , i n such a way that a l l remaining edges can be drawn i n without further crossings. W e shall need the following: Remark: If H t o g e t h e r w i t h chords ( a , b ) , ( c , d ) , ( e , f ) is homeomorphic t o I< 3,3 then t h e p l a n a r i z a t i o n of H obtained from i n t e r s e c t i n g i t s e l f once i f t h e crossed edges both l i e i n a segnent of H c o n t a i n s a homeomorph of K c o n t a i n i n g a t most one 3'3 v e r t e x i n { a , b , c , d , e , Q . Thus i n t h e f i r s t example, H t o g e t h e r w i t h chords ( 9 , 1 0 ) , ( 7 , 8 ) , ( 5 ' , 8 l ) is homeomorphic t o K so t h a t one of the crossed edges must l i e i n t h e segment < 7 , 7 ' , ,8,0>, 3,3 non-spoke edges i n tnis segment being: ( 9 , d ) ,(8',11' ) , ( I 1,101 , ( l o ' , I ' 1 , ( 1 ' , 4 ' 1 , ( 4 ' , 7 * ) . Taking t h e s e i n t u r n , ( 9 , 8 ) is s e p a r a t e d from each of the e d v s i n tne segment <91,12f,...,61) by t h e parallel c n o r d s ( g 8 , 6 * ) , ( 9 , 1 O ) and from t h o s e i n segment <101,1',4',71> by ( 9 , 1 0 ) , ( 1 0 1 , 7 ' ) so t h a t t h e only p o s s i b l e edges c r o s s i n g it are: ( d l , l l l ) , ( l O , l l ) and ( 6 , 7 ) . The o n l y o t h e r possib1.e c r o s s i n s s are simi1arl.y found t o be: ( 8 1 , 1 1 1 )x (11,101, ( 8 I , 1 l 1 ) x ( 6 , 7 ) , ( 1 0 , 1 1 ) x ( 6 , 7 ) and ( l O 1 , l f ) x (4''7'). Of t h e s e the f i r s t and last are disposed of by v i r t u e of Remark (above). As f o r t h e remaining cases, i n (9,8)x (10,111 and ( 9 , d ) x ( 6 , 7 ) t h e r e s u l t i n g graphs are indeed p l a n a r , but i n the unique p l a n e embedding either ( 1 , l ' ) or ( 4 , 4 * j crosses a t least two edges, i n a l l t h e rest, the p l a n a r i z a t i o n of H together with (6,,9' ) ,('7,a), (8!,5') is uniquely embedded i n t h e p l a n e , but t h e n ( 1 l 1 , 2 " ) n e c e s s a r i l y i n t e r s e c t s some edge.

...

I n t h e second i n s t a n c e , H t o g e t h e r with ( 6 , d t ) , ( 6 , 7 ) , ( Y , l 0 ) is homeomorphic with H must l i e i n t h e segment < 6 , 6 ' , lo>. I n tne t h i r d i n s t a n c e , t h e i n t e r s e c t i n g edge must l i e i n segment (10' , 7 ' , 1 1 ' ) s i n c e h e r e H and chords (10',10),(Y,01),(81,111) is K O f the 3' 3 ' v e r i f i e d t h a t s i x non-spokes i n t h e segment of t h e f o r n e r , it is r e a d i l y none q u a l i f y to i n t e r s e c t any other edge and of t h e seven i n t h e latter case o n l y one, ( d , 9 ) x ( 1 0 , l l ) . I n Lhis case, t h e r e s u l t i w graph is indeed p l a n a r b u t the d e l e t e d spokes ( 1 , 1 1 ) , ( 6 , 6 q ) cross a t least twice each i n t h e unique embedding. There r e n a i n s t o c o n s i d e r t h e f o u r t h graph obtained by d e l e t i n s ( 1 , l ' 1, ( 7 , 7 ' which is n o t Haniiltonian. IJe c o n s i d e r i n s t e a d i t s subgraph obtained by d e l e t i n g '7' and i n c i d e n t edges. T h i s graph i( has a c i r c u i t d = <12',9',9, 1 0 , 1 1 , 1 1 ~ , ~ ~ , d , 7 , ~ , 6 8 1 3 ~ ,,2',2,12> ~ 1 4 , 5 , t~h~a t i n c l u d e s a l l v e r t i c e s except 4 1 1 1 f , 1 0 f which , l i e on a c h a i n j o i n i n g v e r t i c e s 4 and 10 on H. I f t n i s chain is n o t i n t e r s e c t e d i n some optimal drawing i n which H is p l a n a r l y embedded, so tnat i t l i e s i n .tnt(A) without loss of g e n e r a l i t y , then a l l cnords ( 9 , d ) , ( 1 1 , 1 2 ) , (2,3),(5,6) must l i e i n Ext (H), y i e l d i n g a t least two c r o s s i n g s . If on the other nand, H i n t e r s e c t s i t s e l f , then one of t h e crossed edges must l i e i n t n e segnent <12,12', 8> and t h e other i n <6,6' , .,2>. But each non-spoke i n t h e first is s e p a r a t e d from each non-spoke i n t h e second by t h e p a r a l l e l edges ( 2 ~ l l 1 1 ) , ( 5 1 , dexcept ~) for ( l l l , d t ) and ( 5 ' , 2 ' ) which are i n t u r n s e p a r a t e d by (d,c)) and ( 3 * , 1 2 ' ) . W e conclude that t h e c h a i n < 4 , 4 1 , 1 f , 1 0 1 , 1 0 > is i n t e r s e c t e d i n either ( 1 1 , 4 ' ) or ( l g , l O t ) . But then any edqe i n t e r s e c t i n 4 one of t h e s e edges must also intersect one of ( ' / l l 4 ! ) , ( 7 1 , 1 0 1 ) i n t h e correspondinr: drawing of P ( 1 2 , 3 ) .

...,

'<3,3so t h a t some i n t e r s e c t i n g edge of

...,

..

...,

The Crossing Number of Generalized Petersen Graphs

239

Case ( i i i ) : Lf e x a c t l y one spoke is i n t e r s e c t e d , tnen d e l e t i n g t n i s spoke, (12,12') s a y , ShOlJld r e s u l t i n a f r a p h whose c r o s s i n g number is 2. F i g r e 3 stiotas m%tt h i s .;rmli p o s s e s s e s 3 xmi.Ltonian c i r c u i L kl.

dow ii t o 3 e t h e r w i t h chords (5,5'),(9,9'),(3,4Jis K so t h a t i f H is p.lanarly emedded i n some o p t i m i drawinq, then one of t h e 3'3'spokes (S,51),(Y,r1) must be crossed. 'Thus, H crosses itself and we assume that i t i n t e r s e c t s itself e x a c t l y once i n two l o o p s . i n t h e induced p l a n a r i z a t i o n , we show tnat two f u r t h e r crossiws cann0.i be avoided. dy Remark 3, one of the crossed edi:es of H , e s a y , must be i n segment S1 = <8',11', ..., ?I>;otherwise, one of t h e spokes (8,dt),(9,9')is i n t e r s e c t e d . 'The other, f , say,rnust be i n s e p e n t S2 = <4,j,...,5'>; otherwise, one or" (5,5'),(d,d') i s crossed. Since p a r a i l e l spokes (5,5'),(6,df)s e p a r a t e e and f for a l l ( e , f ) E ( S \ (d,7)) x ( S 2 \ (4,5)), o t h e r than spokes, we need only c o n s i d e r (4,5)E S1 and (8,'7)E S2. Ln t h e P i r s t case, ( 3 , 4 ) must CI'OSS i n each of t h e three distinct Locations of' v e r t i c e s U and 9. A i W t h e r c r o s s i n g arises f'rom e i t h e r ( 6 , 7 ) o r (1',4'). I n the second case, (7';IO') must cross and a f u r t h e r c r o s s i n g arises from either ( 1 ' , 4 ' ) or ( 2 ' , 1 1 1 j . Vie conclude t n a t H i n t e r s e c t s itself e x a c t l y twice i n such a way t h a t a l l ocher edTes can be drawn i n without f u r t n e r c r o s s i n q s . As before, some crossed edge, e l i e s i n S1 and some edTe f i n S2. If' e crosses f , then by t h e above reaso1-dq, e i t h e r e = (4,s)or f' (b,7). r f e = (4,5) and f E S = , then e and f are s e p a r a t e d by one or o t h e r of the p a r a l l e l p a i r s (d,Y'),(6,7)and (8,8'),(3,4),so t h a t i f 5 and h are t h e crossed edges, t h e n g (4,5) or (5,6), by Remark 1 of 'Theorem 9. ht 5 = ( 5 , b ) and h E S are also s e p a r a t e d by (8,8'),(5,5') which d o n o t s e p a r a t e e andh,so t h a t 5 = e and f is a t d i s t a n c e a t least 3 from h a.long H by Remark 2 o f Theorem 9. l'he set of' edge p a i r s i n S s a t i s f y i n g these c o n d i t i o n s and s e p a r a t e d from (5,6) oy p r e c i s e l y t n e same sets of' p a r a l l e l edqes is seen t o be empty. S i x i l a r l y , r' = ( d , ' ( ) cannot cross any e E < 5 , 6 , ...,S'> U ( 4 , s ) . 'Thus , e crosses y t f, 5 G '11 I (7' ,4', , 8 ' > , and f c?osses n # e , n E U = e n o t e c h a t L fl U z (5',d1),,(4',7')and t h e spoke (4,4') which < 5 ' , b 1 , ...,4>. W we ignore. Lf g ( t I ' , S ' ) , t n e n g cannot cross any edge i n t h e sub-segnent <4,5,. ,3>, s i n c e t h e y are separated by p a r a l l e l c h o r d s ( 2 ' , 11 ), ( 3 , 4 ) and no edge i n 'I is bounded by (3,4). i f ' g crosses e = (2,3), then h is either ( 5 ' , 6 ' ) or (8l,llf)and f E 0, so that # (dl,5f). If h ( 8 ' , 5 ' ) crosses some edge i n <7',7,.. .,ll>, these are seDarated by (2',11') and one of (4',l1),(10,l'l) which bound no ed<e i n < 5 ' , 2 ' , b ;t h u s (8',5') crosses no edge.

...

..

...,

240

S, Fiorini

...,

If q = (49,71), then g crosses no edge i n <4,5, 5'> since these edges are separated by (11,4') and either (3,4) o r (5,5'), which bound no edge in <7',7, df>. Similarly, h z (4',7') cannot cross any edge in <8,9, ...,8l> since they are separated by (6,'/),(8,8') which bound no edge in <4,5, 5'>. If h I (4',7') and f = (7,8) then e and g cannot be separated by parallel edges since h and f are not. rhus (f,h) are either ((2,3),(5,6)) or ((2,3),(4,5)). In the unique planarization of each case (/',lot)is necessarily crossed. l'hus If e, say, is (5I,2l), then {e,gl &! <4,5,...,j r >and {f,h} C_ <7','7 ,..., g E <2,3,...,4> and f = (b1,Il1), h E . But (5',2') is separar;ed from g E <3,3', 4> by (21,111),(3,4)which do not separate f , h, so that g = (2,3). But each h in < 1 , 1 1 , ...,7 l > is separated from (dl,llr) by (2l,1l1) and another parallel edge wnich do not separate e and g, so that neither e nor g is (5f,21). If e = (2,3), then g E <3',9', 4> excluding <3', ...,6'>, since these latter edqes are separated from (2,3)by (31,6v)l(3,4)which do not separate f,n. Thus g is either (6,5) o r (5,4) both of which are separated from e by (6,7),(9,9'). But then no edge other than the spoke (7,7*) qualifies as either f or h, so that neither e nor g lies in (2,3)u <3', ...,61>. Since these edges account for all distinct possibilities in <5',2', 4>, the result is proved. I 1

...,

...,

...,

...,

...,

Comnents and Problems: It is known [3] that (i) if 1 5 k, in 5 n-1 and !un P 1 (mod n), then P(n,m) 5' P ( n , k ) ; (ii) P ( n , k ) '2 P(n,n-k). It foliows from out' conclusions that v (3h+1,3) = ~(3h+l,h)= v(3n+1,2h+l) = v(3h+193h-2) = h+l v (3h+2,3) = ~(3h+2,h+l) ~(3h+2,2h+l)= ~(3h+2,3h-1)= h+2 'Thefollowing table of knom values for u(n,k) can be drawn up: 1

1 2 3 4 5

6 7 8

9 10 11

12 13

2

3

4

5

0

0

0

0

0 0

2 2 0

7

d

3

0

0

0

0 1 0 0

3 3 3 3 0

0 4 1 4 0 0

6

10

11

12

13

14

0

0

0

0

0

3

3

0

4 ? 1 ? 4

5 5 2 2 5

0 0 4

0

3 2 2 3 2 3

*

5 5

6 ?

? 1 ?

? 3 3

6 ? 1

0

0

5

'

0

3

4

5

6

0

?

>

0

5

?

0

3

6

0

0

0

The Crossing Number of Generalized Petersen Graphs

24 1

Regarding e n t r i e s marked (*), t h e f o l l o w i n g can be s a i d : P( k t , t 'The drawing of P ( k t , t ) i n which tne k t - c i r c u i t is p l a n a r l y drawn and t h e t"k-helms" are drawn s u c c e s s i v e l y a l t e r n a t e l y in t h e i n t e r i o r and e x t e r i o r of' t h e k t - c i r c u i t g i v e s-t h e following upper bound for t h e c r o s s i n g number ct:

It is r e a d i l y v e r i f i e d that i f t h i s estimate is v a l i d for a p a r t i c u l a r odd value of t , t h e n it is a l s o va1.i.d for t + l . 'The same cannot be s a i d for even t. ( I t is of i n t e r e s t t o n o t e t h a t a similar s i t u a t i o n o b t a i n s for t h e complete e conclude that v ( 4 k , 4 ) 2k. b i p a r t i t e ~ y a p h s : c f . [ 2 ] 1. W

References: 1.

2.

G. EXOO, F. Harary, J. Kabell, Ine C r o s s i n s numbers o f some Generalized P e t e r s e n Graphs, irlath. Scand. 2 (1981) 184-188. R. Guy, the d e c i i n e and rali of Zarankiewicz's Theorem, Proof rechniguz? Graph l'heory.(F. r k r a r y , ed.)

.i--n

3.

Iy.

Watkins, A 'Theorern on hit Colourings

..., J.C.T.(B) &

(1969) 152-104.

This Page Intentionally Left Blank

Annab of Discrete Mathematics 30 (1986) 243-250 8 Elsevier Science Publishers B.V.(North-HoUand)

243

COMPLETE ARCS IN PLANES OF SQUARE ORDER J.C. Fisher1, J.W.P. Hirschfeld2 and J . A . Thas3 'Department of Mathematics, University of Regina, Regina, Canada, S4S OA2. 2Mathematics Division, University of Sussex, Brighton, U.K. BN1 9QH. 3Seminar of Geometry and Combinatorics, University of Ghent, 9000 Gent, Belgium. Large arcs in cyclic planes of square order are constructed as orbits of a subgroup of a group whose generator acts as a single cycle. In the Desarguesian plane of even square order, this gives an example of an arc achieving the upper bound for complete arcs other than ovals. 1.

INTRODUCTION

Our aim is to demonstrate the existence of complete (q2 - q + 1)-arcs in a 2 2 cyclic projective plane II(q ) of order q . The only such plane known is

PG(2,q2),

the plane over the field GF(q2)

.

These arcs were found incidentally

by Kestenband [S], using different methods, as one of the possible types of intersection of two Hermitian curves in PG(2,q2) . The importance of these arcs, not observed in [S], is Segre's result that for q e en, a complete m-arc in 1 . Thus, this example of a PG(2,q) with m < q + 2 satisfies m 5 q - Jq +

complete arc attains the upper bound f o r q even As a by-product of the investiis the disjoint union of gation, it is shown that a Hermitian curve in PG 2,q') q + l of these arcs. 2.

NOTATION

Let

n

= n(q

2

)

be a cyclic projective plane of order 9'.

identify its points with the elements i of

ZV, v = q4

+

q2

+

cyclic group is generated by the automorphism u with o(i) = i [ 3 ] , 5 4 . 2 . The lines are obtained from a perfect difference set lo =

j = O,l,.. ., v

Ido,dl,., . dq2} as the sets u J ( l o ) , Let b = q2 + q + 1

and k = q

2

- q

are relatively prime, Zv= Zb x Zk. i = (1,s)

where i

In this notation u(i) = (r + 1, s

+

+

For

1;

i

-

l),

+

1, i

E

Zv,

1.

then v = bk.

Since b

and k

in Zv, we write

r(mod b), i L s(mod k)

taken modulo b and the second modulo k . to any arithmetical operation in Zv.

One can so that the

1,

.

where the sum of the first component is The notation extends in a natural way

J.C. Fischer, J . W.P.Hirschfeld and J.A. Thar

244

By the multiplier theorem of Hall [2], q 3 is a multiplier of II ; this 3 means that the mapping J, given by $(i) = q i is an automorphism of Il. Since q6 F 1 (mod v) , so J, is an involution. Indeed, J1 is a Baer involution since it fixes all b points of kZv = [(r,O) If we define r ( q 3 - 1) E 0 (mod b)

:

r

E

3

Zb1; this is because q r - r

=

.

= I(r,s) : r

B

E

Zbl for s

O,l, ..., k

=

-

1,

then o(Bs) = BS+l and the q2 - q + 1 Baer subplanes Bs partition Il. A l i n e of a Baer subplane Bs is a line of Il meeting Bs in q t 1 points. Similarly define = {(r,s) : s

K

whence o(Kr) = K

r+l It will turn out that

3.

iZk 1 for r = 0,1,. . . , b - 1 ,

also partition Il. Thus i = ( r , s ) = Bs n K r is a complete (q2 - q + 1)-arc.

and the Kr Kr

,

COMPLETE k-ARCS LEMMA 3 . 1 :

$(i)

E

=

(1,

k

-

s)

Proof:

i = (r,s) i n

For each

izb

Zk, we have that

x

of i

, 3

q s + s = s ( q + l)k

LEMMA 3.2:

=

fixes the first component r

It was noted in 52 that

Now, for each s in iZk

whence

izv

.

q3s F - s s k

For any l i n e

IK

n

~~1

.t

-

3

0 (mod k) ,

s (mod k )

.

0

BS , w i t h

of the Baer subpZane odd if

(r,s)

E

(r,s)

=

Bs n Kr,

K

is even

if ( r , s ) #

p..

Proof: By lemma 3.1, the involution $J fixes exactly one point of Kr namely the point (r,O) where it meets B o ; the other points of Kr are interwhich implies changed in pairs. If K is a line of Bo it is fixed by $ , that the number of points of .t n Kr outside Bo is even. Thus the parity of 1.t n Krl varies as L n Kr n Bo is empty o r the point (r,O) . For a line .t of Bs, apply the same argument to o-’(.t) , which is a line in Bo. 0 3

245

Complete Arcs in Planes of Square Order Let

LEMMA 3 . 3 :

be an automorphism group that a c t s regularly on t h e

S

p o i n t s of some p r o j e c t i v e p h n e

n(n)

of order

n,

and suppose t h a t

VO,V1, ...,Vt are the orbits of t h e p o i n t s under t h e a c t i o n of a subgroup s . If .t i s a Zine of n(n) and A . = 19.. n v.1 , then 1

G

of

3

A.(A.

j=1

J

-

1) = I G I

1.

-

J

n2 + n elements y of S \ { l } there corresponds Q of 9. for which y ( P ) = Q ; in fact, P = y - 1 ( 2 ) n 2 and Q = 2 n y(L) . If there was another such pair on 2 , then S would not act regularly on the lines of n(n) . Now we count the set Proof:

To each of the

a unique pair of points P ,

in two ways. First, each y

IJI

whence

= IGI - 1 .

other than the identity gives a unique pair

(P%Q),

Second, 9. is a disjoint union o f the sets 9. n V. , 3

and to each pair (P,Q) , P # Q , in 9. n V . there is a unique y in G such J that y ( P ) = Q ; hence IJI = 1 4 . (A. - 1) and so I J I = A . (A. - 1 ) . 0 Aj>l J J j=1 J J We are now ready to prove the main result.

In 14, an alternative proof is

provided that makes use of the properties of perfect difference sets. For

THEOREM 3.4:

k

=

q2

lie i n

-

q + 1 in

B

q

n(q2) .

are t h e

q + 1

2,

>

each o r b i t

is a complete k-arc w i t h

Kr

Furthennore, the l i n e s through tangents t o

Kr

at

(r,s)

Bs n Kr = ( r , s ) t h a t

.

Proof: Fix a Baer subplane B and let II be one of its lines. For each orbit K r j ( j = 0 , 1 , . . . , q ) that meets .9. n Bs, set C I . + 1 = 12 n K I ; J rj for the remaining orbits, set @ . = n K 1 , j = q + 1, q + 2 , ... , b - 1 . 1 j' By lemma 3.2 both a . and B . are even. I

3

By definition,

By lemma 3.3, b-1

1

j =q+1

Bj (Bj - 1)

+

j=1

( a . + 1) a J. = q J

whence subtraction yields b-1 j =O

2

- 9,

J.C. Fischer, J. W.P.Hirschfeld and J.A. Thas

246

B. I

Consequently

E

{0,2}

for

j 2 q

(rj,s) Krj

E

so t h a t

II n Kr j

a t the point

1.

II o f t h e s u b p l a n e B s ,

Summarily, f o r any l i n e (i)

t

aj

= 0,

(!2 n

K

rj

1

either

= 1 and

is t a n g e n t t o

II

i = (r.,s) 1

or (ii)

.t n K,.

1

Bs = 0 and II meets

n

in

Krj

or

0

points.

2

i s a l i n e of e x a c t l y one o f t h e s u b p l a n e s Bs, it f o l l o w s i n more t h a n two p o i n t s ; t h a t i s , Krj is a (q2 q + 1 ) - a r c . From ( i ) it i s c l e a r t h a t , f o r each p o i n t ( r j , s ) o f K,. , I at t h e q + 1 l i n e s of Bs through ( r j , s ) a r e t h e q + 1 t a n g e n t s o f K r j t h i s point.

S i n c e each l i n e of

il

t h a t no l i n e meets

Krj

-

For q

a s i m p l e c o u n t i n g argument suffices t o show t h a t t h e k - a r c

2 4 ,

Kr

i s complete. Assume t h e c o n t r a r y . Then t h e r e i s a p o i n t P through which p a s s q 2 - q t 1 t a n g e n t s o f Kr , one from each of i t s p o i n t s . S i n c e P E K r , for

some

r’ # r ,

q 2 - q + l

i t f o l l o w s t h a t through each p o i n t o f

tangents of

t h e g r o u p g e n e r a t e d by

(because

K br a )

.

Since

Krl

l i n e s i s counted more t h a n t w i c e , whence tangents.

But a s

i ( q 2 - q + 1)’ When

L

Kr

(q

q = 3,

t

has e x a c t l y l)(q2 - q

+

and

Krl

is i t s e l f a k - a r c , none of t h e s e t a n g e n t 2 - q + l )2 has a t l e a s t ;(q

Kr

(q + l ) ( q 1)

,

- q

t

t a n g e n t s , we have

1)

a contradiction for

it must f i r s t b e observed t h a t

p l a n e of o r d e r 9 , Bruck [l].

there are

Krl

are o r b i t s u n d e r t h e a c t i o n o f

Kr

Then t h e o n l y 7 - a r c o f

q

2

4.

i s t h e unique c y c l i c

PG(2.9)

whose automorphism

PG(2,9)

group c o n t a i n s an element o f o r d e r 7 i s a complete arc, [ 3 ] , 514.7. q = 2

The c a s e

i s a genuine e x c e p t i o n : a 3 - a r c i s never complete. 0 Remark A theorem o f Segre [ 3 ] , 510.3, s t a t e s t h a t a complete m-arc i n

q

even, i s e i t h e r an o v a l , t h a t i s a (q + 2 ) - a r c , o r

m

5

q

-

PG(2,q),

Jq + 1 .

So, f o r

q

247

Complete Arcs ira Planes of Square Order an even s q u a r e , theorem 3 . 4 g i v e s an example of a complete (q - Jq + 1 ) - a r c and shows t h a t S e g r e ' s theorem cannot be improved i n t h i s c a s e . §1 0 . 4 , t h e comparable theorem s t a t e s t h a t a complete m-arc i n is e i t h e r a c o n i c , t h a t i s a (q + 1 ) - a r c , o r

m

odd

q ,

q

odd,

This r e s u l t

However, t h e e x i s t e n c e o f a

iq + 1

i s n o t t h e b e s t bound f o r a l l

514.7.

[3],

LINES IN lI(q2)

4.

Any l i n e of

THEOREM 4 . 1 : (i)

q

+

(ii)

d ( k - 1) = j = 1,2,..

('12

.,

(d,O)

A line

one p o i n t .

Zb;

Each o f i t s l i n e s

Bo.

.

Since

of

j

Zk\{O}

i s f i x e d by

$,

r

Zb

p a i r e d with t h e same element

of

j

r. # D

I t remains t o show t h a t

3

t h a n twice among t h e p o i n t s o f

L

.

f o r by t h e

i = (0,s)

, s # 0,

k - 1 differences

that lie in l i n e of

=

Kt

Bs ,

common wi t h i t .

in

(I

q

+

D

generates 1

for

.

of t he p a r t i t i o n i n exact l y

interchanges

( r j , j ) and j

and

k - j

and t h a t no

r.

I

in

Zb\D

are

can ap p ear more

T h i s f o l l o w s from t h e fact t h a t t h e p o i n t s of

Zv : each of t h e

k - 1 differences

Bo

-

j))

.0

g iv e n i n t h e theorem i s e s s e n t i a l l y t h e

is a k - a r c whose t a n g e n t s are t h e l i n e s through

Kr

Zb

.

t ( ( r j , j ) - (rj, k

(r,s)

The d e s c r i p t i o n of p o i n t s of t y p e ( i ) and ( i i ) shows t h a t any

i s a ta n g e n t t o t h o s e

Bo

the other os(Bo)

Bs.

ok

must o c c u r e x a c t l y once, and t h e s e are accounted

The d e s c r i p t i o n of a l i n e of a l t e r n a t i v e proof t h a t

Bs

it f o l l o w s t h a t both

c o n s t i t u t e a p e r f e c t d i f f e r e n c e set f o r

of t h e form

for

o ccu r s a s t h e second component o f

Lemma 3 . 1 shows t h a t

L. L

therefore contains

meets any o t h e r subplane

Bo

of

II

since

q,

i s an element o f a p e r f e c t d i f f e r e n c e s e t

d

Thus each element

e x a c t l y one p o i n t o f

(rj, k - j)

i s an element o f a p e r f e c t

d

is i t s e l f a c y c l i c p l a n e of o r d e r

where

L

, where

3

a c y c l i c group f o r elements

(d,O)

pairs of p o i n t s of the form ( r j , j ) and ( r j , k - j ) $ ( k - 1) , w i t h the r . d i s t i n c t elements of Zb\D .

Bo

Proof:

for

D

c o n s i s t s of

Bo

1 p o i n t s of the form

difference set

a

-

q

shows t h a t

PG(2,9)

odd, [ 3 ] ,

PC(2,q),

q - Jq/4 + 7 / 4 .

5

has been s l i g h t l y improved by t h e t h i r d a u t h o r . complete 8 - a r c i n

q

For

0

or

2

points.

Kt

t h a t meet i t i n a p o i n t o f

Bo;

it meets

The proof i s completed by n o t i n g t h a t

which e i t h e r c o i n c i d e s with

Bo

o r h as no p o i n t s o r l i n e s i n

248

J.C. Fischer. J. W.P. Hirschfeld and J.A. Thas HERMITIAN CURVES

5.

The only known cyclic planes are the Desarguesian ones and, in this section, we restrict our attention to P G ( 2 , q 2 )

.

Lo of Bo and define are incident exactly when i + j (mod v)

It is convenient to distinguish one line

.

R . = o-’(Ro) 1

element of

Then

Lo.

i

and R .

1

In particular, L

theorem 4.1; now, D

B0

= { i = (d,O) : d

H = {

(d/2, s) : d

Hermitian curve and is t h e d i s j o i n t union of the

liii H is odd or even, whence

n

q

6

+

D, s

.

‘b

Zk} i s a

E

1 compZete k-arcs

Bs is a conic or a l i n e of

tI is a disjoint union o f

is an

as in (i) of

DI

E

0 is a distinguished, perfect difference set f o r

iil The s e t

THEOREM 5 . 2 :

n

Kd/2 ’

Bs according a s

k subconics or

q

k sublines

accordingly.

Define the correlations $ : i a . and p : ( r , s ) c-f 2 (r,-s) . is an ordinary polarity for q odd and a pseudo polarity for q even, [ 3 ] , g 8 . 3 . Thus, with J , as in 12, we have that p = $0 = $ 9 . In fact, p is Pro0f:

Then

+-f

$

a Hermitian polarity since the self-conjugate points of p are the q3 + 1 points (r,s) satisfying ( r , s ) + (r,-s) = (d,O) for d in D , From this ( i ) follows. In Bo in Zb. So

the points are

Bo

(r,O)

while the lines are R

is self-polar with respect to

self-conjugate points of the polarity

@

p

(r,O)

and meets H

induced on

B0

by

p

’

.

both with r in the q + 1 These self-

conjugate points form a subconic when q is odd and a subline when q is even. there exists s ‘ such that bs’ t s (mod k) since b and k are Given s , coprime. Thus H n Bs = 0bs’ (13 n Bo) = I(d/2, s) : d 6 D} is a conic or a line of Bs according as q is odd o r even, and the last part of (ii) follows. 0 THEOREM 5 . 2 : q

The tangents t o any complete

( q 2 - q + 1)-arc

in

2 PG(2,q )

,

even, form a dual Hermitian arc. Proof:

See Thas [ 6 ] .

THEOREM 5.3: The tangents t o any o f t h e complete 2 PG(2,q ) form a dual H e m i t i a n curve i f and only i f q

(q2 - q + 1)-arcs

Kr

in

i s even.

Let q be even and consider the arc K O , where D has been Proof: chosen s o that 21) = D (which is always possible since 2 is a Hall multiplier and each multiplier of Bo fixes at least one line of Bo) . Then the tangents

249

Complete Arcs in Planes of Square Order to

II II

with

(d, 0 )

to

d

namely t h e l i n e s o f

in

E

D = 2D

determined by

p

c o i n c i d e s with t h e s e t o f t a n g e n t s t o Now l e t

q

b e odd.

have t h e form (0,s)

,

takes

KO i s {I. : j = ( d , s ) , d 6 U , J these lines a r e the self-conjugate

Thus t h e s e t of t a n g e n t s t o

H.

KO

P

not i n

and so i s odd, t h i s number i s never

1

+

do n o t form a d u a l H e r m i t i a n a r c . 0

Kr

(q2 - q

Each of t h e

Tb'EOREM 5 . 4 :

,

to

H.

- q

q'

Hence t h e t a n g e n t s t o

q + 1 .

(0,O)

(0,O)

S i n c e t h e number o f t a n g e n t s from a p o i n t

h a s t h e p a r i t y of

Kr

to

which t a k e s

t h e set of t a n g e n t s t o

9,

l i n e s of t h e p o l a r i t y

Kr

containing

Bo

obs' ,

Since

D .

(d,-s) ' Zk}. From t h e assumption t h a t

(d,O)

s

(0,O) ,

at

KO

+

i s t h e intersection of t u o

Kr

1)-arcs

Wermitian curves.

Proof:

First, let

contained i n a p(H)

=

H

q

be even.

H,

Hermitian c u r v e

and l e t

H*

by t h e t a n g e n t s t o Now, l e t

q

Kr

Then a s i n theorem 5 . 1 , t h e a r c

which d e t e r m i n e s a p o l a r i t y

Then, a s i n theorem 5 .

Hrl

n Hr2

d2

= Kt

D

in

,

so

Zb,

Let

n Hr21

(b)

If

H

iZk such t h a t

f o r any

k

is a perfect

+

r2,

r1 # r 2 .

Also

s i n c e t h e r e e x i s t unique

dl

0 , q

even, and l e t

ti.

then

q > 2 ,

Suppose m

theorem 1 0 . 3 . 3 , c o r o l l a r y 2 ) , (q2 + 2 ) - a r c .

E

be a Hemitian curve in PG(2,q')

m = 4 if q = 2 . 2 m = q - q + 1 and (i)

=

rl = i d

+

(m+1)-'zw in H , 2 m = q - q + l if q > 2 ;

Pro0f:

D, s

d l - d 2 E 2 ( r 2 - r l ) (nod b ) .

If there is no (a)

(ii)

IHrl

t = ad

where such t h a t

be an m-izric contained in

(i)

E

s c izk

we have t h a t

THEOREM 5.5: K

D,

In f a c t , s i n c e t h e r e e x i s t s r ' kr' Hr = o (Ho). Since D

i s a l s o a Hermitian c u r v e .

and

E

Hence Hr = { ( d / 2 + r , s ) : d

difference set in

is

Kr

Let

b e t h e d u a l H e r m i t i a n c u r v e o f theorem 5 . 3 t h a t is formed Then p(Kr) = H* n h , whence K = p(H*) n H .

be even o r odd.

is a Hermitian c u r v e .

,

.

.

H = Ho = { ( d / 2 , s ) : d

k r ' : r (mod b)

p

>

K

Now, count t h e p a i r s

q'

then -

K

q + 1

.

is compLete. Then by S e g r e ' s theorem ( [ 3 ] ,

is c o n t a i n e d i n an o v a l (P,Q)

such t h a t

P

E

t h a t is a

0, K,

Q

E

0 ,

P

# 9 and

250

J. C.Fischer. J. W.P. Hirschfeld and J.A. Thas

PQ is tangent to H .

There are at most two points P f o r a given Q ,

since

three would be collinear. So

Hence 3m

2q2

5

+

4,

and 3q2

2 ( q 2 + 2 - m)

m

.C

-

3q

+

3

c

2q2

+

. 4

implies that q = 2 .

This

gives the result. (ii)

Suppose K is not complete, then the same argument as (i)

gives q ' whence q2

-

3q

-

1

5 0 ;

-

q

+ 1 5

2(q

that is, q = 2 .

+

l),

U

For q odd, the points of Bo together with the q2 + q + 1 Remark: conics Cr = { (6d + r, 0) : d E D} , r E Zb , form a plane of order q This

.

plane is isomorphic to PG(2,q) via the isomorphism 6 given by 8(x,O) = (gx, 0) For all q , this configuration of conics also appears as the section

.

by a plane

TI

in PG(3,q),

of the q2 + q + 1 quadric surfaces through a twisted cubic T where 71 is skew to T ; see [4], theorem 21.4.5.

REFERENCES [l]

Bruck, R.H., Quadratic extensions of cyclic planes, Proc. Sympos. AppZ.

Math. 10 (1960), 15-44. [2]

Hall, M., Cyclic projective planes, Duke Math. J. 14 (1947), 1079-1090.

[3]

Hirschfeld, J.W.P.

Projective Geometries over Finite Fields (Oxford

University Press, Oxford, 1979). [4]

Hirschfeld, J.W.P., Finite Projective Spaces of Three Dimensions (Oxford University Press, Oxford, to appear).

[5]

Kestenband, B., Unital intersections in finite projective planes, G e m .

Dedicata 11 (1981), 107-117. [6]

Thas, J.A., Elementary proofs of two fundamental theorems of B. Segre without using the Hasse-Weil theorem, J . Combin. Theory Ser. A . 34 (1983), 381-384.

Annals of Discrete Mathematics 30 (1986) 251 -262 0 Elsevier Science Publishers B.V. (North.Holland)

25 1

ON THE MAXIMUM NUMBER OF S Q S ( o ) H A V I N G A PRESCRIBED PQS I N COMMON" Mario G i o n f r i d d o ' , Angelo L i z z i o ' , Maria Corinna Marino'

S u m m a r y . We d e t e r m i n e some r e s u l t s r e g c r d i n g t h e p a r a m e t e r D (v,ul , where D ( v , u ) i s t h e maximum number of S Q S l v l s s u c h t h a t a n y t w o o f t h e m i n t e r s e c t i n u quadruples, which occuring i n each of t h e S Q S l v l s

.

1.

Introduction

A p a r t i a l quadruple system v

is a f i n i t e set having sets c f

elements and

such t h a t every

P

an element

of

s

. If

(PQS)

3-subset of

(P,sll

and

is a p a i r

{ x ,y , z } c p

are

t h a t every

,

s

her

then

DMB

, then

3-subset (P,s)

/PI = v

of

p

a r e two

(P,s2)

I

=(I,

if

.

If

If 2 . (P,s) is a

= v (v-1) (~-2)/24

s

,

they n s

1 s

1

2

=@

if and

(P,sll

s

such

PQS

i s c o n t a i n e d i n e x a c t l y one element o f (SQS)

is t h e o r d e r and i t i s well-known t h a t an v :2

I n what f o l l o w s an 19

PQSs

(DMB)

i s s a i d a S t e i n e r quadruple system

re e x i s t s i f and o n l y i f

4-sub-

is contained i n an element of

l s l l = 1s21

P

i s c o n t a i n e d i n a t most

P

and o n l y i f i t i s c o n t a i n e d i n an element of lP,s2)

where

is a family of

s

a r e s a i d t o be d i s j o i n t and m u t u a l l y balanced and any t r i p l e

,

(P,s)

SQSlv)

or

.

The n u t

SQS(vl

the-

4 (mod. 6 ) .

w i l l b e denoted by

(&,a)

. We

have

.

On o f t h e m o s t i m y o r t a n t p r o b l e m i n t h e t h e o r y o f

SQSs i s t h e

determination of the parameter: D l v , u ) =Max 112 :

1

h SQSlvl ( Q , q l )

,..., l Q , q h l / q i n q j = A ac

i,j

J

i# j

J

,

IAl = u }

.

" L a v o r o e s e g u i t o n e l l ' a m h i t o d e l GNSAGA e c o n c o n t r i b u t o d e l MPI (1983).

' D i p a r t i m e n t o d i M a t e m a t i c a , U n i v e r s i t B , V i a l e A . D o r i a 6 , 95125 Catania, Italy. ' D i p a r t i m e n t o d i M a t e m a t i c a , U n i v e r s i t g , V i a C . R a t t i s t i 9 0 , 98100 Messina, I t a l y .

M. Gionfriddo. A . Lizzio and M.C. Marino

252

I n [2]

J. Doyen h a s p o i n t e d o u t t h i s problem f o r S t e i n e r t r i p l e

systems. I n t h i s p a p e r we p r o v e some r e s u l t s r e g a r d i n g

.

SQSS

for

D(v,ul

2 . Known r e s u l t s Let

be a

(P,s)

d(x) = r

degree

X,YEP

,

X # Y

9

if

FQS

x

. We w i l l

say t h a t an element

belongs t o e x a c t l y

we w i l l i n d i c a t e b y k

tained in exactly

quadruples of

quadruples of

r

(x,ylr s

. We

x,.:. P

a pair

has

. If

s

{x,y~P c

have

con-

.

dlx) =41s1 X E P

The d e g r e e - s e t o f a where

ve d e g r e e (he),

are t h e elements o f

,

hi

for

..., ( hs II,.

2

If An

.

xly,..

(of

K

1x1 )

i s a factor [I] o f

Fi

F . n F . = @ f o r every 1

1

K

3

I < h < IxI-1 ,

F

i,j =1,2,

on K

elements of

ri

c

DS = ( h

w i l l write

1x1

i s called a partiaZ

)

1

P I J

X

F = [ Fl,--.,Fh~

is a family (on

,

ha-

P

b e t h e c o m p l e t e g r a p h on

1x1 X

...

. If

f r = IPI

+...

is a f i n i t e set, l e t

X

I-factorization

where

r

where

P

, we

i =1,2, . . . , p

DS = [ d ( x ) , d l y ) , . . . ]

is the s e t

IP,s)

PQS

.

,

X) a n d , f u r t h e r ,

i #j

.

I t is

h = 1x1-1

.

If

1-factorization (of

On

x. A partial

1-factorization

embedded i n an

1-factorization

if

Y s X

Let XnY=@

G = I GI

, and e v e r y X

. If

and

Y

F={Fl

,..., G u - l }

an

,

then

{1,Z,...,u-Z}

{xl,x2,yl,ye} C X A Y

Fa\ = { F ; , F ; , ..., Fe} h F = {F .,F 1 on l’** k

on a s e t

X

,

i f and o n l y

.

F9‘: E Fg: is contained i n a F .€ F z 3 b e two f i n i t e s e t s such t h a t 1x1 = I Y I = u

,..., F

V-

1

1

i s an

I - f a c t o r i z a t i o n on

I - f a c t o r i z a t i o n on IF,G,cl)

such t h a t

Y

,

a

is

Y

X

and

,

a p e r m u t a t i o n on

i n d i c a t e s t h e s e t of the quadruples

253

On the Maximum Number of SQS(vI I t i s well-known

with

X n Y =0

,

I n 1-31, [4],

,

(X,A)

and

IQ,q) = [ X u Y ] IA,B,F,G,rxI

then

q=AuEwr(F,G,cO

morphism, a l l

that, if

is an

[6]

SQS(2v)

fY,B)

a r e two

, where

Q =XuY

having

m=8,12,14,15

(i.e.

m~ 1 5 q) u a d r u -

These r e s u l t s a r e t h e f o l l o w i n g :

1,2,:,4 1,2,5,6 1,3,5,7

1,2,3,5 1,2,4,6 1,3,4,7

1,4,6,7

1,5,6,7

2,3,5,8 2,4,6,6 3,4,7,9

2,3,4,8 2,5,6,8 3,5, 7,9

5,6,7,9 3,4,8,0 3,5,9,0

4,6,7,9 3,4,9,0 3,5,8,C

4,6,9,0

4,6,8,0

5,6,8,0

5,6,9,0

and

.

pies.

92

,

M. Gionfriddo has constructed, t o within iso-

DMB P Q S

97

SQSlv)

1,4,5,6 1,4,?,8 1,5,7,9 1,6,8,9 2,4,5,7 2,6,7,8 2,6,5,9 2,4,8,9 3,4,6,8 3,5,&,7

1,4,5,7 1,4,6,6 1,5,6,9 1,7,8,9 2,6,8,9 2,4,5,5 2,4,7,8 3,6,5,7 3,5,7,9

3,4,5,9 3,7,8,9

3,4,8,9 3,6,7,8

3,4,5,&

1,2,3,4 1,2,5,6 1,2,7,6 1,3,5,7 1,4,6,7 1,3,6,6 2,3,5,8

1,2,3,5 1,2,4,7 1,2,6,8 1,3,4,6 1,5,6,7 1,3,7,8 2,3,4,a

2,4,5,7 2,4,6,8 3,4,5,6 3,4,7,8 5,6,7,8

2,4,5,6 2,5,7,8 3,4,5,7 3,5,6,8 4,6,7,8

M . Gionfriddo. A . Lizrio and M.C. Marino

254

1,2,3,4 1,2,5,6 1,3,5,7 1,4,6,7 2,3,5,8 2,4,6,8 3,4,7,8 5,6, 7,9 5,6,8,0 5, 7,8,A 5,9,O,A 4,7,9,A 4,8,0, A 4,6,9,0

41

42

7,3,4,5 1,3,6,7 1,3,8,9 1,4,6,8 1,5,7,8 1,4,7,9 1,5,6,9 2,3,4,6 2,3,5,8 2,3, 7 , 9 2,4,5,9 2,5,6, 7 2,6,8,9 2,4,7,8

1,3,4,6 1,3,5,8 1,3,7,9 1,4,5,9 1,5,6,7 1,6,8,9 1,4,7,8 2,3,4,5 2,3,7,6 2, 3,8, 9 2,4,6,8 2 , 5 , 7,8 2,4, 7,9 2,5,6,9

3. The v a l u e o f

1,2,3,4

1,2,3,5 1,2,4,6 1,3,4,7 1,5,6,7 2,3,4,8 2,5,6,8 3,5,7,8 4,6, 7 , 9 4,7,8,A 4,6,8,0 5,6,9,0 5,7,9,A 5,8,0, A 4,9,0, A

1,3,5,7 1,4,6,7 2,3,5,8 2,4,6,8 3,4,?,9 3,4,8,0 3,6,9,0

1,2,3,5 1,2,4,6 1,3,4,7 2,5,6,7 2,3,4,8 2,5,6,8 3,4,9,0 3,6,8,0 3,5,7,8

3, 6 , 7 , 8 5,6,8,0 5,6,7,9 4,5,9,0 4,5,7,8

3,6, 7,9 4,6,7,8 5,6,9,0 4,5,8,0 4,5,7,9

4,

9"

1,2,5,6

1,2,3,4 1,2,5,6

1,2,7,8 1,3,5,7 I, 4 , 7 , 6 I, 3 , 6 , 8 1,4,5,8 2,3,5,8 2,4,5,7 2,4,6,8 2,3,6,7 3,4,5,6 3,4,7,8 5 , 6 , 7,8

~ ( v , q ~ - m lf o r some c l a s s e s o f

1,2,3,5 1,2,4,7 1,2,6,8 2,4,5,8 2,5,6, 7 2,3, 7,8 2,3,4,6 1,3,4,8 1,4,5,6 1,5,7,8 1,3,6,7 3,4,5, 7 4,6,7,8 3,5,6,8

SQSfvl

We prove t h e f o l l o w i n g theorems.

THEOREM 3 . 1 . L e t

(P,sII

,..., ( P , s h )

be

h

D M B PQS

.

If t h e r e

On the Maximum Number of SQSlv) e x i s t an

I x , ~ ) ~i n

, then

IP,s.) 2

It f o l l o w s

Proof.

{ r , y l ~ P such t h a t i t i s

.

h(2k-1

in

ix,ylk

,

(P,s.l 3

..., h l .

f o r every , j ~ I 1 , 2 ,

{ x , ~ , a ~ ~ , a ~ ~ } a { ~ , ~ , a . ~, {~x ,, ya, a~ k~l 3 } aa k.2 .} ~ s i ' let F i IIallJa12}a{a21,a22},.. {akl,ak211 It f o l l o w s

If be the

is a partial

.,Fkl

1

1-factorization of

torization on

I-factors

Fi elements, it follows

2k-1

IF1 = k i 2 k - l

We have

THEOREM 3 . 2 . L e t

(in the case

on

KZk

,.

Since the set o f the

can (at most) be an h<2k-1

be two s e t s such t h a t

and

XnY = @ . F u r t h e r , l e t

F

and

Kzk

on

{1,2

and

X

,..., Z k - 1 1 . If

then

t h e r e e x i s t s an > 2k-1

Dlv,qv-k'(2k-1)l

Proof.

Let

be two

r e s p e c t i v e l y , and l e t

Y

be an

(Q,ql

F

1x1 = I Y I

=2k

I-factorizations

of

be a p e r m u t a t i o n on

CY

containing

SQSlvl

]-fa5

necessarily.

Y

G

on

h =2k-I

A).

and

X

KZk

1-factorization of

the set A = ~ all,al2,aZlJaz2,.. .,ak1"ak2 l is exactly an

.

.,

1-factors F ={F , F 2 , . ,

that

and a p a i r

i€{l,,,.,hl

255

,

I'(F,G,al

. SQS(vl

containing the family

lF,G,aI

.

I t is

If 1

2

...

1

2k-1

...

2

2k-1

[a .+iE a +i a ti 1 2

Zk-1

i

for where ''*

a

2k-1

z 2k-I 1

+i

, then the quadruples of khe families TfF,G,a.l, i =0,1,2, ..., 2lk-1) , form 2k-1 DMB PQS f P , s ) , I P , s I ,... 1

=1,2,.

..,2k-2

2 (k-2 ) l

('9

D(v,q-k

...

3

2

,

all embeddable in an

(Zk-1)) L 2 k - 1

THEOREM 3.3.

If

SQS(v)

.

2k i2

or

Hence

.,

k€N

i s such t h a t

4 (mod. 6 )

,

then

M. Cionfriddo. A. Lizzio and M.C. Marino

256

Proof. I f

is such t h a t

k€N

p o s s i b l e t o c o n s t r u c t an

or

of o r d e r

Zk

s ~ ~ l 2 k lw i t h

( ~ ~ , q b e~ two l

two

SQS

2k - 2

I-factorizations of

. Let

on

and

Q,

Theorem 3 . 2 we c a n c o n s t r u c t e x a c t l y 2 k - 1 2 -k ( 2 k - I ) q u a d r u p l e s i n common. Hence

and

lQ1,ql)

~ = @~ , a n dQ l e t ~ F

Q

K2k

(mod. 6 ) , t h e n i t i s

4

and

G

be

r e s p e c t i v e l y . From

Q2

having

SQS(4k.J

q4k

2

D(4k,qqk-k

TI-IEOREM 3 . 4 .

v -2

For e v e r y

6'11

4 (mod.

or

Proof. Let

. Further, . ., 2 k - l b e

,

k'2

. w =min { v E N : u , 4 k ,

let

F =IF 1

let

i i=I,.

..,2 k - 1

1x1 =

tions

F'=fFII

with

or

F

on a s e t

embedded i n

4 (mod. G ) }

X ' n Y ' = @ and

G' ={GI}

z i=l,...,w

.

If

.

F'

ip

then

i s embedded i n

G

SQS(wl

and

a

a bijection

I - f a c t o r i z a t i o n on

{r,ylEGE

{ g

G'

-1

(x),m

. Further,

i s a permutation of

-1

From Theorem 3 . 2 i t f o l l o w s

Y

l-factoriza-

Y

,v

~2

such t h a t

X' +Yfy

Y'

such t h a t

(y)lEFI

if

,

fX',ql),

{l,Z, ...,w-

11

, IY',q2)

a r e two

,.

, t h e n we c a n c o g

SQS(2wl = [.x'uY'] Iq7,qZ,F',G',al

s t r u c t an

and

X

I X ' I = ~ = m i nC V E N : u >4k,8

Let

lY'I = / X ' I

on

X C X ' , l X f 1 , 2 1 X I = 4 k >-8 ,

such t h a t

is a set, containing

Y'

the

,

X'

IYI = 2 k > d

and

1 two I - f a c t o r i z a t i o n s o f K e k i i=I,. r e s p e c t i v e l y . From Theorem 8 o f 18.1, t h e r e e x i s t s a n

G = { G

.

>2k-l

D(2w,qZM-k'!2k-l))

b e two f i n i t e s e t s w i t h

Y

X n Y =@

and

,

I t follows

,

and

X

k€N

(2k-1)) Z2k-1

containing T(F',G',al 2 D(2w,qgM-k ( 2 k - I l l 2 2 k - 1

COROLLARY, F r o m t h e s a m e h y p o t h e s e s of T h e o r e m 3 . 4 it f o ~ l o w s D(2v,qw-k

2

(2k-1)) >2k-l

,

f o r every

v 'w

, v

:2

or

..

4 (mod.

6).

P r o o f . The s t a t c r n e n t f o l l o w s f r o m p r o o f o f Theorem 3 . 4 a n d f r o m

Theorem 8 o f

181

From p r e v i o u s t h e o r e m s w e h a v e t h e f o l l o w i n g s c h e m e :

.

257

On the Maximum Number of SQS(v)

k

-

w> 4k

2

q2u-k

VLW, v - 2

i2k-1)

o r 4 (mod. 6)

-

2

8

3

14

928

4

16

5

20

6

26

7

28

- 112 q4@- 225 q s 2 - 396 q50'- 637

8

32

964 960

..

........ ........

q16 1 2

- 45

qs2

-

*.

... ...

I t is easy t o see that v

4 . The v a l u e o f

2

(2k-1))

= + m

.

-t+m

for

D(v,qv-rn)

D(v,qv-k

Zim

m=8,14,15

a n d the value o f

D( 8, qg-1 2 )

I n t h i s s e c t i o n we d e t e r m i n e

THEOREM 4 . 1 .

s

i

for

DMB P Q S

(for

and

m =8,14,15

.

D18,q8-121

, > a

D(v,qv-m)

(

a quadruple

Let

i I,. ~

(P,s..J

. .,Jil)

such t h a t

b

be

h

i

=1,2

t h e r e exist t h r e e e l e m e n t s

.....

x,y,z

h ) . If

E P

and

( ~ , ~ ) ~ , ( x , ~ ) ~ , { x= , by E, s~ . } , t h e n

h < 2 . Proof.

and

h=3

hence

From Theorem 2 . 1 i t i s

,

then

.

h < 3

.

If

{~,14,~,~},{2,y,a,b}Es 3 .

I t follows

{z,z,c,~~€s~,

(x,z)>~ -

THEOREM 4 . 2 . I t i s n o t p o s s i b l e t o c o n s t r u c t t h r e e m =8,14,15

quadruples.

DMB PQS

with

M. Gionfriddo, A . Lizzio and M.C. Marino

258

P r o o f . I t i s e a s y t o s e e t h a t i n t h e u n i q u e p a i r s of

with

rn = 1 5

and

rn = 8

DS = [17/2, (617]

and

,

,

and i n t h e p a i r s o f

,

(x,y12

(x,zJ2

,

DS = [(714, (612, (4)41 and a q u a d r u p l e

DS = [ ( 7 / 2 , ( 6 1 3 , ( 4 ) 6 ] , (see

b ={z,y,zl

DS = [(7)8]

, since

and

. If

it is

h =3

.

then

{1,2,3,x}~s w i t h z ~ { 6 , 7 , 8 } But, x = 6 3 implies {2,2,4,81, {3,2,5,71 E s [resp. {1,2,4,6), 3 w i t h { 1 , 4 , 6 , y } ~ s ~ [ { 1 , 3 , 8 , y l ~ s ~ ] and y @ { l , Z x = 8

it follows

. Therefore,

y e {1,2,.,.,81

DS = [ ( 6 1 8 ]

T h e i r d e g r e e - s e t is

P r o o f . I n t h e p a i r s of

or

DS = [ ( 6 ) 4 , ( 4 ) &

( X , Y ) ~

,

(x,z)

with

G = { G ,G ,G 1 1 2 3

X = {1,4,5,8)

. It

I

with

has degree-set

I

F3

E s3]

{1,2,5,8)

,... ' 8 1 . From

,

with

m=l2

m = 12

quadruples.

h a v i n g DS=[f6)6,(4)31 such t h a t :

z,y,z

(see 5 2 ) . Therefore,

Consider t h e l a s t p a i r of DS = [ ( 6 j 8 ]

. Let

1 - f a c t o r i z a t i o n s of

F={F

K4

respectively:

I

x=7]

{ 1 , 5 , 8 , y l ~ s ~ and

b =Ix,y,z)

(Theor. 4 . 1 ) .

Y = {2,3,6,7)

[resp.

the

.

DMB PQS

be t h e f o l l o w i n g and

Fl

F =

D M B PQS

2 ) . The-

with

t h e r e e x i s t t h r e e elements

h =2

m =12

3 '

h =2

and a q u a d r u p l e

2

f o r them i t i s PQS

it is

There e x i s t t h r e e

THEOREM 4 . 3 .

..

{1,3,5,6),(2,3,4,7)~s

§

. Consider

h =2

m =14

such

x,y,zEP

r e f o r e , from Theorem 4 . 1 , i n t h e s e c a s e s i t i s case

m = 14

D M B PQS w i t h

t h e r e e x i s t ( i n every c a s e ) a t l e a s t t h r e e elements that

D M B PQS

I

I

G2

I

G3

DMB

F F 1, 1, 2' 3

on

259

On the Maximum Number of SQS(v) Further, l e t

=( If

=

1 2 3

1(618]

)

1 2 3 3

, we

(P,T(F,G,a

3

.

Further,

r(F,G,a

J

1

)

2 3 1

can v e r i f y t h a t

I )

are three

and

,

D M B PQS

) 1

with

*

l) , m=12

and

it follows t h a t it is not possi-

m

with

7 2 ~ 3 D M B PQS

(P,I'(F,G,a

{1,2,3}

a r e t h e two f a m i l i e s i n d i -

r(f,G,ci2)

cated i n 5 2. Since it is ble t o construct

1 2 3

)

3 1 2 '

,..., 8 1

,

(P,r(F,G,a21)

DS

1 2 3

P={1,2

b e t h e f o l l o w i n g p e r m u t a t i o n s on

a i

=12

and

DS = [ ( 6 j 8 ]

. Hen-

c e , i t follows t h e statement.. THEOREM 4.4. Proof.

and

Let

D(8,q8-121 (X,A)

( i= 1 , 2 , 3 )

(Y,8)

let

I - f a c t o r i z a t i o n s of

K4

b e two

F = f F , F ,F 1 1 2 3

on

X

, where

SQS(4)

and

Y

, ,

SQS(8)

W e have :

.

X={1,4,5,8}

G = { G ,G G 1 1 2' 3

and l e t

b e t h e p e r m u t a t i o n s , d e f i n e d i n Theorem 4 . 3 .

191 t h a t t h e p a i r

i n an

.

. Further,

Y={2,3,6,7}

t h e two

and

= 3

a

be

i

I t i s known

260

M. Gionfriddo, A . Lizzio and M.C. Manno We can see immediately that:

,

q1nq2=qlnq3 =q2nq3

for every

i,j€{I,Z,31,

have

Di8,q8-12) = 3

v = 2

Proof.

n+2

,

u=5-2

Since for

=2

. From

Theorem 3.1, in the case

k =2

, we

. D(v,qv-8) = D ( v , q - 1 4 ) =D(u,q - 1 5 ) = 2 v v

THEOREM 4 . 5 . We h a v e every

3

i # j .

D(8,q8-12) >3

Hence

lqinq.l

n

v =2

, n+2

to construct at least two

u=7.2

,

, and

v =5.2

SQS(vl

quadruples in common (see [ 6 ] ,

n

n

,

with

nL2

[ 7 ] , [13]),

.

v =7.en

qv-8

or

, for

it is possible q -14 V

or

4,115

the statement follows

from Theorem 4.2, directly..

REFERENCES

111 C. Berge, Graphes e t h y p e r g r a p h e s , Dunod, Paris, 1970. 1-21 J. Doyen, C o n s t r u c t i o n s of d i s j o i n t S t e i n e r t r i p Z e s y s t e m s , Proc. Amer. Math. SOC., 32 (1972), 409-416.

[3] M. Gionfriddo, On some p a r t i c u l a r d i s j o i n t and m u t u a Z l y b a l a n c e d p a r t i a l q u a d r u p l e s y s t e m s , Ars Combinatoria, 12 (1981), 123-134. 141 M. Gionfriddo, Some r e s u l t s on p a r t i a l S t e i n e r q u a d r u p l e sys t e m s , Combinatorics 8 1 , Annals o f Discrete Mathematics, 18 (1983), 4 0 1 - 4 0 8 . 1 5 1 M. Gionfriddo, On t h e b l o c k i n t e r s e c t i o n p r o b l e m for S t e i n e r

q u a d r u p l e s y s t e m s , Ars Combinatoria, 15 (1983), 301-314. 161 M. Gionfriddo, C o n s t r u c t i o n of a l l d i s j o i n t and m u t u a l l y b a l a n c e d p a r t i a l quadrupZe s y s t e m s w i t h 1 2 , 1 4 o r 1 5 b l o c k s ,

Rendiconti del Seminario Matematico di Brescia, 7 (1984), 343354. 171 M. Gionfriddo and C.C. Lindner, C o n s t r u c t i o n of S t e i n e r q u a d r u p ? e s y s t e m s h a v i n g a p r e s c r i b e d number of b l o c k s i n common, Di-

screte Mathematics, 34 (1981), 31-42. 1-81 C . C . Lindner, E. Mendelsohn, and A. Rosa, On t h e number of I - f a c t o r i z a t i o n s of t h e c o m p l e t e g r a p h , J . o f Combinatorial

Theory, 20 (B) (1976), 265-282.

26 1

On the Maximum Number of SQSlvl

[ 9 ] C.C. Lindner and A . Rosa, S t e i n e r q u a d r u p l e s y s t e m s - A s u r v e y , Discrete Mathematics, 2 2 ( 1 9 7 8 ) , 1 4 7 - 1 8 1 .

[lo] A . Lizzio, M.C. Marino, F. Milazzo, E x i s t e n c e of v ~ 5 . 2 and ~

n z 3

, with

qv-Zl

and

qv-25

S(3,4,vl

,

b l a c k s i n common,

Le Matematiche 1111 A . Lizzio, S. Milici, C o n s t r u c t i o n s of d i s j o i n t and m u t u a l l y baZanced p a r t i a l S t e i n e r t r i p l e s y s t e m s , B o l l . Un. Mat. Ital. ( 6 ) 2-A ( 1 9 8 3 ) , 183-191.

1121 A . Lizzio, S. Milici, O n some p a i r s of D a r t i a l t r i p l e s y s t e m s , Rendiconti 1st. Mat. Un. T r i e s t e , (to a p p e a r ) . 1131 G . L O F a r o , On t h e s e t order 47. 1141 A .

v =7.2n

with

Jlvl n22

f o r S t e i n e r quadruple systems o f

, Ars Combinatoria, 1 7 ( 1 9 8 4 ) ,

39-

Rosa, I n t e r s e c t i o n p r o p e r t i e s o f S t e i n e r q u a d r u p l e s y s t e m s , Annals of Discrete Mathematics, 7 ( 1 9 8 0 ) , 1 1 5 - 1 2 8 .

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 263-268 0 Elsevier Science Publishers B.V. (North-Holland)

263

ON FINITE TRANSLATION STRUCTURES WITH PROPER DILATATIONS Armin Herzer Fachbereich Mathematik Johannes Gutenberg-Universitat Mainz, Germany

Recently, Biliotti and the author obtained a certain number of results on translation structures with proper dilatations including structureand characterisation-theorems, which here will be reformulated in a different manner, throwing a new light on some of the regarded questions. 1 . GROUPS OF EXPONENT p AND CLASS 5 2 .

Let K be a (commutative) field of characteristic p > 0 with automorphism Y and V a vector space over K. For a subspace W of V we consider mappings f: VxV + W with property ( * ) : namely f is alternating, vanishing on VxW and bisemilinear with automorphism y , i.e. f satisfies the following conditions: (*I (i) f(UrV) = -f(v,u) (ii) f (ul+uz,v) = f (u1,v)+f (uz,v) (iii) f(uk,v) = f(u,v)ky (iv) f(u,u) = 0 = f(u,w) for all U , U ~ , U ~E ~V, V w E W, k E K. Clearly f is bilinear iff y=l.

G = (G,.) is called of exponent p, if xp = 1 for all x G holds, and G is called of (nilpotency) class 5 2 , if the commutator subgroup of G is contained in the center of G: G' 5 Z ( G ) . We define a multiplication a on V by xoy := x+y+f(x,y) for all x,yEV. We write (V,f) for the structure consisting of the set V and the multiplication a on it, where f has property ( * ) . A group

PROPOSITION 1 . G = (V,f) is a group of exponent p and class 5 2 . Proof: The neutral element is 0 , the inverse of x is -xI and an easy computation shows (xay)oz = x+y+z+f (x,y)+f(x,z)+f(y,z) = X O ( ~ * Z ) . Moreover xn = x+...+x (n times) holds and so xp=ol since K has chaPacteristic p . At last for the commutator of x and y we have [x,yl = x-I= y-l- x'y = 2f (x,y), and so G' 5 W 5 Z(G) is valid.

A . Herzer

264

Conversely the following is true: PROPOSITION 2 . Every group of prime exponent p and class 52 is isomorphic to a group (V,f) as defined before.

Proof: Let G be such a group. We define an abelian group (G,+) in the following manner. For p=2 let be x+y=xy; for pf2 we define =

p-l

x+y := xy[x,yl 2 for all x,yEG. Then is a K-vector space for some field K of characteristic p (at least K=GF (p)) Defining p+l f(x,y) := [x,yl 2 for p odd, and f(x,y)=o for p=2, the mapping f: GxG + G ' has property ( * ) with y = l , and G=(G,f) holds.

.

It is easy to construct such mappings f with property ( * I . Let V,W,K, y be as before and vlf...vh a base of a complement of W in V. We choose elements wijEW for l
'

For p an odd prime there is also a connection between such groups and the kinematic algebras: We look for A a local K-algebra with A= K @ M, where M is the maximal ideal of A satisfying x2=0 for all xEM. Then defining f on M by f(a,b)=ab for all a,bEM one can show that the bilinear map f has property ( * ) with y=l, since char K # 2 . On the other hand for A* the group of units of A the factor group A*/K*={(l+a)K*I aEM} is a two-sided affine linearly fibered incidence group1, and is For, since (1+a) (1+b)=1 +a+b+f (a,b) holds , the isomorphic to (M,f (l+a)K* is an isomorphism of groups, " 6 1 ) mapping M --t A*/K*; a Conversely from the preceeding results every group of prime exponent p and class 5 2 has (in at least one way) the structure of such an affine incidence group as cited before.

.

~

2 . GROUPS WITH A PARTITION TI AND A NON TRIVIAL TI-AUTOMORPHISM.

(non-trivial) partition TI of a group G is a system of subgroups of G called components with the properties: (1) G and 1 1 1 are no components (2) every element of G different from 1 is contained in exactly one component. An endomorphism a of G is called a n-endomorphism, if Ua -< U does hold for every component U of n. A

For an abelian group G with partition n we have the following RESULT 1 (And&, [l]). The n-endomorphisms form a ring K without zerodivisors. So in the finite case K is a field (called the kernel of n) and G is a K-vector space, whereas the components are K-subspaces. More general we have

Translation Structures with Dilatations

265

PROPOSITION 3 (Biliotti/Scarselli [ 4 ] , Herzer [ 7 ] ) . Let G be a finite group with a partition n and a non-trivial n-automorphism a. Then G is of prime exponent p and class 22. Now suppose we have a group G = (V,f) for some K-vector space V, and there is some aEK with O+a+l, and a defined by va = va for every vEV is n-automorphism of G for a partition n of G. This forces (x0y)a = (xa)o (ya) and so f(x,y)a = f(xa,ya) = f(x,y)a2Y

.

Thus, if G is non-abelian and therefore f(x,y)#o for appropriate x,y, we have a = a2' which is only possible for y # 1 . (For every odd number n one can construct such an a of period n for an infinite series of finite fields K and suitable automorphism y of K , see 171, pg.387.) In a certain sense also the converse of the above statement is true: PROPOSITION 4 (Herzer 1 7 1 ) . Let G be a finite non-abelian group with partition n and non-trivial n-automorphism a. Then there is a field K, an automorphism y of K, a K-vector space V and a mapping f with property ( * ) , such that G is isomorphic to (V,f). Moreover there is some aEK with va=va for all vEV, and every component is a K-subspace of V.

3. THE GEOMETRIES BELONGING TO THOSE GROUPS

(Most of the following ideas can be found in Biliotti/Herzer [ 3 1 . ) Let P be a set the elements of which are called points, and 8 a set of subsets of P called blocks with the property, that any two points x and y are contained in exactly one block denoted by (x,y) and moreover every block contains at least two elements and is different from P. We call = (Pl6,Il)a arallel structure, if /I is a parallelism of i.e. an equivalence re?ation on 8 the classes of which are partitions of P. A permutation u of P is called a dilatation, if for every block B also Bo and Bo-l are blocks and moreover BHBo holds. A dilatation is called a translation, if it has no fixed points or is the identity map, and is called a ro er dilatation, if it has exactly one ( P , B , ' ) is called a translation fixed point. A parallel structp&= structure, if there is a group of translations of acting transitively on P.

s,

s

s

Let G be a group with a partition n.Then S(G,n) = (G,6,//)is a translation structure, where 8 is defined to be the set of all right cosets of the components and any two blocks are parallel, if they are right cosets of the same component: 6 = {UxlUEn, xEGI; UyIlWz * u = w. A transitive translation group of S ( G , n ) is G acting on the pointset G by right multiplication. PROPOSITION 5. S ( G , n ) possesses proper dilatations if and only if G possesses at least one non-trivial n-automorphism. (So far we have repeated basic concepts, which can be found in Andrg's paper [ I ] and as a short survey also e.g. at the beginning of Biliotti [ 2 1 . ) NOW, by the preceding results, if G is finite and possesses a non-tri-

A . Herzer

266

vial n-automorphism, then G is of prime exponent p and class 52: moreover G has the structure of a K-vector space such that the components are subspaces. Then the concept of parallel structure leads to the following definitions:

s

The parallel structure = (P,B,I() is called affinely embedded, if there is an affine space A such that P is the set of points of and B is a set of affine subspaces of A ( , whereas 1 I in general is not the natural parallelism of A ) . For different points x,y of A the line of A joining x and y is expressed by xvy.

A

Let R be the set of all lines of A and Ilr a parallelism on R. For BEB and xEP define C = n(x,B) by C E B and BllC and xEC. Similarly for RER and YEP define S = nr(y,R) by S E R and RllrS and yES.Let the parallel strucure A = (P,B,W) be affinely embedded in the affine space A as defined before. A parallelism Ilr on R is called compatible with if for every BEB and RER with IBnRl = 1 the following hold: for every xER and every yEB (i) In(x,B)tlnr(y,R) I = 1 (ii) Let xER and yiEB and define z by [ z , )=ll(x,B)nnr(y.,R) for i=l ,2,3. Then y ,y ,y are colfinear fn A if and on$y if z 1 , are collineir In ‘2 “3 (If an affinely embedded parallel structure = (P,B,/l) possesses a compatible parallelism /r on R for BE8 and RER with lBnRl = 1 , the set U n(x,B) = U nr(y,R) xER YEB is a generalized affine Segre variety, and the set { X E B I X I I B , X n R # @ } is the projection of an affine d-regulus, where d is the dimension of B in A, as defined in Herzer [ a ] . )

s,

A,

s

PROPOSITION 6. An affinely embedded parallel structure most one parallelism Ur on R beeing compatible with

s,

s

s possesses at

The parallel structure = (P, B, ll) is called an affine microcentral translation structure (of odd order), if in an embedding in an a€fine space A (of odd order) is a translation structure with translaand tion group T, such that the elements of T also are affinities of there is a parallelism Ilr on R which satisfies for every T E T with t f l the condition XVXT jr yvyr €or all x,yEP.

s

A

s

PROPOSITION 7. For an affine microcentral translation structure the parallelism Ir on R as given by the definition is compatible with

s

s.

5.

Let be a finite affinely embedded arallel structure possessing a parallelism f/r on R compatible with We define the following incidence proposition (D) ( a kind of Desargues’ theorem - 1 : (D) For i=1,2,3 let xi,yi be points no three of which are collinear. If then xlvyl Ilr x vy and j j hold for j=2,3, ( X I J 1 I (Y1,Yj) j so also (X,IX3) (YyY,).

-

u

s

PROPOSITION 8. Under the conditions for given in the preceding section is an affine microcentral translation structure, if and only if (D) is valid for

s

s.

Translation Structures with Dilatations

261

By omitting the more complicated case p=2 (which can be found in [3]) all this together gives the following characterisation:

s

the following statements THEOREM. For the finite parallel structure are equivalent: a)There is an affine embedding of in an affine space of odd order, such that possesses a compatible parallelism Ir on R , for which (D) is valid b)S is an affine microcentral translation structure of odd order c)There is a group G of prime exponent p > 2 and class 5 2 and a partition n of G, such that = S ( G , n ) holds.

s

s

s

Concerning the proof the implications a)*b) and c)-a) follow from the preceding propositions. The implication b)-*c) follows from the fact, fulfilling b) can be considered as an affine two-sided linearthat ly fibered incidence group1, which then is represented by an affine kinematic algebra A= K (3 M as defined at the end of chapter 1.

s

EXAMPLE. Let V be a K-vector space of dimension at least 2, and for a

subspace W of V and automorphism y of K let f: vxv + W have property (*) Define G = (V,f) and n = {vKlo#v€V}. Then the translation structure = S(G,n) is even a microcentral affine translation Sperner space (i.e. all blocks have the same cardinality): For any subfield F of the fixed field of y we can choose A = AG(V/F) as an embedding affine space of (In the last chapter of 131 these special geometries of most simple shape in their class are characterized.) The properties of f imply f(v,u)=o for some fixed u#o and all vEV. So in there is at least cne parallel class of blocks which is also a parallel class of subspaces of A w.r.to the natural parallelism in A , (The analogic thing holds for the lines and their parallelism Ilr.)

.

s

s.

s

s

REMARK. Since the parallel structures of the theorem are the only candidates for finite translation structures with proper dilatations, one could ask how to characterize this additional property. But for a which is affinely embedded and possesses a paralparallel structure for different points x l y l zof the lelism //r on R compatible with same block it is easy to give an incidence proposition (Dxyz), which guaranties the existence of a (proper) dilatation 6 with fixed point x and y6 = z (see [31, 3.8).

s

s,

FOOTNOTE 1. G is an affine two-sided linearely fibered incidence group, if G is the set of all points of an affine space A, such that for every element a of G by left multiplication and right multiplication a on A are induced affinities and moreover all lines through 1 are subgroups of G. For A = AG(V/K) every such grou can be represented by means of a lo= ' x VxEM, see [ 5 1 or [91. cal K-algebra A = K @ M with , REFERENCES

[I] Andrg,

Math.Z.76 (1961) 85-102 and 155-163. [ 2 ] Biliotti, M., Sulle strutture di traslazione, Boll.U.M.I.(5) 14 A (1978) 667-677. [3] Biliotti, M. and Herzer, A . , Zur Geometrie der Translationsstrukturen mit eigentlichen Dilatationen, Abh.Math.Sem.Hamburg (1984) 1-27. J . , Uber Parallelstrukturen I.II.,

268

A . Iferzer

[ 4 ] Biliotti, M. and Scarselli, A., Sulle strutture di traslazione

[5]

[6]

171 181 [91

dotatedi dilatazioni proprie, Atti Acc.Naz.Lincei, Rend. C1.Sci.Fis.Mat.Nat. (8) 6 7 ( 1 9 7 9 ) 75-80. BriScker, L., Kinematische Raume, Geom.Ded. 1 (1973) 2 4 1 - 2 6 8 . Herzer, A., Endliche translationstransitive postaffine Riume, Abh. Math.Sem.Hamburg 48 ( 1 979) 25-33. Herzer, A., Endliche nichtkommutative Gruppen mit Partition ll und fixpunktfreiem WAutomorphismus, Arch.Math.34 (1980) 385-392, Herzer, A., Varietg di Segre generalizzate, Rend.Mat.(Roma) to appear 1986. Karzel, H., Kinematic Spaces, Symposia Mathematica XI ( 1 9 7 3 ) 4 1 3 439 (Istituto Nazionale di Alta Matematica).

Annals of Discrete Mathematics 30 (1986) 269-274

269

0 Elsevier Science Publishers B.V. (North-Holland)

Sharply 3 - t r a n s i t i v e groups g e n e r a t e d by i n v o l u t i o n s

Monika H i l l e and H e i n r i c h W e f e l s c h e i d

The s e t J: = { y E G ) y 2 = i d , y + i d } o f i n v o l u t i o n s of a g r o u p G which o p e r a t e s s h a r p l y 3 - t r a n s i t i v e l y on a s e t M, i n d u c e s , t o a l a r g e e x t e n t t h e s t r u c t u r e of t h i s g r o u p G . F o r example t h e c h a r a c t e r i s t i c of G and t h e p l a n a r i t y of G a r e e x p r e s s i b l e a s p r o p e r t i e s of J In t h i s paper we a r e looking f o r

2

.

s h a r p l y 3 - t r a n s i t i v e permutation-

g r o u p s G which a r e g e n e r a t e d by t h e i r i n v o l u t i o n s . I t i s shown t h a t :

K c Z 2 and r e s u l t s on g r o u p s G w i t h G = J 3 a r e g i v e n . Also a c l a s s of examples of g r o u p s w i t h G = J 3 a r e p r e s e n t e d The q u e s t i o n , f o r what n E N t h e r e e x i s t g r o u p s G s u c h t h a t G = Jn b u t G c Jn-l i s s t i l l open. On t h e o t h e r s i d e t h e r e do e x i s t s h a r p l y 3 - t r a n s i t i v e g r o u p s G which a r e n o t g e n e r a t e d by t h e i r i n v o l u t i o n s ; e . g . a l l f i n i t e such g r o u p s G which a r e n o t isomorphic t o a P G L ( 2 , K ) f o r some K have t h i s p r o p e r t y . G = J 2 n G SPGL(2,K)

fi

F o r u n d e r s t a n d i n g t h e f o l l o w i n g w e need t h e n o t i o n of a K T - f i e l d and t h e b a s i c r e p r e s e n t a t i o n theorem: Definition 1:

F ( + , . , a ) i s c a l l e d a KT-field

i f t h e f o l l o w i n g axioms

are valid: FB 1

FB 2

( F , + ) i s a l o o p ( w i t h i d e n t i t y 0 ) which h a s t h e p r o p e r t i e s : a + x = 0 x + a = 0. ( w e p u t x:= -a) F o r e a c h p a i r of e l e m e n t s a , b F F t h e r e e x i s t s a n e l e m e n t - x f o r each x E F. d a I b E F such t h a t a + ( b + x ) = ( a i b ) +d a,b ( F * , - ) i s a g r o u p ( w i t h i d e n t i t y 1; F*=F { O } )

FB 3 -

a.(b+c) = a-bta-c

KT

o i s an i n v o l u t a r y autornorphism of t h e m u l t i p l i c a t i v e group ( F * , * ) which s a t i s f i e s t h e f u n c t i o n a l e q u a t i o n :

and

0.a = o

for a l l a,b,cCF

270

M.Hille and H Wefehrheid

-

O ( l + U(x) = 1

F:=

f o r all x E F \ { 0 , 1 } .

x

--f

an e l e m e n t n o t

FUI-1.

The t r a n s f o r m a t i o n s of t y p e a a n d B of a:

-

L e t F be a KT-field,

R e p r e s e n t a t i o n - T h e o r e m 2: i n F a n d d e n o t e by

a ( l + x)

8: x

a+m-x

+

F

onto

F:

a+cs(b+m-x)

w i t h a , b , m E F , m $ O a n d a ( - ) = - , o ( n ) = 0 , o ( 0 ) =-, f o r m a g r o u p T~ (F) which o p e r a t e s s h a r p l y 3 - t r a n s i t i v e l y On F'. C o n v e r s e l y each s h a r p l y 3 - t r a n s i t i v e group i s isomorphic a s a permutation g r o u p t o t h e g r o u p s T (F) of a u n i q u e l y d e t e r m i n e d K T - f i e l d . 3 In the following let

G

r e p r e s e n t e d i n t h e form x

+

a

8,:

x

+

-b + n o ( b + x )

case of

,

3:

-1

6(a) = b

Lemma 4:

If

*

F*.

and

S

b E F and

-

I f t h a t i s t h e case

=

x1,x2

- b _ + ~

).

6

6 E G with

2

*

is called a

id

i f t h e r e e x i s t s two d i f f e r e n t p o i n t s

and G

n E S := { s E F*lo(s).s=lI

then

A transformation

pseudo-involution with

are:

G

a E F ( i f char F = 2 , then a * O )

- ;o) I z E F * ) .

n = z.o(z

a s being

.a has t h e o n l y f i x e d p o i n t ;in 2 t h e n .a h a s t h e two f i x e d p o i n t s and p o s s e s s e s f i x e d p o i n t s x 1 , x 2 i f and o n l y i f

*

char F whereas

Definition

x

char F = 2

n E R := { z . a ( z with

T 3 ( F ) The i n v o l u t i o n s of

a0:

I n case of a-2-I

-

be a s h a r p l y 3 - t r a n s i t i v e g r o u p

F

a,b E

6(b) = a 6 then 6

c o n t a i n s a pseudo-involution

4

J

2

Proof: L e t be 6 E G a p s e u d o - i n v o l u t i o n w i t h S ( 0 ) = m a n d 6(-)

Then 6 h a s t h e form 6 ( x ) = a ( m - x ) = o(m) - a ( x ) t h e r e e x i s t two i n v o l u t i o n s 6 , m =

6(0) =

0 = 6(m) =

1 . C a s e : L e t be

,8 ,

E J such t h a t 6 =

e102co,

*

8, (-1

B,B2(-)

=b

B,

B,

(m)

Then 66(z) = 6 S B 2 ( 0 )

= B(0) = z

w i t h m E F*\S.

Bl 8,.

= 0.

Suppose

Thenwehave:

= B2(0)

(0) =

02(4

* 0,-.

= 6R,B2B2(0) = 6 8 , ( 0 ) = 6 B 2 ( m )

=

I~~B,B,(-)

=

Sharply 3-Transitive Groups =

PI(-)

=

z. * u ( m ) * m - z = 6 6 ( z )

-

= z

27 1

a(m).m = 1 * m E S c o n t r a r y t o

o u r assumption m E F * \ S .

6,

2 . Case:

(a) =

* B1 h a s t h e 1 B 2 i n t e r c h a n g e s 0 a n d OJ. T h e r e f o r e (3, h a s t h e form @ , ( x )= n.o[x) w i t h n E S . * u ( m ) -r?(x) = 6 ( x ) = = B15,(x) = - n - a ( x ) . * o ( m ) = - n * m E S s i n c e n E S and -1 l i e s i n t h e c e n t e r of (F*,’) C o n t r a d i c t i o n t o m E F* \ s.

Then w e g e t B 2 ( 0 ) = B, (-1 f i x e d p o i n t s 0 and

m

=$

=

=.

0 = B2(m) = @ ( 0 )

=

-

QD.

B, ( x )

x.

.

3 . Case: @ ,

*

B2(0)

=

= 0

(m)

B1

(m)

= 0.

T h i s l e a d s t o t h e same proof as i n case 2 , i f

w e i n t e r c h a n g e B 1 and B2.

Theorem 5 : L e t G be a s h a r p l y 3 - t r a n s i t i v e g r o u p . Then G = J2 i f and o n l y i f G S P G L ( 2 , K ) and I K I > 2 . Proof: “4‘ PGL(2,K) S J

2

if

I K I > 2 i s commonly known and c a n b e

v e r i f i e d by c a l c u l a t i o n . 2 w*11 Let be G = J Because of lemma 4 t h e g r o u p G c a n n o t c o n t a i n any

.

pseudo-invo1ution.Therefore F * \ S = #. A K T - f i e l d w i t h F* = S i s a commutative f i e l d ( c f . [ 3 1 , S a t z 1 . 6 ) . Now w e t u r n t o g r o u p s G w i t h G = J

3

.

(F,+,*u) i s c a l l e d p l a n a r , i f t h e e q u a t i o n ax + b x = c w i t h a + - b always h a s a s o l u t i o n x E F . D e f i n i t i o n 6: A KT-field

P l a n a r KT-fields a l r e a d y a r e n e a r f i e l d s ; i . e . group.

(F,+) i s a n a b e l i a n

(cf. [ I ] , 5.6).

Theorem 7 : L e t be (F,+,.,u) a K T - f i e l d w i t h I F / > 2 a n d l e t b e G t h e induced s h a r p l y 3 - t r a n s i t i v e group. G~:=

I yEGly(a)

= a

Then

~ f o r ca n a E ~F

~

i s v a l i d i f and o n l y i f F i s p l a n a r and S - S = F* w i t h S : = =

CsEF*ls-U(s) = I}.

M . HiNe and ti Wefelscheid

212

Proof:

W e conduct t h e demonstration i n s e v e r a l s t e p s :

L e t b e F a p l a n a r K T - f i e l d a n d S - S = F*.

A:

1 . Gm -

cJ' 1 0

Proof:

The e l e m e n t s of t h e g r o u p Gmlo:={yEGI y ( 0 ) = 0 , y ( m ) h a v e t h e form:

=m)

= {LIEGI p : = - , k x w i t h k E F * ] . .

Gm10

Because of S - S = F* t h e r e e x i s t n 1 , n 2 E S s u c h t h a t nl' n 2 = k . Then ( x ) = n n (x) = n l * u a ( n x ) = B 1 B 2 ( x ) w i t h B ( x ) = n l . u ( x ) , 1 2 1 2 2 1 p,(x) = u ( n 2 x ) a n d B 1 , B , - E J .

'V ( x ) = n n

A

2*

f o r each a E F .

Gm,acJ2

P r o o f : L e t b e a : x-, a t x . Then G -1

b e c a u s e of

3.

J

ci

c1-I

=

J.

a

= ci G

ci

-1

c c1 J

2

2 a-l

--J

=I0

Gm= { y E G l y ( x ) = c + m-x, c , m E F , m t O } c J

2

Proof: L e t be T:= { T E G I T ( x ) = ~ = , x = ~ } . Then w e h a v e

S i n c e F i s p l a n a r , t h e s e t T c o n s i s t s of t r a n s f o r m a t i o n s o f t h e form: T = { T E G ~ I T ( X= ) b t x w i t h b E F * l

But f o r IF1 > 2 w e h a v e Hence T c J

2

T

.

B . L e t be G a c J 2

Then Gm = y G a y

-1

= a 2 a 1 E J2 w i t h a i ( x )

= ai-x

f o r some a E F . 2 -1

c y J y

= J

2

f o r some y E G w i t h y ( a ) =

L e t now b e c 1 E Gm w i t h a ( x ) = c + m x , m + 1 a n d c1 =

B,(x)=-b.+n *o(bi+x),i = 1,2. i

a n d a 2-al = b .

i

B2Bl

EJ

m.

2

with

Because of B i E J w e h a v e n

i

ES.

W e show t h a t c1 p o s s e s s e s a n o t h e r f i x e d p o i n t d i f f e r e n t f r o m - .

B 2 B 1 (-1

= - b 2 + n 2 u ( b 2 + ( - b +n o ( b l + - ) ) )

m

= a(-)

=,

= -b +n. o ( b -b ) 2 2 2 1 b2 = b l = : b . +. a ( x ) = -b

=)

The s e c o n d f i x e d p o i n t of

=

1

(n2n1-'b

t

ci

i s -b.

1

t

n2n1 - 2 . x )

We h a v e

Sharply 3-Transitive Groups

213

S i n c e e a c h a E G m h a s two f i x e d p o i n t s , F i s p l a n a r w h e r e a s f o l l o w s t h a t (F,+) i s a n a b e l i a n g r o u p .

* a ( x ) = (-b+n2n1-lb)+n2n,-'-x

=)

m

= n n -1

ES.S

2 1 Thus F* = S - s s i n c e m E F * was a r b i t r a r i l y c h o s e n .

Remark:

I f Ga c J2 t h e n G c J3. Wether t h e c o n v e r s e i s t r u e , w e do n o t

know. I n c o n c l u s i o n w e g i v e e x a m p l e s of s h a r p l y 3 - t r a n s i t i v e g r o u p s G w i t h 3 b u t G t. J2. W e u s e t h e f o l l o w i n g g e n e r a l method f o r c o n s t r u c t i n g

G = J

s h a r p l y 3 - t r a n s i t i v e g r o u p s due t o Xerby ( c f . [ 2 ] , p . 6 0 ) : L e t (F,+, - 1 be a commutative f i e l d and l e t a ( x ) = x - ' .

Suppose

A < (F*,.) s u c h t h a t :

2

EF*laEF*}cA

(1)

Q:=

(2)

T h e r e e x i s t s a monomorphism

(3)

i (x) E

{a

xA f o r all

T

E

TI

F*/A

TI:

+

Aut(F,+,')

(F*/A) a n d a l l x E F*.

D e f i n e Oob = 0, and f o r a + O , a o b = a - a ( b ) where a cp

Then ( F , + , o , o ) i s a K T - f i e l d . coupled Dickson-nearfield,

(To b e e x a c t :

(F!+#o)

cp

= a(a.A).

is a strongly

which i n a d d i t i o n i s p l a n a r ( c f . [ 5 ] ) )

( 1 ) i n d u c e s [F*:A] = 2111 where I d e n o t e s a n a p p r o p r i a t e i n d e x s e t .

One can c o n s t r u c t f o r a r b i t r a r y i n d e x sets s u c h K T - f i e l d f o r which t h e i n d u c e d g r o u p G s a t i s f i e s G = J 3 b u t G t. J2, W e i l l u s t r a t e t h i s method of c o n s t r u c t i o n f o r I ={1,2}: L e t b e K a commutative f i e l d and L : = K ( x 1 , x 2 ) a t r a n s c e n d e n t a l

r1

e x t e n s i o n of K . C o n s i d e r t h e f o l l o w i n g automorphisms a 1 , a 2E AutKL

al:

{

X

+

1-Xl

x2

+

x2

k E K

+

k

a2:

+

1-x2

x

+

x1

k E K

--*

k

u 1 and a2 a r e i n v o l u t a r y a n d s a t i s f y u 1 u 2 = a 2 a 1 . Take F:= L ( t l f t 2 ) where t h e t i are t r a n s c e n d e n t a l o v e r L. L e t b e 7i

E A u t F t h e f o l l o w i n g c o n t i n u a t i o n of a 1 f o r h E L [ t ,

/t21:

:

214

M . Hille arid H. Wefelscheid

and define

h, := h2

T , (-)

I

ri(hl) for -r;(hZJ

h, A l s o let be gradi€ = gradi$-:=

f = -hlr h2

h ,h EL[tl,t21 1 2

gradihl

- gradih2 the degree function

of the polynomials h l r h 2EL[tlrt21with respect. to ti. Now we define for € , g E F : grad f grad2f T2 fog:= f m f w ( g ) with fQ:= Then Fr+,oru)is a planar KT-field (to be exact: (F,+,oru)is a planar KT-nearfield; o(a) = a-1 the inverse with respect to ( - 1 1 . We note thatA = Kern cp = {fEF*l gradifZO mod 2 for i = I r 2 } c S and [F* : A] = 4 = 211[. The 3 other cosets are: t,*A, tZ-Ar t,-t2-A The Dickson-group r:= If EAut F / fEF*] consists of the 4 automorcp -c2, T ~ T ~ } . Fany O ~ T €rand any z E Itl,t2,tl.t21 phisms: r = {id, ~~r we have T(Z) = z and therefore Z O U ( Z ) = 1. * z E S and z - A = zoAcSoS for z E {t,,t2rtl-t2).* F* = S o S . This shows that this example satisfies the conditions of Theorem 7, whence the induced group G fulfills G o . c J 2 and therefore G = J 3

.

References [l]

H. KARZEL: Ztmsammenhlnge zwischen Fastbereichen,scharf 2-fach transitiven Permctationsgruppen und 2-Strukturen mit RechtecksaTiom, Abh. Math. Sem. Hamburg, 31 (1967) 191-208

[2] W. KERBY:

Infiilite sharply mul'iply transitive groups. Hamburger Mathematische Einzelschriften. Neue Folge Heft 6, GSttingen 1974, Vandenhoek und Ruprecht

[3] W. KERBY und H. WEPELSCHEID: Uber eine scharf 3-fach transitiven Gruppen zugeordnete algebraische Struktur. Abh. Math. Seminar Hamburg

37(1972) 225-235 [4]

H. WEFELSCHEID: ZT-subgroups of sharply 3-transitive groups, proceedings of the Edinburgh Mathematical Society 1980, Vol. 23, 9-14

[5]

H. WEFELSCHEID: Zur Planaritat von KT-Fastksrpern, Archiv der Mathematik, Vol. 36(1981) 302-304

Fachbereich Mathematik der Universitlt Duisburg D 4100 Duisburg 1

-

Annals of Discrete Mathematics 30 (1986) 275-284 0 Elsevier Science Publishers B.V. (North-Holland)

275

OY T I E G E N E R A L I Z E D CIIROMATIC NUMBER F l o r i c a Kramer and I l o r s t K r a m e r Computer T e c h n i q u e R e s e a r c h I n s t i t u t e s t r . R e p u b l i c i i 109 3400 Cluj-Mapoca ROFIANIA

The c h r o m a t i c number r e l a t i v e t o d i s t a n c e p,den o t e d b y r(n,C) i s t h e minimum number o f colors s u f f i c i n g f o r c o l o r i n g t h e vertices o f G i n s u c h a way t h a t a n y t w o v e r t i c e s of d i s t a n c e n o t q r e a -

ter t h a n p have d i s t i n c t colors. W e q i v e upper 'f(3,G) f o r h i p a r t i t e ( p l a n a r ) g r a p h s and g e n e r a l i z e a r e s u l t

bounds f o r t h e c h r o m a t i c nur.ber

o f S . A n t o n u c c i q i v i n q a lower bound f o r r ( 2 , C ) .

I. INTFODITCTION I n t h e f o l l o w i n g w e s h a l l u s e some c o n c e p t s and n o t i o n s i n t r o d u c e d

i n t h e book L 5 ] o f C.Oerqe. We r e s t r i c t o u r s e l v e s t o s i m p l e g r a p h s , c o n n e c t e d u n d i r e c t e d f i n i t e g r a p h s which h a v e no l o o p s o r mul-

i.e.

b e a s i m p l e g r a p h , where 17 i s t h e v e r t e x s e t and R i s t!ie ec'ge s e t o f G. I f x , y e V , w e s h a l l d e n o t e by q ( x ) the deqree o f t h e v e r t e x x , by b (G)=max L g ( x ) ; x ~ V t3h e maximum

t i p l e edqes. L e t G = ( V , E )

d e g r e e o f t h e v e r t i c e s of G , by d G ( x , y ) t h e d i s t a n c e b e t w e e n t h e v e r t i c e s x and y i n t h e g r a p h G , i.e.

t h e l e n g t h of t h e s h o r t e s t

p a t h c o n n e c t i n g v e r t i c e s x and y i n t h e q r a p h G ( t h e l e n g t h of a p a t h i s c o n s i r i e r e d t o be t h e number o f e d g e s w h i c h f o r m t h e p a t h ) . d e n o t e s t h e c o m p l e t e g r a p h w i t h n v e r t i c e s and K t h e complete n m*n b i p a r t i t e qraph. it

I n 1969 w e h a v e c o n s i d e r e d i n t h e p a p e r s

[111 and 1123 t h e

follow-

in? c o l o r i n g problem of a graph: DEFINITION 1. L e t p h e a g i v e n i n t e q e r , p z 1 . An a d m i s s i b l e k-color i n q r e l a t i v e t o d i s t a n c e p o f t h e g r a p h G=(T',E) f: V

9 {1,2,...,k)

such that

2

=>

is a function

v x , y € V with 1 ~ d C ( x , v ) p f(x) # f(y). The s m a l l e s t i n t e g e r k f o r w h i c h t h e r e e x i s t s an a r ' r n i s s i h l e k-colo-

F. Kramer and H. Kramer

216

r i n g r e l a t i v e t o d i s t a n c e p is denoted by y ( p , C ) and i s c a l l e d t h e c h r o m a t i c number r e l a t i v e t o d i s t a n c e p of t h e q r a p h G . DEFINITION 2 .

L e t G=(V,E) b e a g i v e n g r a p h and p an i n t e g e r , p

W e s h a l l d e n o t e by G P = ( V , E

set

17

2 1.

t h e g r a p h , which h a s t h e same v e r t e x P a s t h e g r a p h G whereas t h e edge set E is d e f i n e d by: )

5Pp.

( x , y ) E Lp i f and o n l y i f 1 s d G ( x l y )

There e x i s t s t h e n t h e f o l l o w i n q r e l a t i o n between t h e c h r o m a t i c numb e r r ( p , C ) and t h e o r d i n a r y c h r o m a t i c number g(C) : )$(PIC)

In t h e papers

= $-(l,CP) =

[ill,

K(C,P). [121 and [133 w e o b t a i n e d some r e s u l t s r e l a t i v e

t o t h e c h r o m a t i c number

r ( p , G ) , e s p e c i a l l y one c h a r a c t e r i z i n g t h o s e

g r a p h s G f o r which we have r ( p , G ) = p + l . The problem of c o l o r i n q a g r a p h r e l a t i v e t o a d i s t a n c e s w a s recons i d e r e d i n 1 9 7 5 by P.Speranza [ l G ] u n d e r t h e name of L s - c o l o r i n g of a g r a p h ; t h e same oroblem was a l s o s t u d i e d by M.Gionfriddo [6], [7] and [f311S.Antonucci [11 and by G.Wegner L171. 2 . UPPER BOUNDS FOR THE CHROMaTIC NUMBER r ( 3 , G ) G.Weqner 1111 proved t h e f o l l o w i n g theorem: T!IEOREY

1. L e t G=(V,F) be a s i m p l e p l a n a r g r a p h s u c h t h a t

W e have t h e n :

s(2,G)

-

A

( C ) f 3.

f 8.

A s t i l l open problem is t h a t of f i n d i n g a p l a n a r g r a p h C, w i t h maxi-

mal d e q r e e 3 such t h a t w e have $ - ( 2 , G ) = 8 o r else t o prove &-(2,G)f7. F i r s t w e s h a l l g i v e some bounds f o r t h e g e n e r a l i z e d c h r o m a t i c number ) ( " ( 3 , G )of b i p a r t i t e g r a p h s . L e t C=(A,B;E) be a q i v e n b i p a r t i t e graph. t h e g r a p h , which h a s t h e v e r t e x s e t A and t h e cdqe s e t EA is d e f i n e d by:

W e s h a l l d e n o t e b y Gp=(A,EA)

( x , y ) E E A i f and o n l y if d G ( x l y ) = 2 . The g r a p h GA c a n b e o b t a i n e d from t h e g r a p h C, by t h e e l i m i n a t i o n o f t h e v e r t i c e s of 13 and by t h e a p p l i c a t i o n of t h e f o l l o w i n g t h r e e operations: a ) I f w e have a " 3 - s t a r " w i t h t h e c e n t e r i n a v e r t e x y of B ( i t f o l l o w s t h a t t h e v e r t i c e s x1,x2,x3

struct in

GA

a d j a c e n t t o y are i n A ) t h e n w e con-

t h e c i r c u i t x1x2x3x1.

h) I f a verte:: y o f B h a s d e q r e e 2 i n G and x1,x2 a r e t h e two v e r t i ces of A a d j a c e n t t o y i n G I t h e n w e r e p l a c e t h e p a t h xlyx2 o f l e n g t h 2 i n C by an edge (x1,x2) i n GA.

c) I f by t h e a p p l i c a t i o n of t h e p r e c e d i n g two o p e r a t i o n s a p p e a r mult i p l e e d g e s between t w o v e r t i c e s x and y, w e s h a l l r e t a i n o n l y one

011the Generalized Chromatic Number

211

of t h e s e e d g e s and delete t h e o t h e r s . Analogously w e i n t r o d u c e t h e g r a p h G B = ( B , E R )

with t h e v e r t e x set 8

and t h e e d g e set EB d e f i n e d by: ( x , y ) E E B i f and o n l y i f d c ( x , y ) = 2 . An upper hound f o r t h e c h r o m a t i c number

o r d i n a r y c h r o m a t i c numbers

and

$-(C,)

r ( 3 , G ) i n f u n c t i o n of t h e

r(CR)

is qiven i n t h e follo-

winq: TlICOREM 2 .

L e t G = ( A , B ; E ) b e a b i p a r t i t e q r a p h . Then

(1) Proof.

T(3,G)

5 pA) + K(CB).

b e i n g t h e o r d i n a r y c h r o m a t i c number of t h e g r a p h

$(GA)

t h a t ( x , y ) E EA w i t h )-'(CB)

,...,

CA

3 Ll,? r ( G A ) ) of i t s v e r t i c e s such i m p l i e s f l ( x ) # f l ( y ) . We c o n s i d e r a l s o a c o l o r i n g

t h e r e is a c o l o r i n g f l :

A

c o l o r s o f t h e v e r t i c e s of t h e g r a p h GB f 2 : B +

..

[bp(GA)+l,

.

such t h a t ( x , y ) 6 EB i m p l i e s f2 ( x ) # f 2 (y) bb(GR)+ 2 , . , y(GA) + ?,G(GB)) The f u n c t i o n f : A U l 3 -3 ( 1 , 2 , r ( C A ) + g ( C B ) j d e f i n e d by

...,

f(x) =

(2)

{

fl(x),

i f xEA

f2(x), if xEB i s t h e n a c o l o r i n q of t h e v e r t i c e s of t h e b i p a r t i t e g r a p h G=(A,B;E) w i t h '$"(GA)

+

r ( G B ) colors. W e s h a l l now v e r i f y t h a t t h e s o o h t a i -

ned c o l o r i n g is an a d m i s s i h l e ( d i s t a n c e 3 of t h e g r a p h

C,,

i.e.

t (C,) ) - c o l o r i n g r e l a t i v e t o t h a t x , y 6 A V B and 15 d G ( x , y ) 3

f-(GA)

6

i m p l i e s f ( x ) # f ( y ) . R e a l l y , i f d G ( x I y ) = l or d C ( x , y ) = 3 it f o l l o w s t h a t one v e r t e x , l e t it be x , b e l o n g s t o A and t h e o t h e r , y , b e l o n g s

.

t o B. Then o b v i o u s l y f ( x ) # f ( y ) On t h e o t h e r hand i f d G ( x , y ) = 2 it f o l l o w s from t h e b i p a r t i t e c h a r a c t e r o f C t h a t b o t h v e r t i c e s x and y a r e i n A , o r b o t h a r e i n R . C o n s i d e r now t h e c a s e s : a ) i f x , y E A and A G ( x , y ) = 2 , i t r e s u l t s t h a t ( x , y ) E E A and t h e r e f o r e f ( x ) = f1(x) # f p = f(y). b ) i f x , y E B and d G ( x , y ) = 2 , i t r e s u l t s t h a t ( x , y ) c E B and t h e r e f o r e f (XI = f 2 ( x ) f f 2 ( y ) = f ( y ) . Thus w e have proved i n e q u a l i t y ( 1 ) .

THEOREM 3 . L e t G = ( A , B ; E )

be a s i m p l e p l a n a r b i p a r t i t e g r a p h w i t h

A(G)$ 3 . Then w e have (3) r ( 3 , G ) 5 8. T h i s bound is s h a r p i n t h e s e n s e t h a t t h e r e are c u b i c p l a n a r b i p a r t i t e g r a p h s f o r which

X(3,C)=S.

P r o o f . From t h e p l a n a r i t y of t h e i n i t i a l g r a p h G and t h e c o n s t r u c t i o n way of t h e g r a p h s GA and GB r e s u l t s t h e p l a n a r i t y of t h e g r a p h s

GA

and CB. By t h e f o u r - c o l o r theorem proved by K.Appe1 and W.Haken([Z],

218

I;: Krumer and ti. Krumer

and [41) w e have t ( G A ) 4 and t h e n immediately i n e q u a l i t y ( 3 ) .

5

L31

'rl:

Fig. 1 i n Theorem 3 we o b t a i n :

$(GB)

5

4.

B y Theorem 2 f o l l o w s

The f a c t t h a t t h e so o b t a i n e d bound f o r $ ( 3 , C ) can n o t be lowered , c a n be s e e n from t h e f o l l o w i n g example: L e t G=(A,B;E) be t h e g r a p h o f t h e

hexahedron, which is a b i p a r t i t e p l a n a r c u b i c g r a p h w i t h 8 v e r t i c e s and d i a m e t e r 3 . I t r e s u l t s t h a t w e have = 8. I f w e renounce t o t h e p l a n a r i t y of x ( 3 r G ) = IAlJBl

be a b i p a r t i t e g r a p h such t h a t

THEOREM 4 . L e t C = ( A , B ; E )

(GI

f

G

3.

Then w e have: p(3,C)

(4)

5

14.

The o b t a i n e d bound i s s h a r p i n t h e s e n s e t h a t t h e r e e x i s t b i p a r t i t e c u b i c q r a p h s f o r which r ( 3 , C ) = 1 4 . Proof. Consider a b i n a r t i t e qraph G=(A,B;E)

w i t h n ( G ) f 3 and t h e

an(' c,., d e f i n e d above. From A ( C )

correspondinq qraphs

L,

3 results

i p m e d i a t e l y A ( C , ) f 6 and A(C,) L, 6 . A well-known t.!ieorem (see f o r i n n t a n c e [ l 5 ] v o l . 1 , S a t z I V . 2 . 1 ) asserts t h a t f o r any g r a p h G , t h e o r d i n a r y c h r o m a t i c number i s a t most one g r e a t e r t h a n t h e maximum d e g r e e A (GI. W e have t h e n ( G A ) 5 7 and f 7. Applying a g a i n Theorem 2 w e have Y ( ? , G ) 14. The f a c t t h a t t h i s bound i s s h a r p ,

F(GB)

5

can be s e e n from t h e example o f t h e well-known

Ileawood g r a p h (see F i q . 2 )

,

which i s a c u b i c b i p a r t i t e g r a p h w i t h 1 4 v e r t i c e s and o f d i a m e t e r 3 . W e ha-

r ( 3 , C ) = I A U B I = 14. A d i r e c t g e n e r a l i z a t i o n of Theorem 4 i s t h e f o l l o w i n q theorem.

ve t h e n :

Fig.

2

THEORE'fi 5 . L e t C: be a s i m p l e b i p a r t i t e qraDh w i t h maxinum clegree

A

=

a (c).

Then w e have: ~

(5) P r o o f . From

A

(

3

f~

~

+ 1A

(

A -

1)).

= max I g ( x ) ; x E A U B) w e o b t a i n immediately t h a t

t h e number of v e r t i c e s y which a r e a t d i s t a n c e 2 from a g i v e n v e r t e x

On the Generalized Chromatic Number

A (A-1)

x i n t h e q r a p h G i s a t most and

b(GB)

-

r ( C B )f

2 n(A-1)

and t h e r e b y

a(a-1)+1. Applying

COROLLARY 1. L e t G = ( A , B ; E )

279

A(c;,) A( A - 1 ) +1 a n d

.\ve h a v e t h e n

g(GA)5

= L

A(A-1)

respectively

now Theorem 2 w e o b t a i n i n e q u a l i t y ( 5 ) .

be a simple A - r e g u l a r

b i p a r t i t e graph of

d i a m e t e r 3. Then w e h a v e : ( A \

16)

+ !B1 5 2 ( 1 +

h ( a - 1).

Examples o f g r a p h s f o r w h i c h w e h a v e t h e e q u a l i t y s i q n i n ( 5 ) and

1) f o r

also in (6) are:

b

= 2 t h e c i r c u i t C6 o f

a=

lenqth 6; 2) f o r

3 t h e a b o v e d i s c u s s e d ifeawood

g r a p h and 3 ) f o r

A

= 4 t h e 4-regular

b i p a r t i t e q r a p h of d i a m e t e r 3 w i t h 2 6 v e r t i c e s from F i q . 3. D(C) = 3

implies again r ( 3 , C ) = I A U R \ = 26. An improvement of Theorem 5 and o f Theorem 4 c a n b e o b t a i n e d i f w e appl y Brooks theorem (see f o r i n s t a n c e [ 9 1 Theorem 1 2 . 3 ) : I f

G is A-colorable

A(C)=A t h e n

unless:

= 2 and G h a s a comnonent w h i c h

(i)

i s a n odd c y c l e , o r ( i i )A > 2 and

K A + l i s a component o f

Fis. 3

G.

w a s d i s c u s s e d i n [ _ 1 1 ] , 1 1 2 1 and L131.Let u s assume t h a t 7 2 . The g r a p h s CA and GB a r e c o n n e c t e d s i n c e C, i s c o n n e c t e d . I f

The c a s e A.2

A

now a t l e a s t o n e of GA, GB

-

s a y , GA

-

is not a K

t h e n by

g(GA)fA ( A - 1 ) and b y Theorem 2 w e h a v e t h e i n e q u a 5 2 A ( A - 1 ) + 1 . T h u s i f e q u a l i t y h o l d s i n Theorem 5 w e \ B \ = n(n- l ) + land C, i s a b i p a r t i t e A - r e g u l a r q r a p h o f

Brooks t h e o r e m , lity

K(3,G)

have ( A 1 = d i a m e t e r 3. bb(GB)5

I f n o n e o f GA,GB

n(n-1) and

is a K

by Theorem 2

A ( A -1) +1 t h e n )$'(GA) f a(n-1), a ( A - 1 ) . W e c a n now

x(3,G)f2

enounce : COROLLARY 2 . L e t C = ( A , B ; E )

A ( G ) =A >

be a simple b i p a r t i t e qraph such t h a t

2 and min ( I A I , \ B \ ) )9(3,G)

In p a r t i c u l a r , i f

A

>A(a-l)+l.

52 4 ( A -

= 3 and I A \ > 7 , \ R 1 > 7

1;(3,G)

5 -

12.

Then w e h a v e :

1). t h e n w e have:

280

F. Kramer and H. Kramer

3 . LOWER BOIJNDS FOR THE CHROMATIC NUMBER 2('$

,G)

F i r s t we s h a l l p r o v e a r e s u l t needed i n t h e s e q u e l . THEOREM 6 .

(i)

L e t G=(V,E) b e a s i m p l e g r a p h w i t h t h e p r o p e r t i e s :

t h e derjree o f e a c h v e r t e x i s a t l e a s t 2 ,

( i i ) t h e d i a m e t e r o f t h e g r a p h D(G)=2, ( i i i ) G d o e s n o t c o n t a i n c i r c u i t s of l e n g t h 3 and 4 .

Then G is a Moore g r a p h . P r o o f . P r o p e r t y ( i ) i m p l i e s t h e e x i s t e n c e of a t l e a s t o n e c i r c u i t i n t h e g r a p h G. By ( i i ) and ( i i i ) r e s u l t s t h a t t h e g i r t h o f G i s 5 . (7 is t h e n a Moore q r a p h by a r e s u l t o f R . S i n g l e t o n l 1 4 1 which a s s e r t s

t h a t a simgle graph with diameter k

2

1 and g i r t h 2 k + l I s a l s o regu-

l a r and h e n c e a Moore a r a p h . I n 1978 S . A n t o n u c c i L 1 1 o b t a i n e d t h e f o l l o w i n g lower bound f o r t h e c h r o m a t i c number y ( 2 , G ) a s a f u n c t i o n of t h e number of v e r t i c e s and t h e number o f e d q e s o f t h e g r a p h G: THEOREY 7. L e t C = ( V , E ) be a s i m p l e g r a p h w i t h n v e r t i c e s and m e d g e s and w i t h o u t c i r c u i t s of l e n g t h 3 and 4 . Then w e h a v e : n

(7)

3

n 3- 4m2 But S . A n t o n u c c i d i d n ' t g i v e a n y example o f g r a p h s f o r which t h i s hound is a t t a i n e d . W e s h a l l p r o v e t h e f o l l o w i n g theorem: TIIE0RE:I 8. The o n l y g r a p h s of d i a m e t e r D ( C , )

f

2 , without c i r c u i t s of

l e n g t h 3 and 4 , w i t h n v e r t i c e s and m e d g e s f o r w h i c h w e h a v e

a r e t h e g r a p h s K1,

K2 and t h e Moore g r a p h s of d i a m e t e r 2 .

P r o o f . The o n l y g r a p h of d i a m e t e r D(C)=O i s t h e g r a p h K1,

f o r which

we h a v e n = 1 , m=O, '$'(2,G)=1.K1 v e r i f i e s t h e n e v i d e n t l y (8). I f D ( G ) = l , G is a c o m p l e t e g r a p h K w i t h n - 2 . The o n l y c o m p l e t e n 2 , w i t h o u t c i r c u i t s o f l e n g t h 3 is t h e g r a p h K2, f o r graph Knl n which w e h a v e n=2, m=1, ) f ' ( 2 , K 2 ) = 2 and t h e r e f o r e ( 8 ) is v e r i f i e d . I f D(G)=2, w e h a v e t o d i s t i n g u i s h t w o cases:

a)

g = min c q ( x ) : x G V }

= 1. Then t h e r e is a v e r t e x al o f d e g r e e 1

and t h e v e r t e x b a d j a c e n t t o a l h a s t o b e a d j a c e n t t o a l l t h e o t h e r v e r t i c e s o f V b e c a u s e D ( G ) = 2 . But a s

G

d o e s n ' t c o n t a i n c i r c u i t s of

On rhe Generalized Chromaric Number

l e n g t h 3, G i s a ( n - 1 ) - s t a r w i t h n

2

28 I

3 , i . e . G=(V,E) w i t h V= Ca1*a2* i=1,2 n-1J W e h a v e t h e n m=n-1 and E= [ ( a i , h ) , =n. R e l a t i o n (El becomes n ( n 3 - 4 ( n - 1 ) 2 ) = n3 The o n l y sol u t i o n s of t h i s e q u a t i o n a r e n =0, n 2 = 1 and n =n =2, none o f w h i c h 1 3 4 c o r r e s p o n d s b e c a u s e as w e h a v e s e e n a b o v e w e h a v e n 'I, 3. The c o n c l u s i o n is t h a t w e can n o t have b)

x = min

$=

,...,

.

.

1.

{g(x) ; x E V 3 2 2 . G is t h e n a Moore g r a p h o f d i a m e t e r 2

by Theorem 6 . A r - r e g u l a r Moore g r a p h o f d i a m e t e r 2 h a s n = 1

+ J2

v e r t i c e s and m = n. x/2 = $(l+x 2 ) / 2 e d q e s . Because D ( G ) = 2 w e h a v e x ( 2 , G ) = n = 1+J2. I t f o l l o w s t h a t

With t h a t Theorem 7 i s p r o v e d . REMARK.

By a well-known r e s u l t o f A.J.Hoffman

and R . R . S i n q l e t o n

(101

a Moore g r a p h o f d i a m e t e r 2 h a s o n e o f t h e d e g r e e s 2 , 3 , 7 o r 5 7 ;

f o r e a c h o f t h e d e q r e e s 2 , 3 , 7 t h e r e i s e x a c t l y o n e Moore g r a p h of d i a m e t e r 2 ( i t is. n o t known w h e t h e r o r n o t t h e r e i s a Moore q r a p h of d i a m e t e r 2 and d e g r e e 5 7 ) .

lower bound f o r b " ( 2 , G ) s i m i l a r t o t h a t o b t a i n e d by S.P.ntonucci can a l s o b e deduced f o r g r a p h s which h a v e c i r c u i t s of l e n g t h 3 o r 4 .

A

TIIFOREM 9 . L e t C=(V,E) be a s i m p l e c o n n e c t e d g r a p h w i t h n v e r t i c e s and m e d g e s i n w h i c h w e d e n o t e by: (i)

c3 t h e number of c i r c u i t s o f l e n q t h 3 i n G ;

( i i ) c:

t h e number of c i r c u i t s o f l e n g t h 4 , f o r w h i c h n o p a i r of o p p o s i t e vertices i n t h e c i r c u i t are a d j a c e n t i n G ; 1 ( i i i ) c 4 t h e number o f c i r c u i t s o f l e n g t h 4 , f o r which o n e p a i r o f o p p o s i t e v e r t i c e s i n t h e c i r c u i t a r e a d j a c e n t i n C and t h e o t h e r p a i r of o p p o s i t e v e r t i c e s a r e n o t a d j a c e n t i n G . If G doesn't

c o n t a i n a n y s u b a r a p h of t h e t y p e K t h e n t h e chroma2,3 t i c number X ( 2 , G ) v e r i f i e s t h e i n e q u a l i t y 3 n (9)

c

3 0 1 2 n +n ( 6 c 3 + 4 c 4 + 2 c 4 )- 4 m

I

**

T h i s bound i s s h a r p i n t h e s @ n s e t h a t t h e r e e x i s t s s r a p h s v e r i f y i n q t h e h y p o t h e s e s o f t h e t h e o r e m and f o r which w e h a v e t h e e q u a l i t y siqn in (9). P r o o f . The p r o o f o f t h i s t h e o r e m c a n b e o b t a i n e d hy a m o d i f i c a t i o n of t h e p r o o f g i v e n by S . A n t o n u c c i f o r Theorem 7 . A s w e h a v e o b s e r v e d

282

F. Kramer and H. Kramer

above w e h a v e

Y(2,C)

=

s q u a r e of t h e q r a p h G.

$(llC2)

=

2

r(C;) ,

where c; 2 = ( V I E ) i s t h e 2

I f w e d e n o t e b y m2 t h e c a r d i n a l i t y of t h e

e d q e set E 2 , t h e n w e h a v e by a Theorem of C . B e r q e r(2,C= ;) g(G2) 7 =

(10)

2

( c 5 J l p.321)

.

n2-2m2 The number o f a l l p o s s i b l e p a t h s of l e n g t h 2 i n t h e q r a p h by t h e sum

2 (q(ii)).

xyz of lenq&'

G

i s qiven

If we introduce corresnondinq t o each path

2 i n C an e d q e ( x , z ) w e s h a l l o h t a i n a q r a p h C " = ( V , E " ) .

C E", h u t t h e r e may h e e d q e s i n E 2 which are m u l t i p l e e d g e s i n El'. L e t a , b EV h e a p a i r of v e r t i c e s , which i s c o n n e c t e d i n O b v i o u s l y E2 C;

by a t l e a s t o n e p a t h of l e n q t h 2 . W e h a v e t o d i s t i n q u i s h t h e cases:

1) a and h are a e j a c e n t v e r t i c e s i n G . Then t h e ec'qe ( a , b ) i s cont a i n e d i n a t l e a s t one c i r c u i t of l e n q t h 3 i n G and t h e o r d e r of mult i p l i c i t y of t h e e d q e ( a , b ) i n E" w i l l be e q u a l w i t h t h e numher of c i r c u i t s o f l e n n t h 3 which c o n t a i n t h e e d q e ( a , b ) . As e a c h c i r c u i t of l e n g t h 3 c o n t r i b u t e s t o t h e i n c r e a s e of t h e m u l t i n l i c i t y o f e a c h e d g e of t h i s c i r c u i t by one u n i t y , w e h a v e t o d e l e t e 3c3 e d q e s from El' i n o r d e r t o make a l l e d g e s a p a r t a i n i n q t o c i r c u i t s of l e n q t h 3 simple e d g e s . 2 ) a and 13 a r e n o t a d j a c e n t i n C. Because G d o e s n o t c o n t a i n any s u b g r a u h of t h e t y p e K 2 , 3 , t h e v e r t i c e s a and h c a n h e c o n n e c t e d i n G by a t most t w o p a f h s of l e n g t h 2. W e d i s t i n q u i s h t h e n t h e s u b c a s e s : 2 a ) a and b a r e c o n n e c t e d i n C: by e x a c t l v one p a t h of l e n g t h 2 . Then ( a , b ) i s obviously a simple edge of t h e graph G " . 2b) a a n d b a r e c o n n e c t e d i n G by two p a t h s of l e n q t h 2 . The e d g e ( a , h ) w i l l b e a d o u b l e edge i n C". But i n t h i s c a s e a and b form a p a i r of o p p o s i t e v e r t i c e s i n a c i r c u i t of l e n q t h 4 i n G . B e c a u s e a c i r c u i t of l e n q t h 4 of t h e tyrJe (ii) leads t o t h e d u p l i c a t i n g of b o t h d i a g o n a l s of t h e c i r c u i t , and a c i r c u i t of l e n g t h 4 o f t h e t y p e ( i i i ) l e a d s t o t h e d u p l i c a t i n q o f o n l y o n e d i a q o n a l , i n o r d e r t o obe d g e s from El' b e s i d e t h e t a i n t h e q r a p h C2 w e h a v e t o d e l e t e 2c:+c: 3c3 edctes a l r e a d y d e l e t e d . W e h a v e t h u s n g(xi) 1 (3c3+2cy+c4). m2 = m + i=1 I t r e s u l t s then:

t(

)-

283

On the Generalized Chromatic Number

2

- -2m-

.

1

( 3c3+2c>c4)

n

T h i s i n e q u a l i t y and. (10) v i e l c l s

As

r(2,C)

w h i c h [r]*

i s an i n t e g e r w e o b t a i n i m m e d i a t e l y t h e i n e q u a l i t y ( 9 ) i n 2 r. An e x a m p l e o f a n r a p h f o r w h i c h w e

denotes t h e smallest i n t e q e r

have t h e e q u a l i t y s i g n i n ( 9 ) i s

4 f o r which we 1 h a v e n = 7 , m = l l , c = 3 , c 4 = l , c0=2, 4 3 D ( G ) = 2 and )f(2,C)=n=7. I t i s e a s y t h e q r a p h from F i g .

to v e r i f y t h a t f o r t h i s qraph we have t h e r e l a t i o n s : Fiq.

i3

n

4

3

'i'

= [343/551*

0

1

n +n ( 6 c 3 + 4 c 4 + 2 c 4 )- 4 m

= 7 = $(2,C,).

ACKNOWLEDCYENT. The a u t h o r s w i s h t o t h a n k t h e r e f e r e e f o r t h e h e l p f u l comments.

The s e c o n d a u t h o r would l i k e t o t h a n k a l s o t o t h e

A l e x a n d e r von H u m b o l d t - S t i f t u n q f o r t h e f i n a n c i a l s u p p o r t d u r i n g t h e y e a r s 1981-1982. REFERENCES [l) A n t o n u c c i , S . ,

G e n e r a l i z z a z i o n i d e l c o n c e t t o d i cromatismo d ' u n

q r a f o , Boll.Un.Vat.Ita1.

[23

Eq

[q

Appe1,K. ,Haken,W., Amer.Vath.Soc.

( 5 ) 15-B

( 1 9 7 8 ) 20-31.

E v e r y p l a n a r map i s f o u r c o l o r a b l e , B u l l .

82 ( 1 9 7 6 ) 711-712.

Appe1,K. ,Ifaken,IJ.,

E v e r y p l a n a r map is f o u r c o l o r a h l e , P a r t I .

D i s c h a r g i n g , I l l i n o i s J.Math. Appe1,K. ,Iiaken,W. , K o c h , J . ,

2 1 ( 1 9 7 7 ) 429-490.

E v e r y p l a n a r map i s f o u r colorable,

? a r t 11. R e d u c i b i l i t y , I l l i n o i s J . M a t h . ti51 R e r q e , C .

[6!

,

Craphes e t hypergraphes

Gionfriddo,M., Mat.Ita1.

S u l l e c o l o r a z i o n i Ls d ' u n g r a f o f i n i t o , B o l l . U n .

( 5 ) 15-A

[7]

Gionfriddo,M.

[8]

Gionfriddo,M.,

2 1 ( 1 9 7 7 ) 491-567.

(Dunod, P a r i s , 1 9 7 0 ) .

( 1 9 7 8 ) 444-454.

, Alcuni

r i s u l t a t i r e l a t i v i a l l e c o l o r a z i o n i Ls

d ' u n q r a f o , Riv.Mat.Univ.Parma

(4)

6

( 1 9 8 0 ) 125-133.

Su un p r o b l e m a r e l a t i v o a l l e c o l o r a z i o n i L2 d ' u n

q r a f o p l a n a r e e c o l o r a z i o n i t s I Riv.Mat.Univ.Parma

(4) 6 ( 1 9 8 0 )

I;. Krarner and H . Kramer

284

151-160.

1 IIarary,F., Graph Theory

(Addison-Wesley Publ.Comp. ,Mass. 1969). - - Hoffman,A.J. ,Singleton,R.R., On Moore graphs with diameters 2 and 3, IBM J.Res.Deve1op. 4 (1960) 497-504. [l
[lo1

rlq -

.

Annals of Discrete Mathematics 30 (1986) 285-290 0 Elsevier Science Publishers B.V.(North-Holland)

285

A CONSTRUCTION OF SETS OF PAIRWISE ORTHOGONAL F-SQUARES OF COMPOSITE ORDER Paola L a n c e l l o t t i and Consolato P e l l e g r i n o

D i p a r t i mento d i Ma t e m a t i ca V i a Campi 213/B 41100 MODENA (ITALY)

I n t h i s n o t e complete systems o f o r t h o g o n a l F-squares a r e c o n s t r u c t e d i n which t h e number o f symbols i s v a r i a b l e and t h e o r d e r n i s a prime power. F u r t h e r , we g i v e an e x t e n s i o n o f t h e MacNeish theorem by c o n s t r u c t i n g systems o f o r t h o g o n a l e F-squares o f composite o r d e r n = p1 elp2e2 p, m having a v a r i a b l e number o f symbols: such c o n s t r u c t i o n improves t h e r e s u l t s t h a t have been reached so f a r .

...

1

-

DEFINITIONS AND PRELIMINARY RESULTS

A square m a t r i x

F = [aij]

o f order n

, defined

,i s

on a s e t A

s a i d t o be

an

F-square o f t y p e (n,1) (and we s h a l l w r i t e s h o r t l y F(n,A) i f each element o f A appears A t i m e s i n each row and i n each column o f F Given t h e F-squares F1(n,’xl)

.

and F2(n,A2)

d e f i n e d r e s p e c t i v e l y on t h e s e t s A1 and A2

, we

s h a l l say t h a t t h e y appears ;\1x2

a r e o r t h o g o n a l i f a f t e r superimposing them each p a i r (x,y)€A,xA2 times. I n such case we s h a l l w r i t e F 1 ( n y ‘ x 1 ) ~ F p ( n y ’ x 2 ) . Given t h e F-squares

Fl(n,’xl)

, F2(n,’x2) ,

...

, Ft(n,’xt)

, we

s h a l l say t h a t t h e y

.

f o r m an o r t h o g o n a l system i f

If

ItAl,‘x

2,...,~tll

z

Fi(n,’xi)LF.(n,’xj) for all i,j=1y2y...yty i#j J we s h a l l say t h a t t h e o r t h o g o n a l system has a v a r i a b l e

1

number o f symbols, I n

3

1

t h e f o l l o w i n g theorem has been proved:

THEOREM 1 .l.The maximal number

..., Ft(n,’xt) i=1,2, ...,t ) :

t

o f orthogonal

F-squares

s a t i s f i e s t h e f o l l o w i n g i n e q u a l i t y (we s e t

i

mi

i=1

-

t

An o r t h o g o n a l system o f F-squares

<

(n-1)

2

F1(n,.“,),

F1 (n,hl),

n = l .im i

. F2(n,X2),

...

, Ft(n,At)

complete i f

5

mi

-

t

=

(n-1) 2

.

i=1 Complete systems o f

F2(n,’x2), for

F-squares a r e c o n s t r u c t e d i n

[2 ] , [3 ]

.

is

called

P. Laricellotti and C. Pellegritlo

286

-

2

CONSTRUCTION OF COMPLETE SYSTEMS OF F-SQUARES

,...,Xr

We s h a l l say t h a t a p a r t i t i o n . d =I XI,X2 is

1 Xi\

h-regular i f

@ = O(

&)

o(ai)

,..,, r ) .

(i=1,2

= h

t h e mapping f r o m A

to

1,

1 o f an m-set A=Ial,a2 ,...,am) I f I,= I 1 , 2 ,..., r 1 we denote by

d e f i n e d by

i f and o n l y i f

= j

aie x

j '

The mapping o w i l l be c a l l e d t h e c a n o n i c a l mapping a s s o c i a t e d w i t h t h e p a r t i t i o n Sg Given an F-square F o f t y p e (n,h) d e f i n e d on a s e t A and t h e c a n o n i c a l mapping o a s s o c i a t e d w i t h an h - r e g u l a r p a r t i t i o n .d o f A , we r e p l a c e each element x o f F by ~ ( x and ) o b t a i n a n F-square o f t y p e (n,ih).We s h a l l c a l l such a square t h e descendant o f F by o and denote i t by o ( F ) .

.

OBSERVAT 10'1 I f F i s an F-square o v e r t h e s e t w i t h a n 1-regular p a r t i t i o n o f A

PROPOSITION 2.1. L e t d e f i n e d on t h e s e t s

F1 Al

A

and @ i s t h e c a n o n i c a l mapping a s s o c i a t e d we can i d e n t i f y @(F) w i t h F

.

, F2

be F-squares o f t y p e (n,x,)

and

A2

; let

o1

and (n,h2) r e s p e c t i v e l y ,

be t h e c a n o n i c a l mapping a s s o c i a t e d

w i t h an h l - r e g u l a r p a r t i t i o n dl = I X1 ,X2,,,.,Xrl o f A1 and l e t o2 be c a n o n i c a l mapping a s s o c i a t e d w i t h an h 2 - r e g u l a r p a r t i t i o n d2= { Y1 ,Y2,..., of

A2 ; i f

F1

1F2

then

o,(F1)

1a2(F2) .

constitute a partition

element

(a,b)cXi

F2 t h e p a i r ol(F1) with

x Y.

J

hlh2

yS}

...,

(i=1,2, r ; j=1,2, Xi x V j A1 x A2 ; f u r t h e r , s i n c e each times i n t h e s u p e r i m p o s i t i o n o f F1 w i t h

PROOF. L e t us observe f i r s t o f a l l t h a t t h e s e t s

...,s )

the

-regular o f

appears ~~h~ x Isappears i 1 x 2 hlh2 t i m e s i n t h e s u p e r i m p o s i t i o n o f and t h a t proves t h e a s s e r t i o n .

( i y j ) 61,

e2(F2)

PROPOSITION 2.2. Given an o r t h o g o n a l system .F o f F-squares, i f one r e p l a c e s one p a r t i c u l a r square o f 9 by a s e t o f descendants which a r e p a i r w i s e o r t h o g o n a l , one obtains a n o r t h o g o n a l system again. PROOF. I t f o l l o w s i m m e d i a t e l y f r o m P r o p o s i t i o n 2.1. and f r o m t h e o b s e r v a t i o n p r e c e d i n g it . PROPOSITION 2.3. L e t F be an F-square o f t y p e (n,x) d e f i n e d on a s e t A ; l e t Q be t h e c a n o n i c a l mapping a s s o c i a t e d w i t h an h - r e g u l a r p a r t i t i o n d ={ X1,X2,

...,X r }

of

A

and l e t

B={Yl,Y2,

partition

(I) holds, t h e n

lxin

YJ

be t h e c a n o n i c a l mapping a s s o c i a t e d w i t h a

...,Ys yj

I

1 of A

? -- r-

=

k-regular

; then i f the c o n d i t i o n

(i=1,2

,...,r

; j=1,2

,...,s )

o(F) I Y ( F ) and c o n v e r s e l y .

PROOF. Superimposing

@ ( F ) and

times i n each row and hence

Y(F) each p a i r

nAlXi

n Y J. 1

( i 7 j ) e 1, x

Is appears A I X i n V . 1

times altogether; i t follows t h a t

J

Sets of Painvise Orthogonal F-Squares

Xi n Y i s the j and t h a t proves t h e a s s e r t i o n .

i f and o n l y i f t h e c a r d i n a l i t y o f t h e s e t s

Y(F)

Q(F) f o r each

287

.,r ; j=1,2,.

i=l,Z,..

..,s

same

Two mappings Q and 'Y s a t i s f y i n g t h e hypotheses o f P r o p o s i t i o n 2.3. t o g e t h e r w i t h t h e c o n d i t i o n ( I ) w i l l be c a l l e d o r t h o g o n a l and denoted b r i e f l y b y Q _L Y

I

PROPOSITION 2.4. L e t (G,t) be an a b e l i a n group; l e t H, K be sobgroups o f G and Q , Y t h e c a n o n i c a l mappings o f G o n t o t h e q u o t i e n t groups G/H ang G/K respectively. I f (11) then

Q

H

t

K

IXty : X E H

, y6K

G 1

=

1Y .

PROOF. S i n c e H n K i s a subgroup o f G , i t s u f f i c i e s t o p r o v e t h a t f o r each X E G / H and f o r each Y a G / K t h e c o n d i t i o n X n Y E G/H.nK h o l d s . I n f a c t i f X = a t H and Y = b t K , t h e n i t f o l l o w s f r o m ( 1 1 ) t h a t X = y t H f o r some element y c K and Y = x t K f o r some element x e H I f Z E X n Y t h e n we have

.

y

2

x' = x

t

t

y'

with X'

-

= (X t

y'

-

x = y'

Y E

.

HflK

Hence

z

-y

t

y ) E (X

y)

t

HfIK

t

and t h a t means X A Y c_(x t y ) t

(x

S i m i l a r l y one shows t h a t PROPOSITION 2.5. d e f i n e d on a s e t

t

y)

t

HAKSXnY

For each p r i m e power

A

o f cardinality

.

HllK

q m=qk

and t h a t proves t h e a s s e r t i o n .

and f o r each F-square

, there

F

e x i s t s r = (qk

o f t y p e (n,A)

-

l)/(q

-

1)

I q' PROOF. L e t us c o n s i d e r a k-dimensional v e c t o r space V o v e r GF(q); f o r any two d i s t i n c t ( k - 1 ) - d i m e n s i o n a l v e c t o r subspaces W1 , W2 o f V we have W1 @ W2 = V.

p a i r w i s e o r t h o g o n a l descendants o f F of t y p e

(n,Aqk-l)

d e f i n e d on t h e s e t

-

Since t h e r e a r e r = (qk l ) / ( q - 1 ) d i s t i n c t ( k - 1 ) - d i m e n s i o n a l v e c t o r subspaces o f V , i t f o l l o w s f r o m t h e p r e v i o u s p r o p o s i t i o n t h a t F admits r p a i r w i s e k-1 o r t h o g o n a l descendants o f t y p e (n,Aq ). PROPOSITION 2.6.

Let 9

be a complete system o f

t

F-squares o f o r d e r

one o f t h e squares of t h e system i s d e f i n e d o v e r a c a r d i n a l i t y prime power q , t h e n t h e r e e x i s t s a complete system o f F-squares o f o r d e r n , PROOF, An immediate consequence o f P r o p o s i t i o n 2.2.

m = qk

s = t - 1

t

n ; if for

a

k

(q -l)/(q-1)

and 2.5.

The P r o p o s i t i o n 2.6. p e r m i t s , i n a s i m p l e way, t o c o n s t r u c t complete systems e i t h e r o f known t y p e o r w i t h a v a r i a b l e number o f symbols. F o r i s t a n c e we can o b t a i n complete systems o f b o t h t y p e s by r e p l a c i n g one o r more l a t i n squares, w r i t t e n i n t h e f i r s t column o f t h e f o l l o w i n g t a b l e , w i t h t h e r e s p e c t i v e descendant w r i t t e n beside each o f them.

P. Lancellotti und C. Pellegrino

288

3

-

0 1 2 3

1 0 3 2

2 3 0 1

3 2 1 0

0 0 1 1

0 0 1 1

1 1 0 0

1 1 0 0

0 1 0 1

1 0 1 0

0 1 0 1

1 0 1 0

0 1 1 0

1 0 0 1

1 0 0 1

0 1 1 0

0 1 2 3

2 3 0 1

3 2 1 0

1 0 3 2

0 0 1 1

1 1 0 0

1 1 0 0

0 0 1 1

0 1 0 1

0 1 0 1

1 0 1 0

1 0 1 0

0 1 1 0

1 0 0 1

0 1 1 0

1 0 0 1

0 1 2 3

3 2 1 0

1 0 3 2

2 3 0 1

0 0 1 1

1 1 0 0

0 0 1 1

1 1 0 0

0 1 0 1

1 0 1 0

1 0 1 0

0 1 0 1

0 1 1 0

0 1 1 0

1 0 0 1

1 0 0 1

EXTENSION OF THE MACNEISH THEOREM

MacNeish i n 1922 proved the f o l l o w i n g theorem THEOREM 3.1.

(cf. [ 4 ] ):

n be el e2 pmem then there e x i s t s a s e t o f min(pl ,p2 , n = p p2 pairwise orthogonal l a t i n squares o f order n

el e2

I f we l e t the prime decomposition o f a number

...

.

...

e

,p,

3.P.Mandeli and W.T.Federer i n 1983 ( c f . [ 5 ] ) gave a method f o r constructing orthogonal systems o f F-squares o f composite order n w i t h a v a r i a b l e number o f symbols, using a technique which i s an extension o f the MacNeish theorem f o r l a t i n e el e2 then the system consists o f the F-squares squares. If n = p1 p2 p, defined as follows:

...

c

... c

c

...

LJ m '

j = 1, 2,

... B

LJ

j = n 1' n1tl,

On1n2.. n.mm-l

B LAitl

0

On1n2., n.mm-l

c

Lil 8

LA 2

0 0 "1

LJ

we have denoted by

"m

... c

j = n

LJ"m

...,n2-1

n.+l,...,n i' i

j = nmm-lynmm-,tly

LJ "m

Yi =

nl-1

1 Lni

2 Lni

, ...

it1-l

...,nm-1

;

n.-1 L

'

"i

a complete system o f l a t i n

Sets of Pairwise Orthogonal F-Squares

squares o f o r d e r

ei

ni

:= pi

0

and by

, by

On

289

t h e square m a t r i x o f o r d e r

n

with a l l

@

t h e d i r e c t sum of m a t r i c e s . By P r o p o s i t i o n 2.6. e. 1 i s p o s s i b l e t o r e p l a c e each square o f o r d e r pi w i t h a system o f t

e n t r i e s equal t o

F-squares. i=2,3,

...,m

If

s

denotes t h e g r e a t e s t o f t h e number

si=(pi

it

e. el '-p, )(

we can t h e n s t a t e t h e f o l l o w i n g Theorem

THEOREM 3.2.

I f t h e p r i m e decomposition o f a number n i s q i v e n by e el e2 em el e2 ( p1 < p2 < < p, rn ) t h e n t h e r e e x i s t s a system n = p1 P2 * * . P,

...

t = s + p 1 F-squares of o r d e r n w i t h a v a r i a b l e number o f symbols; 1 such system c o n t a i n s t h e system o f p a i r w i s e o r t h o g o n a l l a t i n squares c o n s t r u c t e d by MacNeish.

of

EXAMPLE. If n = 2 3 - 3 3 - 3 1 we c o n s t r u c t a system o f

76

F-squares which c o n t a i n s :

a ) t h e system o f 7 p a i r w i s e o r t h o g o n a l l a t i n squares c o n s t r u c t e d by MacNeish; 3 2 b ) 23 F-squares o f t y p e (n,2 . 3 ) ; 3 2 c ) 46 F-squares o f t y p e (n,2 - 3 - 3 1 ) . ACKNOWLEDGEMENTS. Work done w i t h i n t h e sphere o f GNSAGA o f CNR, p a r t i a l l y supported by M P I . REFERENCES J.P.Mandeli, F.C.H.Lee, W.T.Federer, On t h e c o n s t r u c t i o n o f o r t h o g o n a l F-squares of o r d e r n f r o m an o r t h o g o n a l a r r a y (n,k,s,2) and an OL(s,t) s e t ; J . S t a t i s t . P l a n . I n f e r e n c e 5 (1981) 267-272. A.Hedayat, D.Raghavarao, E.Seiden, F u r t h e r c o n t r i b u t i o n s t o t h e t h e o r y o f F-squares d e s i g n ; Ann. S t a t i s t . 3 (1975) 712-716

.

W.T.Federer, On t h e e x i s t e n c e and c o n s t r u c t i o n o f a complete s e t o f o r t h o g o n a l F ( 4 t ; 2 t , Z t ) - s q u a r e s d e s i g n ; A n n . S t a t i s t . 3 (1977) 561-564 H.F.MacNeish,

E u l e r ' s squares ; Ann. Math. 23 (1922)

221-227

.

.

J.P.Mandeli, W.T.Federer, An e x t e n s i o n o f MacNeish's Theorem t o t h e c o n s t r u c t i o n o f s e t s o f p a i r w i s e o r t h o g o n a l F-squares o f composite o r d e r ; U t i l . Math, 24 (1983) 87-96

.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 291 -296

0 Elsevier Science Publishers B.V. (North-Holland)

29 1

RIGHT S-n-PARTITIONS OF A GROUP AND REPRESENTATION OF GEOMETRICAL SPACES OF TYPE "n-STEINER" Domenico Lenzi Dipartimrnto di Matematica Universith degli Studi LECCE (Italy)

SUMMARY. In [9] we gave a generalization of thr notions of subgroup and of S-partition of a group, in order to obtain a description of all the linear spaces having a transitive group of collineations, by means of a method that generalizes some methods used in particular cases l i k e transitivr projective planes, transitive linear spaces with a parallelism (in the sense of AndrB) and others (see bibl.iography) . after some considerations on gromrtrical spaces, rxtrnd our method, by means of the concept of right S-n-partition of a group ( s e e def.6), in order t o obtain a description of all geometrical spaces (E,Q), where &jcz,y(E), of type "n-Steiner" (i.e.: if P1 , . . . , Pn+l arr n arbitraty and pairwisr different elrmrnts of E , then there is a unique Be%? such that {P1,...,Pn+l}c B) having a transitive group of automorphisms. Hrre,

IVO

N.1.PRELIMINARIES AND RECALLS. Let G be a group and S a subgroup of G . Wr shall say that a subset Q of q ( G ) is a partial right S-covering of G if the following properties hold:

REMARK 1. Onr can immediately verify that: (a) S a c A , for every AeB. lndeed S c Aa-l,by (2) and(1). (b) Let I:=AzBA. T h e n , by ( 2 ) , A1-l 2 I for every AeB and i e I ; thus A 2 I 1 and hence I is a submonoid of G . This ensurrs that t h e set of t h e subgroups of G included in I has a maximum

292

D.Lenzi element (c)

H 1= {ieG: Ii=I }

= {ieG: iI=I} = { ieG;VAeQ ,iA=A}.

If we set &G:={Ag:AeQ , geG1, then the group of the right translations determined by the elements of G is a group of automorphisms of the geometrical space ( G , 0- G ) . Moreover Q coincides with the such that leB; set of the elements B E & G as a consequence, for xeG and B e Q G , xeB if and only if B x - l e 8 .

(d) For every A , A ' e Q , let us set AiA' when A'=Aa-', where a is a suitable element of A. In such a manner we define an equivalence relation on & . Now. for an arbitrary right S-covering & o f G , let: E:=ISx:xeC} (the set of the right cosets of S in G ) ; < B > : ={Sx:Sx (for Be(tG) and W : = { < B > : B e Q G I .

B}

If we set (Sx)*y=S(xy) (for every x,yeC). then G becomes a transitive generalized automorphism group of the geometrical space (E,g)(i.e.: the function that associates to every geG the permutation 8 acting on E like g is an homomorphism from G into the automorphism group of ( E , O ) ) . Conversely. every geometrical space (E' a')having a transitive generalized automorphism group G can be obtained (but for an isomorphism) in the previous manner. Let indeed P be a fixed element of E' and S:= {geG:Pg=P1 (the stabilizer of P in C); if we associate to every Q=PgeE' the right coset Sg. then we obtain a bijective function f from E' onto the set E * of the right cosets of S in G ; as a consequence we can construct a geomrtrical space (E*, a * ) (by setting g*:=W'f): furthermore G becomes in a natural way (by setting (Sx)*y:=(Sx)f-'yf=S(xy)) a transitive generalized automorphism group of (E*, a * ) ,Now let us B; : = s x y B ~ (for S ~ every B*e W*) and : = { B* :leBL} U It is easy to verify that Q is a partial right S-covering of G and (E*,B*) can be obtained from Q. in the previous manner.

.

If ( E . 9 )

is a geometrical space let u s set, for PeE. Wp=IBeg:PeBl B(the closure of P in W ) , Then (E,g) is said to be

and = a To space when, for every X,YeE, x=y implies that X = Y ; moreover, ( E . B ) is said to be a T1 space when. f o r every XeE,IX(=l. REMARK 2. Obviously, if ( E , B ) has a transitive generalized automorphism group and P is a fixed element of E , then (E,g)

293

Right S-n-Partitions of a Group

is a To space if and only if, for every X e E , X = P ; moreover, (E,.B) is a TI space if and only if

x=P P =

{

implies that PI.

Now let us fix a subgroup S and a right S-covering The following propositions hold. PROPOSITION 3 . For every geG, A70.(Ag)=Apo-A ( s e e (b) in remark 1).

(Z

of G.

if and only if geHl

PROOF. T h e assert is obvious, since ,4za(Ag)=(A2fiA)g=

Ig. Q.E.D.

PROPOSITION 4. Let (E,.%?) b e the geometrical space associated to Q with respect to S ; then: n A. i) ( E L B ) is a TI space if and only if S =A€&

ii) ( E 3 ) is a To space if and only if S=H1 (see (b) remark 1).

in

PROOF. i) It is obvious. ii) For every e 3 , S g e < B > if and only if geB; then the closure of Sg in B is the stit of the right cosets o f S included in ge'&B B= A2m,(Ag). As a consequence, is equal to the closure of S in if and only if (Ag) = Apa A ; whence the thesis by proposition 3 and remark 2.

Q.E.D. In [9] ( s e e thror. 9 ) we proved that a group G is a transitive generalized group of automorphism (collineation) of a linear space if and only if it admits a right generalized S-partition. We can say that a subset of 9 ( G ) is a right generalized S-partition of G if it is a partial right S-covering of G 'and the following properties hold: (3) AYaA = G ; (4) VA1,A2eQ: (5)

Al#A2

a

AlnA2

= S;

wa.

By (1),(2) and (4) it is easy to verify that, for every element A of a right generalized S-partition 0 of G , the following property holds (cfr. 191. def.2):

(6) S

c

A and

( V x , y e A : A ~ - ~ # A y -= l Ax-l n Ay-l=S).

Wr observe that the foregoing conditions can be reformuled, by virtue of the following PROPOSITION 5 .

Let

Q

c

9 ( G )

and

I:=

Aga

A ; furthermore let

D.Lenzi

294

property (2) hold, and:

= A1nA2

A1 # A 2

( 4 ' ) VA1,A2eC2:

= I.

Then I i s a s u b g r o u p o f G. can suppose

We

.

of Q

101 #

1. Now l e t A1.A2 different elements -1 -1 -1 -1 T h e n , f o r e v e r y i , j e I , i j eAlj nA2j But A , j - ' # A 2 J ,

PROOF.

.

h t ~ n c t( ~b y ( 2 ) and ( 4 ' ) )

A1j

N.2.

THE

THE

OF

CASE

-1

nAZj-'=I, therefore i j - l e I .

GEOMETRICAL

OF

SPACES

Q.E.D.

TYPE

"n-STEINER".

Let u s g i v e t h e f o l l o w i n g

DEFINITION

6.

shall

We

say

a

that

fl+ o f

subset

thr

power

? ( G ) o f
-

( 7 ) Vgl , . . . ,g n e G ] A € & : { g l ( 8 ) VA1,A2e& : A1 # A2

, . . . ,9 , ) s

;1

set

pcirtial

A.

A1 n A 2

is

tht.

of m 5 n r i g h t c o s e t s of S i n G (whrre m

union

is drpen-

d i n g o n A1 a n d A 2 ) .

I t is easy t o v e r i f y t h a t i f ( E ' , B ' ) i s a geometrical space o f t y p e " n - S t r i n e r " ( s e e summary) h a v i n g a t r a n s i t i v e g e n e r a l i z r d automorphism group G , t h e n t h e set I'lc P ( C ) f r o m w h i c h ( E ' , 2 8 ' ) c a n b r o b t a i n e d (*I i s a r i g h t S - n - p a r t i t i o n of G . Convrrsrly,

('I b e

let

a

right

S-n-partition

This

of

G;

then

the

i s of t y p e " n - S t r i n e r " .

associated gromrtrical space (E,%)(sre N.l)

an immediate consequence of t h e f o l l o w i n g

IS

PROPOSITION different

7.

If

right

Sh l , . . . , S h n t l

are

S in G,

c o s e t s of

n

then

arbitrary there

and

pairwise

is a unique s u b s r t n

B of

G of

w i t h AeQ

t y p e Ag,

and geG,

PKOOF. By ( 7 ) a n d ( l ) , t h e r e i s AeQ s u c h c A,

hence Ihl

, . . . ,1

~

~ , t h u~ s

lil S h l c

such t h a t -1

..

B.

-1

t h a t { hlhn+l.. ,hnhntl,l)c n+l ~U S h l 1 4 Ahntl ~ (since

SA=A). On

the

other

gl,g2eG) (see

the

hands, i f n+l then S l!lShlh-:tl second

part

Algl n A2g22Y#

4. Alglh,iln of

(c)

in

Shl

(with

A2g2h,:l.

remark

1)

Al,A2e@

As a c o n s e q u e n c e -1 -1 Alglhn+l=AZg2hntl

b y v i r t u e o f (8), a n d h r n c e A1g1=A2g2.

*

( ) Cut f o r a n isomorphism;

srr N . l .

and

Q.E.D.

295

Right S-ri-Partitionsof a Group

REMARK

8.

Wr

obsrrvr

that

is a right

if

-

t h e n ( b y ( 8 ) and (2))

Ax-' m < n right cosets i s d e p e n d i n g on A ) .

(9)

S-n-partition

of

G,

f o r e v e r y AeQ t h e f o l l o w i n g p r o p e r t y h o l d s :

Vx,yeA:A~-~#Ay-l

n Ay-I

of

of

the

is

in

S

G

iinion

(whrrr

m

s h a l l s a y t h a t a s u b s r t A of C i s a r i g h t S - n - b l o c k o f G i f S c A and p r o p e r t y ( 9 ) h o l d s ( * * ) ; m o r e o v e r i f A i s a r i g h t S - n - b l o c k o f G b u t n o t a r i g h t S - ( n - 1 ) - b l o c k , we s h a l l s a y t h a t A is a proper r i g h t S-n-block of G.

W r

O b v i o u s l y , e v e r y r i g h t S - o - b l o c k i s a subgroup of G . PROPOSITION 9 . A n e c e s s a r y ( a n d t r i v i a l l y s u f f i c i e n t ) c o n d i t i o n f o r a s u b s e t A of G t o b r a r i g h t S - n - b l o c k o f G i s t h a t : (10) S

c

A,

and (11)

Vgl.g2

e

G

union of m

5

# Ag2

Agl

:

-

Agl n Ag2

is

the

n r i g h t c o s e t s of S i n G .

PROOF. I t i s e n o u g h t o p r o v e t h a t t h e c o n d i t i o n ( 1 1 ) i s n e c e s s a r y . H r n c r l r t A b r a r i g h t S - n - b l o c k o f G , l e t Agl#Ag2 ( w h e r e

gl.g2eG)

and l e t t e A g l g i l n A .

Then A g , g , ' t - l # A t - l

and t g 2 g i 1

-'-'

e A n Ag2g;t C o n s e q u e n t l y , by ( 9 1 , Aglg2 t n At-' (and a l s o Agl n AgZ) i s t h e u n i o n of m 5 n r i g h t c o s e t s o f G. We

hrncr S in

Q.E.D.

conclude

by

observing

that

every

d r t r r m i n r s i n a n a t u r a l manner I n f a c t one c a n c o n s i d e r t h e set =

e

{ A1

: A1

e b(G)

=

Aa-',

a

with

right

S-n-block

A

of

G

r i g h t S - n - p a r t i t i o n of G . 0. : = Q1 1 . ~ Q 2 , w h e r e f f l =

aeA}

and

C2

is

the

class

of t h r s u b s e t A 2 o f G s u c h t h a t :

1)

s

Az;

j j ) A 2 i s t h r u n i o n of n + l p a i r w i s e

d i f f e r e n t r i g h t cosrts

of S i n G ; j j j ) VA1e

Consrqurntly, (**) In

for

I(il

rvrry

:A2

L

group

A1. C

and

for

rvrry

subgroup S of

G,

191 and 1101 w r c a l l e d r i g h t S - b l o c k s t h e r i g h t S - 1 - b l o c k s ( c f r . t h r p r r v i o u s p r o p e r t y ( 6 ) ) and p r o v e d s e v e r a l a l g r b r a i c a l and g r o m r t r i c a l p r o p r r t i r s .

D.Lerizi

296

wr

can have several (trivial) geometrical spaces of typr n-Strinrr, since if m 5 n+l then the union A of m pairwisr different right cosrts of S in G such that S 4 A is a right S-n-block of G . B I B L I OG RA PHY

1.11

..

Andre' J., Ubrr Parallrlstrukturrn , Tril I , Tril 1 1 , 85-102,155-163,240-256, Tril 111, Tril IV, Math. Z.E(lY61). 311-333. [2] Biliotti.M., S-spazi ed 0-partizioni. Boll. U.M.I.(5) 14-A(1977), 333-342. [3] m l i o t t i , M . , Sullr strutturr di traslazionr, Boll.U.M.1. (5) G-A(1978), 667,677. 1_4] Biliotti, M., Strutturr di Andre rd S-spazi con traslazioni, Grom. Drdicata, lo (1981), 113,128. 151 Biliotti, M., Strutturr di Andre rd S-spazi con traslazioni 1 1 , Ann. di Mat. pura ed appl., (IV), Vol. CXXXV.(1983), 151-172. 1-61 Biliotti, M., Hrrzrr A., Zur Gromrtrir drr Translationsstrukturrn mit rigrntlichrn Dilatationen, Abh. Math.Sem.Hamburg 53 (1983). 1-27. 1:7] Karzrl, H., Bericht uber projrktive Inzidrnzgruppen, Jbrr. Drutsch.Math. Vrrein. E(1964) 58-92. [ S J Lingenberg,R.. Uebrr Gruppen projectivrr Kollinrationrn, wrlchr rine prrsprktive Dualitat invariant lassen, Archiv. der Math. E(lY62) 385-400. 191 Lrnzi, D., Rrprrsrntation of a linear space with a transitive group of coll inrat ions by a general izrd S-part it ion of a group, Atti Convrgno "Grom.combin." La Mrndola-Italy (1982), 471-482. 1,lOJLrnzi, D . , On a charactrrization of finitr projrctive planes having a transitive group of collinrations, to apprar. llllScarsrllj, A., Sullr S-partizioni rrgolari di un gruppo finito, Rrnd. Acc.Naz.Lincri, LXII (1977). 300-304. 11121 Zappa,G., Sui piani grafici finiti transitivi e quasi transitiv i , Ricrrchr di Matematica,II (lY53), 274-287. 1-13.1Zappa,G., Sugli spazi grnrrali quasi di traslazionr, Le Matematichr, 19 (1Y64), 127-143. 11141 Zappa, G.. Sulle S-partizioni di un gruppo finito, Ann.di Mat.2, (1966). 1-24. 1151Zappa, C . , Partizioni ed S-partizioni dri gruppi finiti, Symposia Math. I(196Y), 8 5 - 9 4 .

Annals of Discrete Mathematics 30 (1986) 297-302 0 Elsevier Science Publisliers B.V. (North-Holland)

297

ON BLOCK S H A R I N G S T E I N E R QUADRUPLE SYSTEMS Giovanni LO FAR0 ("1 Dipartimento di Matematica dell'Universitl, Via C. Battisti 90, 98100 MESSINA, Italy

Abstract , We detarmine, f o r a l l u :4 or 8 (mod. 121 , the s e t of a l l those numbers k f o r which there exist two S t e i n e r quadruple systems of order u on the same s e t t h a t share e x a c t l y k bZocks.

Introduction. A Steiner quadruple system

set and

(I

(SQS)

(Q,qJ

is a collection of four element subsets of

that every three element subset of The number (mod. 61

is a pair

IQI = v

where

Q

Q

is a finite

(called blocks) s u c h

Q belongs to exactly one block of q

is called the order o f the SQS((Q,q!

is an obvious necessary existence condition for an

and

.

u E 2 or

4

SQS of order u

( S Q S l v ) ) . On the other hand, in 1960 Hanani proved 181 that this condition is also

sufficient. Therefore in saying that a certain property concerning SQS(v) true for all If

iQ,ql

u

it is understood that u - 2 or is an S Q S i u )

A quadruple system

and

then

141 = 7,

4

(mod. 6)

=uiv-l)(U-2)/24

is

.

.

(Q,qj has a proper subsystem if there exist sets R C Q

rcu such that ( R , r )

A . Hartman 191 proved that for every

proper subsystem of order

Irl < 141

is an SQS with

8

vL16

.

there exists an SQSlV)

with a

.

Earlier results on subsystems of

SQSs

can be found in 131.

In their excellent survey of Steiner quadruple systems, C . C . Lindner and A. Rosa 1111 rise a series of questions. One of them is: "For a given u

k2qv

is it possible to construct a pair of

SQS(ujs

having exactly

,

for which

k blocks

in common?". Denote by I ~ v=)IO, I, 2,.

J(v) ={k :

j

SQSSIV) (Q,qll, (Q,q2)

. .,qv-14,qu-12,qv-8,qu}

, v8'

.

such that

1q1nq21 = k ) ;

G. Lo Far0

298

So the author's best knowledge, the only results concerning this problem are: ;J f l O ) = { 0 , 2 , 4 , 6 , 8 , 1 2 , 1 4 , 3 0 1 ;

(i) J ( 4 ) = I ; J ( 8 ) = { 0 , 2 , 6 , 1 4 }

.., 77,?9,83}

J ( 1 4 ) - 3 1 ( 1 4 ) - (48,50,52,57,58,60,62,63,. {q

-8=83;q14-12=79;q14-14=77 14

;

-15 = 7 6 I n J ( 1 4 1

"14

=0

(ii) J ( u ) E I ( u ) , for all 028 (iii) J ( V ~= ~ ( u , i for all

u

=

n+l n ~ ; 5 - 2 ;7 . 2 n ,

n22

.

The aim of this paper is the investigation of the block intersection problem for SQSs of order 2u

obtained by doubling suitable systems of order

. In

2,

.

particular it is obtained that J l Z v ) = 1 ( 2 u l

Although he was not able to prove it yet, the author feels that the following conjecture makes sense.

Conjecture:

J(U)

=I(vl

, for

all

.

v216

.

2. Prel i minaries

In this section we describe two constructions for quadruple systems o r order 2u

which are the main tool used in what follows. C o n s t r u c t i o n A (well known e.g. see 1111).

(X,al

Let

..., Fu-1 3

and

tY,b)

and

G = I G 7, G

be any two SQSS(vl with X n Y = B

,..., Gv-l}

vertices) on

complete graph on permutation on the set

{7,2

be any two

x

( 2 ) If

,x E X 1 2

and only if

(x # x 1

)

2

{x ,x ) € F i 1

2

l X u Y l Ia,b,F,G,al

by

K

U

'I

F

2' ' ' * (the

be any

,..., u-21 . , as

S=XuY

follows:

a or b belongs to s ; yl,y2EY

and

,

(yl#y2)

{ y l , y Z ~ E G j and

It is a routine matter to check that iS,sl

I-factorizations of

F={F

X and Y respectively, and let

Define a collection s of blocks of (1) Any block belonging to

. Let

(S,sl

fx

then

a(<) =j

is an

.

3:

1'2

,y ,y )ES 7

2

if

S Q S f 2 u I , We will denotc

,

1 .

C o n s t r u c t i o n B (compare 13 )

iQ,ql

Let and let

cp

be an SQSlu)

, Q'

be a bijection from Q

a finite set such that onto

Qr

IQI

=I&'\ , QnQ'=@

with x'.=cp(x.) , for every z i E Q z

.

299

Block Sliaririg Steiner Quadruple Systems Obviously,

(Q',q')

is an

qlcq

we define a collection p(ql)

I (z ,xq,xi,x i ) ; (x ,t ,xL,x'); A

( 3 ) { i s .x ,x7,x'); 1 2 2 4

(x',x',x',x 2

-

,x ,x ',a- ' );

(x

1

-

4

J.(x',x',x'

3

1

3

);(x',x',x',x

3:

2

3

4

(2

,z ,x 1, x 1); (x z ,x ',x 'I;

6 2 3 I}cp(q ) 1 1

4

.XI);

2 , 4 1 3

'

(x',x',x',J:

4 1 1 2 3 if and only if

, for every x l , x 2 t Q , x l # x 2 (P,p(q

It is a routine matter to check that note

(P,p(ailI

iP,pI

.

I;

4

by

.

1

( ( Q u Q ' l , ( q , q l ) ) If q i = O

)I

.

. W e will de-

is an SQS(2vl

then we denote

(P,p(@II

by

w i t h blocks i n common.

3 . SQS(2vls

In this section we will determine J ( 2 v )

two

;

4

( 4 ) i x , . ~ ~ , x ' , x ' i t p ( )q 1 1 2 1

X={Z,2

Take

P = Q u Q ' as follows:

1

ix ,x9,,x ,x'I; ix ,x .x ,xLI; (x ,x .x

1 2 4 3 " 1 3 4 ' 2 (3: ,x , x 7 , x I E q-q1 ; 1

of blocks of

d 4 1 4 2 3 2 3 1 4 if and only if (x*,x2,x3,x41 '? I'

1 3 (s,u,x 4, . r1' , . r2' l ~ c p ( q 1 ) 1

(Q,q.1) where

(=cp(q I) belongs to p(qlI

q1 or 4;

(1) Any block belonging to

(2)

obtained from

.

q ' = V ( q ) ={vie/ : e c q l If

(the S@S(v)

SQS(v)

,..., v} , F

v an even positive integer, and

X where F = {FlJF2,.. .aFv-Il and

2-factorizations of

We will say that

.

G have k edges in common if and

and

In 1121 C.C. Lindner and W.D. Wallis gave a complete solution to the interse5 tion problem for

I-factorizations F

there exist two

....--v iv-1 I

kEi0,1,2,

Q={I,Z

Take F={F

F

I-factorizations. In particular, they showed that for any

,..., F v- 1 1

I-factorization of

be Q'

and

THEOREM 3 . 1 . Assme

Proof. Let

Q' = { 7 ' , 2 '

(called the

3

H C J(2v)

,..., 0 ' 1

~

2

{xi,x

.8If

) E F

h

be an

i

4 , then

Q n Q ' = 0 and let

F'={F'

F'2J...J~i-ll

. If

is a

F) where

.

X={O,1,2

SQSiv)

with

1-factorization derived from

. iQ,q)

.

1-factorization of

{ x i , x i } E F ' . if and only if

then

G with k edges in common for every

and

- n } - in-l,n-2,n-3,n-51

,..., v}

v>8

v (v-1) ,..., -=nl--in--Ian-2,n-3,n-5I 2

k€H

then there exist two

l-factoriza-

G.Lo Faro

300

e

If

is the i d e n t i t y permutation, it i s a routine matter t o see t h a t

[ Q U Q ' I[q,q',F,G',e]

and

((Qu&!'l,(q,@))

k

SQS(2v)s with e x a c t l y

a r e two

b l o c k s i n common. The statement f o l l o w s .

I t i s w e l l known 111 t h a t i f 32'2~

,

then

there

y

t i o n of

(a

a r e even p o s i t i v e i n t e g e r s and

1 - f a c t o r i z a t i o n of order

exists a

1 - f a c t o r i z a t i o n of o r d e r

y

1 and

2-factorization

c o n t a i n i n g a sub-

z

n

of o r d e r

is a

I-factoriza-

K ).

2

...,n+98,n+lOO,n+104,n+112) Proof. L e t

. If

v iv-1) n =, v> 16

THEOREM 3 . 2 . Let

(Q,q)

1 , then k E J ( 2 v )

be a n

SQS(v)

Let

= (T,z)

SQSS(16) ( ( , ? U R ' ) , (?,'$)) F={FIJFg,

...,Fv-2 1

SQS(8)

(R,rJ

as a subsystem.

((QuQ'), (q,0)) = ( P , p )

SQS(2vl

F"2,.. ,,F"}

be two

7

I-factorizations

.

Q and R r e s p e c t i v e l y w i t h F$GPi , f o r e v e r y i = l , 2 , , . . , 7

of

SQS(2vl

l y easy t o s e e t h a t t h e

SQS(l6)

Ir,r',F",F"',i]

IRuR'I

IQuQ'I

con-

.

F"={F"

and

..

.

containing an

It i s s t r a i g h t f o r w a r d t o s e e t h a t t h e tains the

k E {n-I,n-Z,n-3,n-5,n+l,n+Z,.

Iq,q',F,F',i]

=(P,s)

It i s equal-

contains the

.

=(T,t)

Obviously, we have:

= 28 k E J(16) =1(16)

Let

such t h a t If SQS(2vls

.

(i",al

SQS( 6 )

I t i s p o s s i b l e t o c o n s t r u c t two

(T,b)

and

lanbl = k ,

fp-zlua=p'

Is-tlUb=s'

and

with exactly

v (v-1 I

--28+k

, then (P,p'l

(P,s')

and

are two

b l o c k s i n common.

2

T h i s completes t h e proof of t h e theorem., LEMMA 3 . 3 . O _< h _
Proof. Let /q

2

is:

I =k .

(Q,qi

_

v Iv-1 ) (v-2) =qv 24

b e an

SQS(V)

. If

and t a k e

1 =q -81k-hl qlJq2

C l e a r l y , t h e number o f d i s t i n c t b l o c k s of

Z(k-hl +6(:c-h) =8(k-h) REMARK. L e t

q

1

and then

= 8 and q2Cq

,

then

2v

Ip(q11np(q

2

with

q14q2C.q

(P,p(ql))

)I

ZEJ12vl

=qgv-8(k-h)

and

.

, lql/ = h (P,p(qz))

.

)q21 = t L q v ; from Lemma 3 . 3 , i t f o l l o w s

,

30 1

Block Sharing Steiner Quadruple Systems that

q

2V

- 8 t ~ J ( d v l , f o r every

THEOREM 3.4.

Assume

. If

v,lG

v (v-1) + 98 2

I7 + 4-71

q -12--

e

(Q,qJ

(q

I

1

that

be a n

=t+12(
SQS(v.i

. By

I p i r ) n p ( q I 1 =y 1

2V

SQS(16) ((RuR'),(F,P)) ; t h e r e f o r e

.

(P,p(r)) qZV

The s t a t e m e n t f o l l o w s b y c h o o s i n g

THEOREM 3 . 5 . Assume

Proof.

v,lC

.

If

w i t h a sub-SQS(16)

SQS(2ul

a r e two

(P,p(q

and

))

1 J(2v)

-8(t-i?)-(140-k)€

lEI(2u)

and

contain the

, for

then

LFqZV-l4C SQS(u!

w i t h a sub-SQSS(2u)

. Thus,

q

,

with

SQS(2u)s such

kE~l17,118,...,124)C.1~16/

I t i s w e l l known 111 t h a t i f t h e r e e x i s t s a n

then t h e r e e x i s t s an

then

rCqlCq

and l e t

I

.

It i s straightforward that both

kEJ116) =Iil61

(R,ri

(P,p(q I !

and

,..., 7,

.

8

SQS(8)

(P,p(rll

-8(t-21

, then l € J ( 2 v )

h+2

-0

containing an

Lemma 3.3

2
qZv-7. = 8 t + h , h=O,1,2

Proof. We s t a r t n o t i c i n g t h a t i f

Let

.

,...,

i:E{O,1,2

a11

.. lEJ(2v) ,

w i t h a sub-SQS(u),

t h e r e e x i s t s an

SQS(2Vj

.

It i s easy t o check t h a t

-(140-h.)€ J ( 2 v )

21)

for a l l

h c J ( 1 6 ) =I(161

.

T h i s c o m p l e t e s t h e proof..

Combining t o g e t h e r ( i i i ) and Theorems 3 . 1 ,

3.2,

3.4,

3 . 5 we g e t o u r main

result: THEOREM 3.6.

J l v ) = I ( v ) for a l l

vE4

or

6 (mod 1 2 )

.

REFERENCES

111 A. CRUSE, On embedding incomplete s y m e t r i c l a t i n squares, J. Comb. T h e o r y S e r . A ( 1 9 7 4 ) , 19-22. 121 J. DOYEN and A . ROSA, An u l ' h t e d bibliography and surV?y of S t e i n e r systems, Annals of D i s c r e t e Math. 7 ( 1 9 8 0 ) , 317-349. 131 J. DOYEN a n d M. VANDENSAVEL, Non isomorphic SOC. Math. B e l g i q u e 3 3 (1971), 393-410.

S t e i n e r quadruple systems, B u l l .

141 M. GIONFRIDDO, On the block intersection problem f o r S t e i n e r quadruple stems, A r s C o m b i n a t o r i a 1 5 ( 1 9 8 3 ) , 301-314.

sy-

G. Lo Far0

302

L I N D N E R , Construction of S t e i n e r quadruple SySi%ms having a prescribed number o f blocks i n common, D i s c r e t e Math. 34 ( 1 9 8 1 ) ,

151 M. GIONFRIDDO and C.C. 31-42.

161 M. GIONFRIDDO and G. LO FARO, Cn pear.

Steincr

systems

171 M. GIONFRIDDO and M.C. MARINO, On S t e i n e r Systems U t i l i t a s Math., 25 ( 1 9 8 4 ) , 331-338.

Si.?,4,14!

S(3,4,20)

,

and

to

ap-

S(3,4,32),

181 H. H A N A N I , On quadrupie systems, Canad. J. Math. 1 2 ( 1 9 6 0 ) , 145-157.

19

I

Quadruple systems contuining AG/3,2)

A. HAR'I'EIAN,

, Discrete

Math.

,

39 (1982)

( 3 ) , 293-299. L I N D N E R , On the construction of non-isomorphic S t e i n e r quadrupZe stems, C o l l o q . Math. 29 ( 1 9 7 4 ) , 303-306.

1101 C . C .

1111 C . C . LINDNER and A. ROSA, S t e i n e r quadruple systems t h . 2 1 ( 1 9 7 8 ) , 147-181. 112

1

-

651-

n survey, D i s c r e t e Ma-

C . C . LINDNER and W.D. WALLIS, A note on one-factorizations having a p r e s c r i bed number o f edges i n common, Annals of D i s c r e t e Math. 1 2 ( 1 9 8 2 ) , 203-209.

1131 G. LO FARO, On t h e s e t J ( v ) for S t e i n e r qumdmpZc systems of order v=7-2" with n > 2 A r s Combinatoria, 1 7 ( 1 9 8 4 ) , 39-47.

,

1141 G. LO FARO, S t e i n e r yuadrupie systems having a prescribed nwnber o f quadrup l e s i n cononon, t o appear. 1151 G. LO FARO and L. PUCCIO, S u l l ' i n s i e m e Steiner, t o appear.

J(14)

d e i sistsmi d i quaterne di

("1 Lavoro e s e g u i t o n e l l ' a r n b i t o d e l GNSAGA e con finanziamento MPI ( 1 9 8 4 , 40%).

Annals of Discrete Mathematics 30 (1986) 303-310

0 Elsevier Science Puhlishers B.V (North-Holland)

303

ROOTS OF AFFINE POLYNOMIALS

Giampaolo M e n i c h e t t i (-) D i p a r t i m e n t o d i Matematica, U n i v e r s i t P d i Bologna, I t a l y

INTRODUCTION.

Let F = GF(q) be a G a l o i s f i e l d of o r d e r q = p h , where p i s a p r i m e ,

and l e t K = GF(qn) be an a l g e b r a i c e x t e n s i o n of a g i v e n d e g r e e n > l . An a f f i n e

pziynon;iaZ (of K[x]over F ) i s a polynomial of type (1)

P ( x ) = L(x) - b , b € K ,

with

...,un-1 }

I f a b a s i s {uo,ul,

n- 1 of K o v e r F i s f i x e d then we can p u t x = 5 x . u . .

i =O Hence, t h e d e t e r m i n a t i o n of ( e v e n t u a l ) r o o t s of t h e polynomial ( 1 ) i n K can be

1 1

reduced t o t h c d e t e r m i n a t i o n of s o l u t i o n s o f a l i n e a r s y s t e m of e q u a t i o n s i n i n d e t e r m i n a t e s xi and w i t h c o e f f i c i e n t s i n F ( c f . [ l l , Chap.11). This procedure i s , however, t e d i o u s a l s o i n t h e most simply c a s e s and does n o t d e c i s e "a p r i o r i " how many r o o t s i n K e x i s t . I n t h i s p a p e r , we prove t h a t the e q u a t i o n ( 1 ) h a s r o o t s i n K i f and o n l y i f the following system of l i n e a r equations

(3)

+...+

llYl

loyo

+

l:-lYO

+ l:yl

= b

+. . .+ 1 ~ - 2 y n - l = bq

..................... n- 1

1qn-ly0 + 1;

1 n-1Yn-1

y1 +.

.. +

I

.

,

.

n- 1

n-1

1;

yn-l = bq

has s o l u t i o n s . Moreover, i f ( 3 ) i s s o l v a b l e and i f r is t h e rank of t h e m a t r i c e s b e l o n g i n g t o ( 3 ) , then qn-'

g i v e s us t h e number of s o l u t i o n s of ( 1 ) i n K. Besides

t h i s , we show t h a t t h e r o o t s of ( 1 ) a r e e x p r e s s i b l e as f u n c t i o n s of c e r t a i n s o l u t i o n s of t h e l i n e a r s y s t e m ( 3 ) . I n p a r t i c u l a r , t h e o b t a i n e d r e s u l t s a r e u s e f u l a l s o i n t h e c a s e t h a t t h e c o e f f i c i e n t s of t h e polynomial ( 1 ) are n o t c o n s t a n t ( c f . f . e . [2] ).

(-) This r e s e a r c h was s u p p o r t e d i n p a r t by a g r a n t from t h e M.P.I.(40% f u n d s ) .

304

G.Menicketti 1. An n x n matrix of the type

... an-1

.

A( .a ,al ,. . ,an-

)=

is called autocirculant. Each row of A(ao,al,...,an-l) is obtained by the previous one

if we permute the elements under the cyclic permutation

apply the field automorphism a : K

-+

(0

1. ..n-1) and then

K , a I+ aq.

The sum and the product of autocirculant matrices are autocirculant matrices. In addition, if

A

is autocirculant, the transpose

it is

also autocirculant.

Let

T = and Aq(aO,al

...,0 )

&(O,l,O,

,...,an-1)

=

A(aq,aq, ...,a:-1). 0 1

It is easy to verify

T A- T-l , Aq= -

(4)

A has rank r, 1 5 r G n - 1 , i f and only if i t s (colwnns) are l i n e a r l y K-independent and i t s (r+l)-th row (column) is a K-linear cornbination o f t h e preceding POWS (coLwmzs). LEMMA 1. An autocirculant matrix

first r

FOWS

Proof. The transpose of an autocirculant matrix is itself autocirculant. Thus it is sufficient to prove the statement for the columns of A.

The assertion is an obvious consequence of the following observation, Let

A=

(%,,A1,...,$I-l).

~f

s-1 (5) --s A = E kiAi , kfK, i=O s-1 0 < s < n-1, then A' = C kqAq and therefore -s

i=o

1-1

s-1 s-1 1 k . A . = c k!A. ,k!E K. 1 kqii+l + kq s-liz0 l-li=o 1-1 1 i=O 2 3 n-s-1 and use previous If we raise both sides of ( 5 ) t o the powers q ,q ,...,q

s-1

s-2

&+I=. 1=0 1 kqAi+l

=

arguments, we see that the columns &+2, K-linear combination of

%, A1,...,s A .n

& + 3 , , , , , &-l

can be expressed as a

PROPOSITION 2 . I$& i s an autocirculant m a t r i x o f rank r then the homogeneous

Linear system (6)

-AY

=

0 ,y =

(Yo Yl

Y,-l)t

..,

t zl,,. z 0 0) w i t h zr # 0 . Furthermore, n-r and LL' t = 2. given A' = A(zo, zl,. . , z r' 0,. , O ) , w i h m e rank(&')= Proof. If one has A = (A+, . , 411) then Lemma 1 guaranties the

has s o l u t i o n s of the type 5

.

= (zo

.. 4,. .

n-1 existence of the elements a ! E K such that A -K

=

T a!A. holds. And, this proves the

i=o

1-1

Roots of Affine Polynomials

305

f i r s t p a r t of t h e a s s e r t i o n .

(s,A; ,..., A&). n- 1 By d e f i n i t i o n , one h a s % T-lzq, - ..., r-(n-1IzqLet Att=

-A'= 1

- and

therefore

= z

A= -l'

A?= 0

If we c o n s i d e r t h e q-th power of deduce

0=

=

TAAi.

Thus

and t a k e i n account a l s o ( 4 ) , w e

h i = 0.Analogously,

0,. , . ,A$-l=g,

we prove A A ' =

--2

and have f i n a l l y

hi,. . . , g-l)= Ah'

A($,

=

0.

A;,

We deduce, i n p a r t i c u l a r , t h a t e v e r y element Ai,, ..., -nA ' €Kn i s a 1 s o l u t i o n of t h e l i n e a r s y s t e m ( 6 ) and h e n c e , i t f o l l o w s r a n k ( A ' t ) , < n - r . To see t h e i n e q u a l i t y r a n k ( A ' ) > , n - r ,

w e remember t h a t t h e m a t r i x which a g r e e s

i n t h e f i r s t n-r rows and l a s t n-r columns w i t h

..

A'

is non-singular.0

LEMMA 3 . T4e elements w .€ K , i = 0,1,. ,s , are l i n e a r l y F-independent if und n-1 t n o n l y if the vectoras w . = (wi wq w! ) EK , i = 0,1,. ,s, are Zinearly

.. .

-1

..

K-independent . Proof. Let us examine t h e c o n d i t i o n S

1 k.w.

(7)

i =O

1-1

=

0,kiEK,

under t h e h y p o t h e s i s t h a t t h e e l e m e n t s wi a r e F-independent. I f w e suppose t h a t a t l e a s t one c o e f f i c i e n t k i , then w e can d e t e r m i n e k E K such t h a t h = kk 0

0'

f o r example ko, i s n o t z e r o ,

t r ( h o ) # 0 ( ' ) h o l d s . From ( 7 ) ,

we

obtain S

T: h.w. 1-1

= 0 , h . = kk

-

i=O

I

i'

s

j

R a i s i n g t h e l e f t s i d e o f t h i s e q u a t i o n t o t h e powers qJ we o b t a i n ? h: i =O

...,n-1.

j = 0,1,

particular

wi

=

0,

S

I f we add t h e s e n e x p r e s s i o n s , w e f i n d C ( t r ( h i ) ) w i = i =O

2;

in

S

Y ( t r ( h i ) ) w i = 0, t r ( h o ) # 0 , i=O i n contrast with the hypothesis.

It i s e v i d e n t how t h e second p a r t of t h e t h e s i s may be proved.! COROLLARY 4 . Y7ie n x n matrix

nm-singular

2

=

(%,El,... ,%-,),

if and onlg if { u o , ul,. . .

...

n-1

)t,iz li= ( u i :u uqi , u ~ - ~is} a b a s i s o f the vector F-space K.!

For any polynomial ( 2 ) , t h e s e t Z(L) = { x € K : L(x) =

01

is o b v i o u s l y a v e c t o r subspace of K . Moreover, i f Z(P) = { x E K : P ( x ) = 01

(8)

(l)

Z(P)

=

xo+ Z ( L ) , x0E Z ( P ) .

t r ( x ) = t r ( x ) = x + xq + F

...

n-1

+ xq

,v

XEK.

# d then

G. Menichetti

306

Given an a f f i n c polynomial ( l ) , l e t

A(P)

= G(L):

= A(lo,ll,...,ln-l).

If rank(A(L))

PROPOSITION 5.

...,w

Proof. Let {wo,wl,

r then d i n $ Z ( L )

=

=

n-r.

] b e a b a s i s of Z(L) and l e t V C K n be t h e s o l u t i o n

space of t h e homogeneous l i n e a r s y s t e m

A(L)JI-=

(9)

0,11. =

.. . yn-l)t.

(yo y1

From Lemma 3 , i t f o l l o w s t h a t the v e c t o r s w.= (w. wq -1

1

1

.,. w:

n-1

,...,s ,

)t,i=O,l

a r e l i n e a r l y K-independent and i t i s e a s i l y v e r i f i e d t h a t each of them i s a s o l u t i o n of ( 9 ) . Thus,

Let

(11)

<

di%Z(L)

(10)

A' be

di%V = n-r.

an a u t o c i r c u l a n t m a t r i x which s a t i s f i e s t h e c o n d i t i o n s

-

i \ ( L ) A f t = 0 , r a n k ( A ' ) = n-r

2 = (%,,ul, ...,-n-1 u )

( c f . P r o p . 2 ) and l e t

be an n x n non-singular m a t r i x ( c f .

Coroll. 4 ) . From ( l l ) , we deduce

A(L)(A'

t

g)

=

With t h e o b s e r v a t i o n

0,rank(&' t 2) = n-r. t h a t Attg = (I&,?; ,...,$I&), u! -1 i

can conclude t h a t u ! E Z ( L ) ,

O,l,

=

(lo),

( c f . a l s o Lemma 3 ) . From t h i s and immediately

...,n-1

.o

(u! u!

=

1

and d i % < u ; ) , u i

'... u!'

n-1 ) t , we

1

,...,u'n-1

> = n-r

t h e P r o p o s i t i o n 5 f o l l o w now

COROLLARY 6 . Suppot~e rank(A(L)) = r .

Tf

-z =

0 O...O)

zl...z

(zo

t

is a

s d l u i i o n of the Zineura system ( 9 ) f o r any choose of t h e basis I u o , u l , . , , , u

n-l

1

o f tlie vector F-space K, the eZements n-r

(12)

x. 1

= z

u + O i

29,

n-r

us

n-r+l n-r+l + zq uq + r-1 1

... +

n-1

n-1

u4

2;

,

i=O,l,

...,n-1,

f o m n s e t of generators of Z(L). Hence, oiie has n-r (12)'

Z(L) = { x = z k O

+ zqr

n-r

n-1

+

kq

... + zq1

n-1 : kEK}.

kq

Proof.

The C o r o l l a r y f o l l o w s from t h e proof o f t h e p r e v i o u s P r o p o s i t i o n i f n-r n-r+l n- 1 one o b s e r v e s t h a t A't= A(zo,O 0,z: .z:-~ zq ). 1 t PROPOSITION 7 . If = ( z o z l . . . z ~ - ~ E) K" is a sokction of the linear system

,....

,...,

0

z

(13)

A(L)y =

b, y

= ( y o y1

...

=

(b bq

... bq

n-1

then, for every v E K w i t h t r ( v ) # 0 , (14)

x = (z v 0 0

+

2 z:-lvq

+ zq vq n-2

2

+...+

n-1 n-1 zq vq )/tr(v) 1

It,

Roots of Affine Polynomials

307

is a root of the poZynomiaZ ( 1 ) . P r o o f . Let

A(?)

-At (5)=

=

A(zo,zl,,s.,zn-l).

(2, -'zq, T 2 z-q

Raising L ( L ) z =

b

2

Then

,..., -T - ( n - l ) z q -

n- 1

).

2

- = Lq o r A(L)(X-'L~)

t o t h e power q , we o b t a i n A q ( L ) z q =bq=(bq bq

Using ( 4 ) , we h a v e , t h e r e f o r e , -TA(L)L1zq

=

2.

...bq

n-1 b)t.

Iterating this,

we f i n d

Thus, i t f o l l o w s , t A(L)A ( 2 )

(b b

=

... b).

Now, t h e r i g h t m u l t i p l i c a t i o n o f t h i s e q u a t i o n by n-1 ( t r ( v ) ) b ,I1= (v' v'q., , v f q

A ( L )1'=

=

( v vq...

n-1 vq ) t gives

lt

COROLLARY 8. The poZyizorniaZ (1) hus r o o t s i ? i K ,if and onZy if rank(A(1,)) =

rank(A(L) Ib-)

=

r . If t h i s c ondition holds, one izus IZ(P)I

= qn-r.

Proof. I f ( 1 ) h a s a r o o t x E K t h e n r a i s i n g b o t h s i d e s of t h e e q u a l i t y 0

t o t h e powers q , q

2

,..., q n-1 , we

find

n- 1

... + 1:-2x: ..................... n- 1

n-1

Thus,

&=

xi +

xo+ 1;

:1 (xo

= bq,

+ '10 xq0 +

l:-l~o

X:

...

...

n-1

+ :1

n-1 n-1 xq = bq 0

n- 1 xq ) t is a s o l u t i o n of ( 1 3 ) . From h e r e and from Prop.7, 0

it

f o l l o w t h a t (1) h a s r o o t s i n K i f and o n l y i f ( 1 3 ) h a s s o l u t i o n s . Taking i n account (8), t h e l a s t p a r t o f t h e a s s e r t i o n f o l l o w s from Prop.5.u In p a r t i c u l a r , we find the following RESULT (Dickson 1 3 ) ) .

c r n d onz$

If d e t ( A -( L ) ) #

T k ma[) L : K

-t

K, x

+

L ( x ) is n p r m u t a t i o n on K f,f

0.

Moreover, we o b s e r v e t h a t i f d e t ( A ( L ) ) # 0 , t h e o n l y r o o t x E K of t h e 0 polynomial ( 1 ) can be determined u s i n g Cramer's r u l e , t h a t i s

x 0= d e t ( b, ($,

Al.... ,&-l)

4 ,...,$-l)/det(%, =

A19 .

-

*

sS-1)

3

A(L).

I n g e n e r a l , t h e a f f i n e s u b v a r i e t y of R c o n s i s t i n g o f t h e s o l u t i o n s of polynomial ( 1 ) is given by (8) w i t h xo and Z ( L ) e x p r e s s e d by ( 1 4 ) and ( 1 2 ) ' respectively

.

308

G.Meniclietti From C o r o l l a r y 8 , we deduce t h e f o l l o w i n g u s e f u l

OBSERVATION. A polynomiaz (1) W i t h d e g ( L ( x ) )

=

q d , 0 ,< d

corripZetcZy redueible in K if and on2y if r a n k ( L ( 1 ) j b )

=

<

n-1, is

rank(A(L)) = n-d.

Another consequence i s t h e f o l l o w i n g

PROPOSITION 9 . Tuo a f f i n e poZynomiaZs, (1) and P ' ( x )

=

L'(x)

-

b ' , have

common u>oots in K if and onZy ?'f t h e equalions of t h e Zinear sistems (13) and y= b ' m e compatible.

A(L') -

Proof. If x E K i s acommon r o o t of b o t h P(x) and P'(x) t h e n x

4

O

(xo :x

=

n-1

... x:

) t i s a s o l u t i o n f o r both l i n e a r systems i n t h e a s s e r t i o n .

Conversely, i f t h e e q u a t i o n s of b o t h systems a r e c o m p a t i b l e , we f i n d , by ( 1 4 ) , a common r o o t f o r t h e given polynomials.[ I t i s easy t o prove t h a t , when t h e c o n d i t i o n of t h e p r e v i o u s p r o p o s i t i o n i s

s a t i s f i e d , t h e s e t of common r o o t s f o r P(x) and P ' ( x ) i s an a f f i n e s u b v a r i e t y of K whose dimension i s n - r ' ,

(-Gi1:

,-;I

where

(-:

A(L)

r ' = rank

-

A(L)

=

)

b

-I - :. A(L')I b'

rank

--

Now we want t o use t h e p r e v i o u s r e s u l t s t o d i s c u s s t h e e q u a t i o n

m

xq

(15)

- x

=

b , b E K , 1 ,< m , < n-1

.

F i r s t we observe t h a t , given d = (n,m) and k = n / d , t h e i n t e g e r s i m + j ,

i

= O,l,

...,k-1,

j = O , l , . . .,d-1,

a r e p a i r w i s e incongruent modulo n.

I n t h i s c a s e , t h e l i n e a r s y s t e m (13) becomes

'+m Y2m+ j

- 'm+j

'( k-1 ) m+ j

-

= bq'

...................

(16)

'i

-

(k-2 )m+j (k-1 )m+ j

=

'+(k-2)m bqJ

=

bq

j + (k-1 ) m

,

j = 0.1,

and t h u s i t s e q u a t i o n s a r e compatible i f and o n l y i f

...,d-1, im j k-1 j+im k-1 C bq = ( C bq )' = 0. i=O

i=O

From t h i s , we deduce t h a t (15) has some r o o t s i n K i f and o n l y i f (17)

k-1 im C bq = t r F , ( b ) = 0,

i =O d where F' = GF(q )

(')

C_

GF(qn)

(2),

The i n t e g e r s h d , h = 0 , 1 , k-1

modulo m and t h e r e f o r e

I: bq i =O

... ,k-1, im

and i m , i = 0,1,

k-1 =

C bq h=O

hd

.

...,k-1,

a r e congruent

Roots of Affine Polynomials m

-

L(x) = xq

309

x implies obviously

d Z ( L ) = GF(q ) , d = ( n , m ) .

(18)

T h e r e f o r e , w e can d e t e r m i n e a r o o t x E K of (15) u s i n g P r o p . 7 and supposing t h a t C

(17) i s s a t i s f i e d . From (16), by s u c c e s s i v e s u b s t i t u t i o n s , we f i n d

i-1 yim+l

j

hm

C bq

= yj + (

, i

)q

1,2

=

,...,k-1,

...,d-1,

j = 0,1,

h=O and by (17) yim+j =

A. 1

k-1

j

hm

C bq

(

)'

h=i

, X.EK,

i = 0,1,

J

...,k-1,

j = O,l,...,d-l

Let u s c o n s i d e r t h e p a r t i c u l a r s o l u t i o n =

'im+j

k-1

-

j

hm

C bq

(

, i

)q

=

O,l,

...,k-1,

j

=

...,d-1.

X = 0, j = 0,1, j From ( 1 4 ) w e o b t a i n

obtained f o r

n-1

x tr(v) 0

h

h

k-ld-1

E 24 vq n-h

=

h=O d- 1

Hence, s e t t i n g v = wq

, we

=

i=O

' j=o

qn-(im+j)

n-(im+j)

"4

'im+j

have

k-1 d-1 n-(im+j) n-(im+j)+d-1 x t r ( w ) = C C z;m+j wq 0 i=O j=O where t r ( w ) = t r ( v ) # 0. I f we o b s e r v e t h a t n-(im+j) = -

zSm+j

k-1 n+(h-i)m k-i-1 rm Cbq = C bq , h=i r=O

t h e n , s u b s t i t u t i n g i n t o t h e p r e v i o u s e q u a l i t y , one h a s

x tr(w) 0

k-1 k-i-1 rm d-1 n-im+(d-1-j) Z C bq C wq i = O r=O j =O k-1 k-i-1 r m d-1 n-im+s = - C C bq C w q i = O r=O s =o k-1 k-i-1 rm d-1 s n-im = - C Z bq ( C w ' ) ~ i=O r=O s=o =

-

.

From h e r e , p u t t i n g d-1

a =

C

s W

~

s=o

and o b s e r v i n g

k-1 d-1 tr(w) =

c

i=o we deduce

I: wq j=O

im+j

im

k-1 =

c

i=O

a'

= trF,(a),

...,d-1,

0,1,

h=i

.

G.Menichetti

310 k-1 C i=O k = C h=1 k-1

xOtrFI(a) =

-

c

= -

k-i-1 rm n-im C bq aq r=O r m hm h-1 C b q aq

r=O h-1 ~

hm

rm b

q aq

.

h = l r=O T h e r e f o r e : The equation (15) has r o o t s i n K = GF(qn) i f and only i f b s a t i s f i e s

the condition ( 1 7 ) . If such condition i s s a t i s f i e d , the s e t of r o o t s i s the a f f i n e subvariety ( 8 ) in which Z ( L ) i s given by (18) and x

=--

k-1 h-1 rm hm C C b q a' trF,(a) h=l r=O

,

t r F l ( a ) # 0.

I f ( k , p ) = 1 ( p = c h a r K ) then t r F l ( l ) = k # 0 and t h e r e f o r e , we can s e t a = l . The p r e v i o u s r e s u l t a l l o w s u s t o determine t h e r o o t s of a second d e g r e e e q u a t i o n i n a f i e l d K of c h a r 2 . I n f a c t , f o r q = 2 , m = 1, w e f i n d t h e w e l l known c o n d i t i o n t r ( b ) = 0 i n o r d e r t h a t t h e e q u a t i o n X'

+ x

t

b = 0

h a s a r o o t in K = GF(2").

Moreover, from (18) and ( 1 9 ) , we deduce t h a t t h e r o o t s

of t h e above e q u a t i o n a r e

n-1 x

C

=--

h-1

C b

2r 2h a

and

xo+ 1 ,

t r ( a ) h=l r=O where a E K i s a f i x e d element w i t h t r ( a ) # 0.

REFERENCES

[ l ] Berlekamp, E . R . ,

AZgebraic coding theory (Mc Graw Book Company,New York,1968).

121 B i l i o t t i M. and M e n i c h e t t i G . , On a g e n e r a l i z a t i o n of Kantor's l i k e a b l e planes, Geom. D e d i c a t a , 1 7 (1985) 253-277.

[ 3 ] Dickson, L . E . ,

Linear Groups w i t h an e x p o s i t i o n o f t h e Galois fieZd theory

(Teubner, L e i p z i g . R e p r i n t Dover, New York, 1958).

Annals of Discrete Mathematics 30 (1986) 31 1-330 0 Elsevier Science Publishers B.V. (North-Holland)

O n the parameter

n(v,t

31 I

for Steiner t r i p l e systems ( " )

-13)

Salvatore Milici ("") Abstract. L e t D ( v , k l ([l], [ 8 ] ) b e t h e maximum number of S t e i n e r T r i p l e S y s t e m s of o r d e r v t h a t con b e c o n s t r u c t e d i n s u c h a way t h a t an3 t w o of t h e m h a v e e x a c t l y k b l o c k s i n common, t h e s e k bZocks b e i n g moreover i n each o f t h e STS(v), Let t v =vlv-11/6 I n t h i s p a p e r we prove t h a t D ( v , t v - 1 3 ) = 3 f o r everg ( a d m i s s i b l e ) v,1:

.

.

.

1 . Introduction a n d definitions. A PnrtiaZ T r i p l e System

a finite non-empty set and

(PTS)

(P,P) where

is a collection of

P

2-subset of

called blocks, such that any

P

is

P

3-subset of

,

P

is contained in at

.

P

most one block of

is a pair

Using graph theoretic terminology, we will say that an element

of

x of

P

P

has d e g r e e

if x

d(x) =h

. Clearly

belongs to exactly

. We will

d ( x ) =31PI X E P

of a

PTS

(P,P) the

. If i = I , . .. , s

are the elements of degree

h

,

i

for

,

where

then we will write Two balanced

PTSs

A set of

s

r

1

Ih I

i~

(P,PII if

(DMB)

in a block of

p

Pl

PTSs

+... =h

and

call the d e g r e e - s e t

.

DS = I d ( x ) , d ( y i , . . ]

n-uple

,

where

elements of i , we will write DS =

there are

+ r = IPI S

r

.

If

blocks

h

P

(DSl

..

x,y,.

having

r . = I , for some

i

,

i '

IP,P21

are said d i s j o i n t and m u t u a Z l g

= 0 and a 2-subset o f P is contained 1 2 if and only if it is contained in a block of p2 P n P

(P,P1),iP,P2i,.

. ., ( P , P s l

is said to be a set of

( A )

Lavoro eseguito nell'ambito del GNSAGA (CNR) e con contributo finanziario MPI (1983).

(*$<)

Dipartimento di Matematica dell'Universit5, Viale A . Doria, 6 9 5 1 2 5 - Catania.

*

S. Milici

312 s

if

DMB P T S s

i,jE{1,2,3

,..., s l

s e t of

DMB PTSs

s

Two

and

(P,Pi)

and

.

isomorphic if

A degree set

(P,P;I

i s associated w i t h every

DS

..

(P;P

such t h a t

(S,B)

PTS

IS1

=v B

c o n t a i n e d i n e x a c t l y one b l o c k o f

DS

a r e n o t isomor-

.

and e v e r y

. It

2-subset of

3 (mod 6 ) ( v

or

IBI = t

blocks i s

V

STS(v))

is

S

i s well-known t h a t a r . e c e s

s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e of an

u :I

..., PSl .

v ( o r more b r i e f l y a

4 S t e i n e r t r i p Z e s y s t e m of o r d e r

is a

(P;PI,Pz,

Pi,. . .,P ' I

,P 2 , , ,P s ) and ( P ; P '1, 1 p h i c , i t i s p o s s i b l e t h a t t h e y have t h e same However i f

.

.,PL) are 2 by t h e same isomor-

(P;P;,P',.

..,s .

i =1,2,.

a

(P;PI,P 2 J , . . , P )

and

i s isomorphic t o

IP,Pi)

phism, f o r e v e r y

. .,Psi

(P;Pl,P2,.

f o r every

DMB

. We d e n o t e by J

i # j

(s+l)-tuples

are

(P,P.)

STSIv)

is

a d m i s s i b l e ) and t h a t t h e number o f t h e

=v(v-l)/6 ,

We w i l l p u t

S

A(a,Ia,b,cll

=A(a)-{b,cl

with

u Pi .

{a,b,c} E

i=l

The

i s s a i d t o b e embedded i n t h e t r i p l e s y s t e m

P T S (P,?)

provided t h a t

(S,Bl

Given an i n t e g e r D(v,k)

exactly

k

each o f t h e

,

v

0 -k <

STS(vls

t o, l e t u s denote by

t h a t c a n b e c o n s t r u c t e d on

i n s u c h a way t h a t any two o f them h a v e

b l o c k s i n common, t h e s e Dlv,k)

.

PCB

such t h a t

k

t h e maximum number o f

a set o f c a r d i n a l i t y

D(u,k)

and

P S S

systems. In

k

Ill,

b l o c k s b e i n g moreover i n Doyen a s k e d t o d e t e r m i n e

( s e e a l s o IS]).

Much r e s u l t s a r e a l r e a d y known i n t h e c a s e k f o

, same r e s u l t s a r e c o n t a i n e d i n 1 3 1 , 1 4 1 ,

p a r t i c u l a r , i n these papers

u

a d m i s s i b l e and

k = t -m V

D(v,k)

with

k =O

1 8 1 . For

1 5 1 , 1 6 1 , 1-71. I n

h a s been d e t e r m i n e d f o r e v e r y

m(12

.

313

Parameter D(v, tv-13)for Steiner Triple Systems

I n t h i s paper w e prove t h a t

f o r every

Dlv,t -13) = 3

vz15

2,

.

2 . Preliminar r e s u l t s . W e now g i v e some p r o p e r t i e s w h i c h w i l l b e u s e d i n t h e r e m a i n i n g

sections. 1x1

Let

I n any (2.1)

b e t h e g r e a t e s t i n t e g e r lesser t h a n o r e q u a l t o

fp;P , P 2 , . . . , P

IP

1

I

=IP21

rn=IPil

=...

,

= I P s l 24

(i=2,2,

..., s)

If

(2.3)

d(x) > 2

(2.4)

s c 2 d l u l -rl - 2

(2.5)

If

R

s <2r-2

,

p =min

; we w i l l p u t

IPI ~6

,

then

m22h

{dlx) : S E P } 5

, where

and

n=IPI

;

h =muz { d l s ) : X E P }

(2.2)

.

w e have (31:

)

1

s

n=

IA(u,{u,u,w}l

is a block such t h a t

and

n22h+1 ;

and

s 5 2 ~ - 1;

-A(v,{u.u,w~ll ;

with

l R n M p I '2

P

=2,3

,

then

. 3

Lemma 2 . 1 .

3 '

3 B = nA(j,Il,2,3}1 j=l

i = I, 2 , 3 , 4 i)

1x1

In a

j=l

(P;P ,P2,P ,P ) 1 3 4

if

f o r some

REPi

, then: = 161 = 4

Proof. L e t 4,5,b',aI

.

and

X = uA(j,{l,2,3})

~ e tR = I I , ~ , ~ } = M

and

ii)

017

REP1

,

1x1

and

= 5

IBl = 2

without loss of generality. If

Al2) = {1,3,4,5,6,b}

I

2 4 5 4 5

... . * .

3 a 6 6 b

1 2 4 1 5 u 1 6 3 2 3 5

2 6 b

...

I ...

A l l ) ={1,2,3,

, we h a v e n e c e s s a r i l y

p4 1 1 1 2 2

.

1 2 5

1 1 2 2

4 6 4 3

... ...

3 a h 6

1 2 6 1 4 5 1 3 a

2 4 5 2 3 b

...

*..

S. Milici

314

or ii) -p4

PI

1 1 1 2 2

2 4 5 4 5

...

1 1 1 2 2

3 a 6 6 b

6 3 a 5 b

... ...

... I n c a s e i ) we o b t a i n wc o b t a i n

2 4 5 3 4

. This

3eM4

i s impossible. I n case i i )

and t h i s complete t h e p r o o f o f t h e

A ( 3 ) ={lJt',4,5,a,b}

lemma. Lemma 2 . 2 . (P;P1,P

2

Let

R={1,2,a}

, P 3 , P 1 .J , i f

i) A(l,{1,2,all

,

= 3,4

i i ) if

Pi , f o r some

RE

j =1,2

x

x2EAla)

'I

or

...

1 1 1 2 2

6 7 2 5 6

, then

and

,

j =1,2

then necessariZy

,

I

a 5 7 6 7

In a

If

~ = i l , 2 , a l E P without loss of g e n e r a l i t y . 1 =A/Z,{I,Z,a}i ={4,5,6,7} , we h a v e n e c e s s a r i l y

A(l,{l,Z,a}l

2 4 6 4 5

4 .

.

Proof. Let

1 1 1 2 2

aEM

x.EAlj,{Z,2,a~l 3

, for every

x . e A(3-jJ{1,2,a})

i

3 =1,2,3,4

,

a r e s u c h that

xl,x2EP

and

1,2EM

a n d ~ A l a , { 1 , 2 , a } i n A ~ j , { 1 , 2 , a } 1l

=.4(2,{1,2,u}~

f o r every

3

,with

a 5 4 a 7

...

I

1 1 1 2

6 7 2 4

4 a 5 7

26a

...

1 1 1 2 2

4 5 2 5 7

7 a 6 4 a

...

,

I

or

1 1 1 2 2

2 4 6 4 5

a 5 7 6 7

...

1 1 1 2 2

2 6 7 5 7

4 a 5 6 a

...

1 6 4

i 7 a 1 2 5 2 4 a 2 7 6

...

1 1 1 2 2

2 4 5 4 6

7 a 6 5 a

...

Parameter D(v, tv-13)f o r Steiner Triple Systems

p2

pl

12a 1 1 2 2

4 6 4 5

5 7 6 7

p3

1 2 4 1 6 5

1 1 1 2 2

1 7 a 2 5 a 2 6 7

2 4 5 5 7

315

p4 6 a 7 4 a

1 1 1 2 2

2 4 5 4 6

7 6 a a 5

or

... I ... I ... I ...

1 1 1 2 2

2 4 6 4 5

1 1 1 2 2

0 5 7 6 7

...

2 6 4 4 6

5 a 7 0 7

...

I t i s a r o u t i n e m a t t e r t o see t h a t

1-

I

p4

p1

1 4 x1 1 1 5 a

1 2 5 1 1 4 6

1 2 6 1 4 a

1 2 0 1 4 x

1 6 x

l a x

1 5 x

1 5 6

1 2 4

1 2 a 1 5 6

1

2 5 6

2 5 x ,

1

2 4 a

2 5 a

2 f i x 2

2 4 x

or

1

2 5 s

L1

2 4 6

2 a x

tlence complete

3.

STS

.,

2

X ~ , X , , E

ACa)

ii

... . At

2

2

1 6 a

1 1 6 x 1

2 5 a

2 4 2

2 4 6

2 6 x

...

...

this point t h e

l a x l 2 4 5

2 6 a

2 a x

...

...

2

p r o o f o f t h e lemma i s

D(u,t

. Let

2

2

1

w i t h b l o c k s i n common

I n t h i s s e c t i o n we w i l l p r o v e t h a t u21.i

1

P = {1,2,..

.,8,a,b,cl

following three sets o f

13

and l e t

PI ,

t r i p l e s each:

U

-131 2 3

f o r every

P2

P3

and

be the

S.Milici

316

,

P

Lemma 3 . 1 .

Let

1

1 l 1 2 2 3 3 a a b b c c

=

2 a 4 a 4 b 4 5 7 5 6 5 6

3 b c c b c a 6 8 7 8 8 7

Dlv,tv-131

S=I1,2

P

1 3 a 1 4 b 2 3 4

3 3 c 4 a 5 6 5 6 5 7

b a b c b 7 8 8 7 6 8

v

= l E

for

2 b a

P

,..., 8 , 9 , 0 , a , b , c , d , e } 1 1 1 1 2 2

T = c 9 d c e O

Clearly,

,

2 3

1 2 c

l a c 1 2 2 3 4 a a b b c c

=

2

1 2 4

(S,P~UT)

, i

5 6 7 8 5 6

3

=

3 4 a a b b c 0

b c 5 6 5 7 5 6

15

.

and

9 0 d e e 9

=1,2,3

.

I t follow t h a t

v 231

and f o r e v e r y

, are three

STSllSls

t h a t a n y two o f them i n t e r s e c t i n t h e same b l o c k - s e t IT1 = t - 1 3

c a 8 7 6 8 7 8

T

such

with

.

-13) ' 3 I n [ Z ] , J . Doyen a n d R . M . 15 W i l s o n h a v e shown t h a t a n y S T S f v ) c a n b e embedded i n t o an S T S ( u I

f o r every

u,Bvtl

Lemma 3 . 2 .

Let

D(15,t

. Then

Df19,t

S={1,2

19

D(v,tv-13)

-131 2 3

f o r every

2 3

.

,..., 8 , 9 , O , a , b , c , d , e , x , y , Z , t l I l 2 2 2 2 2

d e 5 6 7 8 d

x y 0 9 t 2 Y

2 e x

3 3 3 3 3 3 4 4

5 6 7 8 9 0 5 6

x Y d e 2 t Y 2

and

6 e z

v231

.

Parameter D(v. tv-131 for Steiner Triple Systems

,

(S,P,u F)

Clearly,

z

,

i =l,2,3

are three

STS(19)s

t h a t a n y two o f them i n t e r s e c t i n t h e same b l o c k - s e t

.

I F 1 = t1 9 - 1 3 Lemma 3 . 3 .

Let

Then

21

y d c y a e

x f c l = x l g

y 3 g y 4 h

x 1 x x x

Clearly,

2 5 7 3 4

y b f

h 9 0 8 6

y y y y d

6 5 2 1 e

(S,P.uL/ z

9 0 8 7 f

,

i =1,2,3

8 h 9 0 4 g h 3 0 6

and

d d d d d e e e e e

1 5 2 3 7 7 8 4 l Z

,

are three

b h O

e 3 5 f 8 g

c g o

f 7 h

c 9 h

f f f $ a

g 2 5

1 2 3 4 9

9 0 6 5 0

h 1 6 9 3 7 4 8 0

u g h b 9 g

STS(2l)s

t h a t a n y two o f them i n t e r s e c t i n t h e same b l o c k - s e t

. Then

[LI =tZ1-13 Lemma 3 . 4 .

Let

S

= i 1,2, ,

2,

hl

x z u z f g o

. ., 8 , 9 , O , x ,

y , z , a, b , c, d , e , f. g , i , p , q , F, s , t , M

0 3 d e u i

2 5 9 2 6 0 2 7 e 2 8 d 2 x s

l z r

2 z i 2 t r

3 3 3 3

2f-s 2 P g

3 u g 3 i f

1 1 1 1 1 l

y t s r P q d

5 6 7 8 x y

I t s '

Ifs

b 9 e

I P q

2 Y U

3 5 x 3 7 2 3 8 t O 9 d e

P q r s

8 x q

Clearly,

(S,P.uH)

, i

4 4 4 4

5 Y 6 x 7 t 8 2

4 4 4 4 4

o 9 d e i

4 u f

BYP

, are t h r e e

=1,2,3

.

Then

D(27,t

27

-13) ' 3

.,

I

q P s r g

ST'S(271s

t h a t a n y two o f them i n t e r s e c t i n t h e same b l o c k - s e t /MI =tg7-13

with

.

-13) ' 3

a 0 9 a d e

a a a a a a b

L

such

,.

D(21,tZ1-131 ' 3

D(27,t

with

.

,..., 9 , 0 , a , b , c , x , y , z , d , e , $ , g , h ~

x y z x d a x e b

F

such

.

~ ( 2 1 , t -13) 1 3

S=Il,2

>3

D(19,tl3-13)

317

M

such

with

and

S . Milici

318

Lemma 3 . 5 .

D125,t

-

Let

25

,...,

P = { I , ~

t h r e e s e t s of

-13) > 3

. -

PI

andlet

9,0}

- ,P 2 , P g

be the following

t r i p l e s each:

13

0 1 2 0 3 4

0 5 6

P

S = {1,2,.

Let

2 4 b Z a l

Z t d 3df 3 e s

Z e f 2 g r

1

2 h s

3 g t 3 h l

Z i n

3 p i

Clearly,

=

8 9 9 1 2 1 4 1 4 3

3 6 2 4 5 5 6 3 5 6

-

P

3

= 8 8 8 9 9 9

1 2 3 1 3 5

8 O b 8 a i 8 c s

7 0 a

? b e ? c h 7 d s

8 n d 8 e g 8 f l 8 h p

7 f i 7 g n 7 1 t

9 O c

c i Z

g i g

9 9 9 9

c t f

4 4 5 5 5 5 5

a t b S d h e Z

c n r c d e 4 4 4 4

9 g P 9 s n 7 p r 8 r t 9 r i

iS,PiuQ)

,

a nd

i =1,2,3

,

a d t f

c l e s

are three

.

Then

D(25,t

25

-13) 2 3

..

h n a p e f g

r p b d n h s

S i t O d i 6 b c O e p 6 d r O f g 6 e i

6 f p 6 g Z 6 h n

STS(25/s

t h a t a n y two o f t h em i n t e r s e c t i n t h e same b l o c k - s e t

IQi = t Z 5 - 1 3

4 5 6 2 4 6

..,S,O,a,b,c,d,e,f,g,h,i,l,n,p,r,s,tI

3 5 c 3 a n 3 b r

2 c p

2

7 7 8 7 7 8 8 9 9 2

Q

O h t O l n O s r

such

with

I n c o n c l u s i o n , b y Lemmas 3 . 1 , 3 . 2 , 3 . 3 , 3 . 4 a n d 3 . 5 w e o b t a i n t h e following theorem.

Theorem 3 . 1 .

4.

Dlv,t - 1 3 1 V

D(v,t -13) 2 3

f o r every

for every

11215

v 215 ,

.

I n t h i s s e c t i o n we w i l l p r o v e t h a t t h e r e d o e s n o t e x i s t a

3 19

Parameter D(v, tv-13)f o r Steiner Triple Systems (p;P , P , P ,P I 1 2 3 4

with

m=13

. Further

w e w i l l d e t e r m i n e D ( v , t -131 V

IP;P , P , P , P i

From P r o p e r t y 2 . 3 , t h e e x i s t e n c e o f a M2 = @

.

m =I3

, can have t h e following parameters:

I t i s e a s y t o see t h a t a

and

DS= [(4/$,3] ;

2)

n =I1

and

DS= L(416,13/5]

3)

n =12

and

DS = [ ( 4 1 3 , / 3 ) 9 ]

4)

~1

=23

and

DS= [ ( 3 I l 3 ]

Proof.

T h e r e i s no

.

If

where

, or

or

{ 7 , 8 , 9 } E Pi

,

Fi(A)

,

(P;P ,P2,P 1

3

,P I

M4 = f l , z

. f o r some

,

=

M

[5,(4/4J(3)6]

2

,

= @ and

or

;

.

={XI,

i =l,2,3,4

DS

implies

4

, with

3

D S = [5,4,f3Il0-1

Suppose t h a t t h e r e e x i s t s a

o s = [ f 4 1 31 Let M 3 9’ A = A ( x l = {1,2,3,4,5,6}

2

3

DS = [ ( 5 ) 3 , ( ~ 1 ; ~ 1

[(5),, (4)2, ( 3 ~ ~ 1 o r,

Lemma 4 . 1 .

A

1

n =10

=

2

IP;P , P , P , P , I

1)

DS

1

with

4

(P;PlJP2,P

,..., 9 1 , ,

i = 1,2,3,4

DS

3

=

[14/9J3]

,P I

with

4

P=M

.

3

u M4

and

then necessarily

t h e r e are f o u r d i s t i n c t

1 - f a c t o r s on

.

If such t h a t

{7,8,9} @ P

REA

P . =

z

with

.

i Let

,

then necessarily there e x i s t s a block

7 8 1 7 9 2 8 9 3

7 3 4 7 5 6 8 2 6

(a= 4 , B = 5 , y = 6 1

Particularly, i f

, we

R={l,2,3}

or

have

8 4 5 9 1 5 9 4 6

(a = 6 , B = 4

{7,8,9} $ PI

x l a

1 2 3

2 2 1 3 x 3 y

,y

=5)

we obtain

.

,?€Pi

.

S. MiIici

320

X I 6 x 2 4

or

x 3 5

Let

F =IFi

I

i =1,2,.

..,5}

b e the

€PI

I-factorization on

ven by:

I t follows that

i) L e t

Alx) = F , 1

Afxl = F

. Then,

I

or

A(xl = F

up t o isomorphism, we o b t a i n

pl

p2

x 1 4 x 2 5 x 3 6

x 1 2 x 3 4 x 5 6

7 8 1 7 9 2 8 9 3

7 8 5 7 9 4 8 9 1

7 8 3 7 9 6 8 9 2

7 3 4 7 5 6 8 2 6

7 1 3 7 2 6 8 2 3

7 1 4 7 2 5 8 1 5

8 4 5 9 1 5 9 4 6

8 4 6 9 2 5 9 3 6

8 4 6 9 2 3 9 4 5

1 2 3

1 4 5

2 3 6

--

5 .

A

gi-

Parameter D(v, t,,-13) for Steiner Triple Systems

p3 x 21 63 1 x 4 5

where

7 8 5 7 9 1 8 9 2

7 8 6 7 9 3 8 9 4

7 8 3 7 9 1 8 9 4

7 8 2 7 9 6 8 9 5

7 2 3 7 4 6 8 1 4

7 1 4 7 2 5 8'1 5

7 2 6 7 4 5 8 1 2

7 1 3 7 4 5 8 1 4

8 3 6 9 3 4 9 5 6

8 2 3 9 1 2 9 5 6

8 5 6 9 3 6 9 2 5

8 3 6 9 1 2 9 3 4

1 2 5

3 4 6

1 3 4

2 5 6

r

jiE{1,3,4},

i i ) Let

A(x) = F

5

.

i

€{1,2,4}

,

8.€{1,2,3}.

z

T h e n , up to isomorphism, w e obtain

x 1 6 x 2 4 2 3 5 7 8 1 7 9 2 8 9 3

7 8 5 7 9 3 8 9 1

7 8 4 7 9 6 8 9 2

7 3 4 7 5 6 8 2 6

7 1 6 7 2 4 8 2 3

7 1 5 7 2.3 8 1 6

8 4 5 9 1 5 9 4 6

8 4 6 9 2 6 9 4 5

8 3 5 9 1 3 9 4 5

1 2 3

1 3 5

2 4 6

32 I

322

S. Milici

i: /

where

jl

p4

2 3 5

7 8 2 7 9 6

7 8 3 7 9 5

8 9 1

8 9 4

7 1 5 7 4 6 8 1 6

7 1 3 7 4 5 8 3 4

7 1 6 7 2 4 8 1 2

8 3 4 9 2 4 9 3 5

8 3 5 9 1 2 9 5 6

8 5 6 9 2 4 9 3 5

8 5 6 9 1 3 9 2 6

1 5 6

2 3 4

1 2 6

3 4 5

7 8 5 7 9 1 8 9 6

7 8 2 7 9 3 8 9 4

7 2 3 7 4 6 8 1 2

E I1,3,41

,

,

r E {1,2,41

i

s

.

E {1,2,3}

i

I t i s a r o u t i n e m a t t e r t o s e e t h a t , i n i ) and i i ) , t h e r e i s no a

(P;P ,P ,P ,P 1 2 3 4

Lemma 4 . 2 .

with

DS= [3,f41

T h e r e i s no

(P;P1,P

2

1 .

9

J P ,P 3

Suppose t h a t t h e r e e x i s t s a

Proof.

DS = 1 , ( 4 1 6 , ( 3 1 5 ]

. Let

M

3

= {1,2,.

.. , 5 )

4

I

with

( P ; P ,P , P ,P ) w i t h 1 2 3 4 and , M4 = {a,b,e,d,e,t}

P=M UM 3 4 .

A t f i r s t , suppose t h a t t h e r e e x i s t s a block E P

R ={1,2,31

1

UAli,{l,2,3}1 i= 1

( Y (= 3 , 4

with

RGM

3

. Let

w i t h o u t l o s s o f g e n e r a l i t y . A p p l y i n g Lemma 2 . 1 ,

i3

obtain

=4,5

.

Let

3 U A f i l

Y = P -

, we

i=l

IYnM4( ) l

. Then

.

DS = I . ( 4 J 6 , ( 3 1 5 ]

( P I (2 1 4

i t must b e

we

obtain

. This

is

impossible. Now, s u p p o s e t h a t t h e r e e x i s t s a b l o c k

I

.

R

such t h a t

=2 Let { 1 , 2 , a ) E P I , X = A ( l ) u A ( 2 ) u A f a ) - { 1 , 2 , a } 3 Y = P - { A ( l ) u A ( 2 1 u A ( a l } , I t follows t h a t 1x1 = 6 , 7 , 8 ,

lRnM

A t f i r s t , suppose

1x1

=8

. Let

X-Alal

and

= { z 1 , z 2 1, from Lemma

323

Parameter D ( v , tv-13)for Steiwer Triple Systems

x EA(i,{l,Z,u}l , f o r some i = 1 , 2 7' 2 ~Ali,{1,2,a?InAla,{1,2,a}l I < 2 This i s impossible.

2.2 we obtain

and hence

x

.

1x1

Now, s u p p o s e

we h a v e

.

Y ={3,4}

since otherwise

.

=6

flence

n I M4

1A(3,{3,4,t}lnA(4,13,4,t1)

4

,

Let

Y =I31

Let

{3,4,t}EP

- { u , t l ) l 1.2

1x1

Now, s u p p o s e

follows t h a t

.

e

5

n =I1

Since

Al41

- {a,ti)l

.

,

4 e Y

,

n=Il

,

(3,41C_P

w e have

,

a $ A(41

we m u s t h a v e

. Let

yEM4-{a,t}

,

and

{b,c,d}GA(3)

(otherwise

R

2

{c,d,t}EA(a)

e,dEM

),

5 with

={24,u,r} E P

or

(c,d)I=R2

(cr,B)

.

= (c,d)

If

.

Since

nor or

.

and

B =d

and with

< 2

, w e have

34Ala)

e,d#A(llnA(21

.

Let

A t t h i s p o i n t we h a v e

,

otherwise

.

S , a E A ( l ) t ~ A ( 2 ) and

a =c

we

x,aEA(4)

Ib,t,e}cA(l)nA(z)

c,dEA(l)c,A(2/

u =u = e

if

I

It

i t follows that

xcEAf41

Further, since

{u,u,r} C{5,b,c,d,el

u =v =5

I t follows t h a t

.

1

it follows t h a t

1 c,d$A(l)uA(2)

neither

If

,

1,414, { 4 , x , y 1 ) n A ( x , { 4 , x , y I )

{4,a,b?,{4,c,d}EP

.

.

,

j =2,3,4

.

(2

=@.

4 i =1,2,3,4

4EA(lluA(2)uA(al

.

n

is impossible.

f o r every

i

.

IAi3,13,4,tl)nA(4,{3,4,t}ll

,

IA(l,{l,Z,ul)

. This

={3,4,5}EP with R C M Since 1 , 2 , a $ A ( 3 ) 1 j 13 a ~ A ( 4 1 and 1,2#A(4/ In f a c t , l e t x E { l , 2 ]

must h a v e

1 ,4t h e n

it follows t h a t Y n M

, o t h e r w i s e , f o r some

R

~

i t follows t h a t

and

~3

1M41 2 7

T h en

I

IPl

tEA(llnAl2lnA(ai

from P r o p e r t y 2.4

. Since

=7

then, since 1

with

{3,4,tlEP1

. Further,

m >13

nA(2,{2,2,ul)n(M

Y n M4 # @ i m p l i e s

Since

Crisp.

$A(lI uA(2)

u =d

with

B =c]

and

, then

.

={5,e,d} From P r o p e r t y 2 . 4 i t i s n o t p o s s i b l e t h a t 2 IA(x,{x,5,y}lnA(S,{x,5,y}ll < 3 for x=l,2 and y E M 4

R

{5,a,t} E P

1

or

{5,a,bI

E

P

. The n

I '

T h i s i s i m p o s s i b l e a n d t h e p r o o f o f t h e lemma i s c o m p l e t e . . Lemma 4 . 3 .

f P ; P ,P2,P ,P I 1 3 4

T h e r e i s no

Proof. S u p p o s e t h a t t h e r e e x i s t s a DS =

is,

/4)4, (3)6]

.

Let

M

3

= { I , z , . ..,6'1

with

DS = [ 5 , ( 4 / q , ( 3 / 6 ] .

fP;P , P ,P ,P I 1 2 3 4

with

,

, M4 ={a,b,c,dl

M

5

and

P =

uMi

i=3

*

A t f i r s t , suppose t h a t t h e r e e x i s t s a block

REE4

3

.

Let,

5

=Is}

3 24

S. Milici

R={1,2,3}EP

I

obtain

w i t h o u t l o s s o f g e n e r a l i t y . A p p l y i n g Lemma 2 . 1 we

2

3 uA(i,{1,2,3]1)

. Let

=4,5

,

Y=P-(Afl)uAf2luA(3))

i=l

then (necessarily)

with

IY( = 3

,

YcM3

lPll 2 1 4 .

since otherwise

I t follows t h a t

1 2 3 2

x 4 5 x 6 d 4 6 a

2 x b 2 . . 3 x c

P = l x a 1 . .

d ~ A ( 4 J u A I 5 l, otherwise

with

$P,

d ~ A ( 4 )

(otherwise 14,d,c) E P

Since

and

1

b E

n

and s o

Now, s u p p o s e

E P

deM3

1 3

. This

. It

I ZZ),

.

it fol-

, it follows t h a t

i s impossible.

follows t h a t

d ~ A ( 4 1 n A f S J

1

(2

. Since

.

Ali) i=l

IA(5,{5,6,b}lnA(b,{5,6,b}ll

{b,IJc},{b,3,a}EP2

15,c,al

IA(5,{5,c,a}lnAlc,{5,c,a}) 3

lows t h a t

3 . .

1A(6,{6,dJ+lInAIdJC6,d,x})I

. Necessarily

Suppose, f i r s t , (4,b,d)

4 . . 5 6 b 5 . .

,

a4A151

other-

3

wise

{4,d,cl

E

,

Pl

c

nA ( i l

E

and s o

{lJc,b},~2,c,a},{3,dJb} E P

i=l

and

~ A f l , { I , a , x } ) n A f a , ~ l , a , x ~ () (2

it is not possible t h a t (4,bJd},{a,2,d3,{a,3,b}

.

If

,

a$A(5/

for Property 2.4

IAf4,[4,6,a})nA(a,{4,6,al)l E

P

1

{l,b,c}E P

and hence

<3

with

1

1

. Then, c E M

3 '

This i s impossible. Now, s u p p o s e t h a t t h e r e e x i s t s a b l o c k

R

such t h a t I R n M 3 1 = 2 .

X = A ( l ) u A ( 2 l u A l a l - { l J Z J a } and 1 ' Y = P - ( A ( l ) u A ( 2 ) u A ( a ) l . I t follows t h a t 1x1 =6,7,8

Let

{1,2,a} E P

1x1

4 . 2 we o b t a i n

Now, s u p p o s e t h e n we h a v e nA(a) n(M4

1x1

. =6

since otherwise

. This

and

. Since

. Hence

Y ={3,41

- {all I 2 2

1M41 > 4

f 8

m >23

YnM

4

{3,4,b} E

= @ implies

. Further,

I A ( 3 ) n . 4 ( 4 1 n (M4

Pl

with

since

- {a,b)) I '

. From

Lemma

lPll 214

,

bEA(l)nAf21n IA(Z)nA(2) A

2

,

it follows

i s impossible.

Now, s u p p o s e

1x1

=7

. Since

n =I1

,

i t follows t h a t Y n M 4 = @

.

325

Parameter D(v, t,-l31 f o r Steiner Triple Systems

,

Y={3}

Let

then, since

I t follows t h a t with

R E P

J-

{3,4,x}

IA(3)nM

RGM

.

3

1

.

In fact, let

. If

y =5

zE {1,2}

,

and

ce

,

P

{5,d,b} E

cA(41

. Then

c,b,d€Af5)

for

6 =b

X, b ) ,

5

.

.

nAla, {1,2,u}) n(M4

E P

I '

with

, f o r some

' 2

a ~ A ( 4 )

.

and hen-

This is impossible.

I t follows t h a t

6EA(I)r\A(2)AA(a)

- { a ] )1

.

Let

{4,x,c},I4,a,d},IS,c,~},

[z,c,b,a} C A(51

t h e r it follows t h a t

lA(y,{y,x,c~})(3

{ I , 6 , c ) , { 2 , x , d ) , { 2 , 6 , b } E Pl

{3,x,c},{3,5,dI

and

and 6 E I x , ~ }

y =4

I t follows t h a t

n ~ ( 2 , { 1 , 2 , a l ) n (M4uM5)I ' 2

Now, s u p p o s e {x,c,d,a}

w e must have

x ~ A l y ) we m u s t h a v e

and

{u,x, 6 } , { I ,

IAf1,{1,2,a})

with

5 2

i t follows t h a t a ~ A ( y )

a , r ~ A ( y )

~ E J M- { a } ) u M 4 {3,5,xl, {3,c,d} E PI

Suppose, f i r s t , x,c,d~A(4)

if

with

nA(x,{y,x,a})1(2

,

1,2,a$A(3)

.

a#A(y)

lA(1,{1,2,u})n

.

I '2

]A(3,{3,y,6jInA(y,13,y,6}1I

for

and so

.

,

j =2,3,4

. Necessarily

A(3! ={4,5,x,b,c,d}

y =4,5

for

f o r every i =1,2,3,4

(3,4)EPi

, o t h e r w i s e , f o r some

=2

{ 3 , 4 , b } ~ P ~ Since

1,2 $ A ( y )

,

A(41 ={3,x,a,b,c,d}

n A f 2 , { 1 , 2 , a } ) n ( M 4 u M5)

and

3

Let

4 PI , o t h e r w i s e

Let

n = I 1

{4,5,c,z}

and hence

.

i =1,2

EA l a l

. Fur-

IA(i,{l,2,a})n

T h i s is i m p o s s i -

b l e a n d t h e p r o o f i s complete.. Lemma 4 . 4 .

P ,P

1 ' 2

3

Suppose t h a t t h e r e e x i s t s a

Proof. DS = [ f 5 1

T h e r e i s no ( P ; P

(4)2,(3/7]

2'

.

Let

M

3

= {I,Z

,P

4

with

DS=[(5)2,(4)2,(3)7].

(P;P ,P2,P ,P ) with 1 3 4 71 , M = { a , b } , M ={z,y} 4 5

,...,

5

and

u Mi

P =

n = 11

, Since

, it follows t h a t

(x,yl

C_P

i=3

A t f i r s t , suppose t h a t e x i s t s a block ~ = { 1 , 2 , 3 }€ P I

,

3

i3

=

1x1

=4,5

.

1 3 14

P I I, -> 1 4 ,

if

if

Let

3

Let

UAli) i=l

Y = P -

i =I

lyl =

.

w i t h o u t l o s s o f g e n e r a l i t y . A p p l y i n g Lemma 2 . 1 , we

u A(i,{l,2,3})

obtain

RGM

I '

=5

1x1 = 4

.

Further, it follows

YGM

3

. Then

otherwise

326

S. Milid

1x1 = 4 , w e

If

I

=i:lI

and hence If

obtain

2 x b

3 y b

4 6 y

5 6 b

2 a y

3 x a

4 5 x

4 7 a

5 7 y

,

we o b t a i n

. This

=2

and

Y={4,5,61

,

Y={4,5,6,7}

l a b

lA(7,f4,7,a}lnA(a,14,7,a})l IXI=5

,

X={a,b,x,y}

6 7 x

is impossible.

.

X=I7,a,b,z,yl

It

3

follows

Ali)

7 E

,

(i,j , k l C A ( 7 1

otherwise

with

i,j

E {1,2,31

i=l

k ~ 1 4 , 5 , 6 l or

and

i,j

j

E {{1,2,31

If

z E A ( ~ J and

-{ill

z =b

and h e n c e

,

with

' 2

IA(7,{7,i:,yllnA(k,{7,k,y}il

r i l y w e have

i

1x1

#8

. It E

P

,

=2

1x1

y E{7,a,x,y}

R

such t h a t l R n M

with

1x1

#6

i24

.

4

s

i24

for

1x1

Further, it follows 6 E M u M

,

and hence

f 7

,

YEM

and hence

.

3

I

=2.

Y =P-(A(I)U

Lemma 4 . 2 w e o b t a i n

since otherwise

bEA(l)nAI2JnA(a)

n ~ l b , t 3 , 4 , b I ) n ~ ~'2 I

with

,

. From

=6,7,8

.

with

X = A ( l l u A ( 2 ) u A f a l -{1,2,a}

If follows

follows

I

,

E {l,2,31

we o b t a i n

R = { 1 , 2 , a } E PI

{3,4,bI

Then n e c e s s a -

,

IAii,{i,b,y}InAIb,{i,h,y}iI

uA(2)uA(a)l.

.

y E {a,b,z,yl

Now, s u p p o s e t h a t t h e r e e x i s t s a b l o c k Let

and h e n c e

k E{1,2,31

with

z # A ( j )

z E {a,bl

and

and

E {4,5,61

, with Y={3,4), 3 IA(3,{3,4,bI) n

since otherwise

{3,4,63

E

IA(3,{3,4,61)nA(4,{3,4,6}1nMi)

PI ( 2

, for

T h i s c o m p l e t e s t h e p r o o f o f t h e lemma., Lemma 4 . 5 . Proof.

There is no

(P;P ,P ,P ,P 1

2

Suppose t h a t t h e r e e x i s t s a

DS= [(5j3,(3j8]

, Let

M

3

={1,2,.,.,8}

3

with

4

DS =

(P;P ,P ,P ,P 1

,

M

5

2

3

4

[ (S13,

I

= { a , b , c } and

(3Jg]

.

with P=M

3

u M

5 .

327

Parameter D(v. t,-13) for Steiner Triple Systems

Since

n =I1 Let

with

ct,B,y

it follows t h a t

{a,b,c} $P,

,

i n3

i=l

(x,ylC_P

,

1

for

x , y E ia,b,cl

.

w i t h o u t l o s s o f g e n e r a l i t y . Then

.

~{4,5,6,7,8}

Since u = B = y = 4

,

(A(i,{1,2,3}j

, f r o m Lemma 2 . 1 we o b t a i n

‘ 3

;

1 2 3

1 l I 2 2 3 4

l a b 1 2 2 3 3

4 a 4 4 b

. Necessarily

{ 1 , 2 , 4 ) E P2

A t f i r s t , suppose

c c b a c

2 a b a 3 4 a

1 1 1 2 2 3

4 3 c b c b c

2 4 3 4 3 4

a b c c b a

...

1 1 1 2 2 3

I

...

2 4 3 4 b a

1 2 b 1 4 3

c a b 3 a c

...

...

l e a 2 3 a 2 4 c 3 b c

...

I

1

...

or

-

ii)

p3 1 2 3

1 2 4

l a b

l u c

1 2 2 3 3

1 2 2 3

4 a 4 4 b

...

c c b a c

3 3 c 4

...

Pan

p4

#@

and

b a b c

...

PI

2 4 b 4 3 a

... ...

a 3 c c b c

1 1 1 2

f0

A

. Then

2 4 3 3

1 2 c

b a c a

1 1 2 2 3

2 4 c 3 4 b

... I *.. I

PI n F 5 # @ and

I n case i ) we h a v e have

1 1 I 2 2 3

PI n a

3 4 4 b b . * * * * *

F3 # 0

(P; P

a b 3 a c

1

. In

case i i ) we

,P 2 ,P 3 ,P 4 )

cannot

exist.

Now s u p p o s e

{I, 2 , 4 }

4P

j

f o r every

j =2,3,4

. Necessarily

S.Milici

328

i) 1 l 1 2 2 3 3

2 a 4 a 4 4 b

3 b c c b a c

p4

pZ

p3

1 2 a 1 4 3

1 2 b 1 4 a

l 2 2 3

1 2 2 3

b 4 3 a

c c b c

...

4 c 4 b

2 b a 3 b c

2 3 4 3

c 0 b 4

...

...

...

...

a , .

3 c a 3

7 1 1 2

. I .

_ I

or I

I

I

ii) 1 l 1 2 2 3 3

2 3 a b 4 a 4 4

e c b a b c

p2

p3

1 Z a 1 4 b 1 3 c

1 2 b 1 3 4

2 4 c

p4

2 4 a 2 3 c 3 a b

1 1 1 2

l e a

2 3 b 3 4 a

... ...

...

2 4 3 3

e a b 4

2 b a 3 a c

... ...

... ...

P n P # @ i n c a s e i ) and

I t follows t h a t

i i ) . Then a

I

(P;P ,Pz,P 1

1

4

3

, P 4)

P l n P 2 # @ in case

c a n n o t e x i s t s and t h e p r o o f i s comple-

te.

Lemma 4.6. or

DS=

1413,(319]

=

RcM

or

1 5 , 4 , ( 3 1 1 0 -I 3

.

.

Suppose t h a t t h e r e e x i s t s a

Pro0 f DS

(P;P ,P ,P ,P I 1 2 3 4

T h e r e i a no

Let

2 . 1 we o b t a i n

{2,2,3}

€PI

I3

UA(i,{1,2,3}) i=l

(XI = 4 + k

and

1x1 # 4 ,

otherwise

(YI = 5

(YI =4

Let

with

1

,P

2’

P ,Pql 3

1

10

with

e v e r y case e x i s t s a b l o c k

w i t h o u t l o s s o f g e n e r a l i t y . A p p l y i n g Lemma

clearly and

(p;P

. In

DS = [ ( 4 ) 3 , ( 3 ) 9 ]

DS = [5,4, ( 3 1

vith

=

YAMi=@

Y={4,5,6,7}~M 3

for

=4,5

.

3

Let

Y

=P- U A ( i l , i=l

for

(Y(= 5 - k

and

1x1

k =0,1

. Then

l P , l 224

. Since

i=4,5 m=13

.

I t follows t h a t

we h a v e

1x1

=5

. ,

we o b t a i n t h a t ( x , y ) C P 1

Parameter D f v , tv-13)for Steiner Triple Systems

with

.

xJy E Y

Let

2 = {z E

I t follows t h a t

Z nM with

3

IZI =3

and hence

zl f z 2

=a . zl

E

Z

M

,

otherwise

IA(4,{4,5,z

1

~

.M A t~ t h i s

3

={1,2

PI w i t h

{ 4 , 5 , z I l , {6,7,z

2

I5

{4,5,z1))

p o i n t we h a v e

,..., 9 ) ,

E

2

1

E

3'

M

5

.

s , y E }'l

P

1

, with

. Observe

lAfl,{l,zl,y}l nafzl,{l,zlJyII

DS = ( 1 4 )

and hence

{zJy,x}

1 ) nA(zl,

O t h e r w i s e we o b t a i n

= {8,9]EM3

Let

.., 7 1 : 7

P - {1,2,3,.

329

I

that =O

3 nA(i,{lJ2,311 =

=@,

i=l

.

13J9]

4 M

4

and

={a,b,c]

U Mi '

P =

i=3

Then

or

I n c a s e i ) we h a v e ii),

l ~ ( l , { ~ , O , ~ } ) n ~ f u J { ~ ,5u2, ~ , }In l l case

.

l ~ f ~ , { l , 8 , a } ) n A f u J { l J 8 , 0 } 5) l2

( P ; P ,P , P , P ) 1 2 3 4

Then a

can-

n o t e x i s t s and t h i s c o m p l e t e s t h e proof., Lemma 4 . 7 .

Proof.

T h e r e is no

3

.P 4

with

DS=[(3)13]

.

The s t a t e m e n t f o l l o w s i m m e d i a t e l y f r o m Theorem 2 . 1 o f [ S ] .

Theorem 4 . 1 . Proof.

(P;Pl,P2,P

D(v,t -13) = 3

for every

V

A p p l y i n g Lemmas 4 . 1 , 4 . 2 ,

obtain that a

I F ; PI, P2,.

..,P

)

e x i s t , Then, s i n c e t h e e x i s t e n c e of o f them i n t e r s e c t i n

tu-13

4.4,

4.5,

4.6 and 4.7, we

m=13

and

s > 3

4.3,

with s

.

v z 1 5

STS(vls

blocks (these

,

t -13 V

cannot

s u c h t h a t a n y two blocks occurring,

330

S.Milici

moreover, in each of the (P;PIJP2,...,PSl v L15

every

.

S T S f v l s ) implies the existence of a

, from Theorem 3.1 we obtain D ( v , t V - 1 3 1

=3

for

REFERENCES

111

J . Doyen, C o n s t r u c t i o n of d i s j o i n t S t e i n e r t r i p l e s y s t e m s , Proc. Amer. Math. SOC., 32 (1972), 409-416.

L2-1

J . Doyen and R . M . Wilson, E m b e d d ings of S t e i n e r t r i p l e s y s t e m s , Discrete Math., 5 (1972), 229-239.

[3]

S . Milici and G. Quattrocchi, Some r e s u l t s on t h e maximum numb e r of S T S s s u c h t h a t any two of t h e m i n t e r s e c t i n t h e same b l o c k - s e t , preprint.

14.1

G. Quattrocchi, A l c u n e c o n d i z i o n i n e e e s s a r i e p e r DMB P T S e o n e l e m e n t i d i g r a d o 2 , Le Mate matiche (to appear). S. Milici and

Z ' e s i s t e n z a di t r e

[5]

G . Quattrocchi, S u l m a s s im o numero d i D M B P T S a v e n t i 1 2 b l o c c h i e i m m e r g i b i l i i n u n STS , Riv. Mat. Univ. Parma (to

appear). 161

G. Quattrocchi, SuZ p a r a m e t r o D ( 1 3 , 1 4 1 di S t e i n e r , Le Matematiche (to appear).

1.71 G . Quattrocchi,

SuZ p a r a m e t r o

D(v,tv-lOl

p e r S i s t e m i d i Terne

,

19 ' v < 33

per

S i s t e m i di T e r n e di S t e i n e r , Quaderni del Dipartimento di Mate-

matica di Catania, Rapport0 interno. 181

Rosa, I n t e r s e c t i o n p r o p e r t i e s of S t e i n e r s y s t e m s , Annals Discrete Math., 7 (1980), 115-128.

A.

Annals of Discrete Mathematics 30 (1986) 331-334 0 Elsevier Science Publishers B.V. (North-Holland)

33 1

A NEW CONSTRUCTION OF DOUBLY DIAGONAL ORTHOGONAL LATIN SQUARES Consolato P e l l e g r i n o and Paola L a n c e l l o t t i D i p a r t i m e n t o d i Matematica V i a Campi, 213/B 41 100 MODENA ( ITALY)

.

We g i v e a new s i m p l e c o n s t r u c t i o n o f p a i r s o f d o u b l y d i a g o n a l o r t h o g o n a l L a t i n squares o f o r d e r n, DDOLS(n), f o r some n=3k i n c l u d i n g t h e case n=12.

A p a i r o f d o u b l y d i a g o n a l o r t h o g o n a l L a t i n squares o f o r d e r n, DDOLS(n), i s a p a i r o f o r t h o g o n a l L a t i n squares o f o r d e r n w i t h t h e p r o p e r t y t h a t each square has a t r a n s v e r s a l b o t h on t h e f r o n t d i a g o n a l D1 and on t h e back d i a g o n a l D2 The r e a d e r i s r e f e r r e d t o t h e monograph [I] by J.Denes and A.D.Keedwel1 for t h e d e f i n i t i o n s which a r e n o t g i v e n here. W.D.Wallis and L.Zhu proved t h e The problem was posed by K . H e i n r i c h and e x i s t e n c e o f 4 DDOLS(12) i n [ Z ] A.J.W.Hilton i n [ 3 ] .

.

.

Let

Q

be a L a t i n square o f o r d e r n based on t h e s e t be t r a n s v e r s a l s o f

S, T

t o t h e element o f of

r)

. We

Inoccupying t h e c e l l

In occupying t h e c e l l

In={O,l

form a permutation

(k,i)

(h,i)

,..., n-1)

and l e t

Inas f o l l o w s :

on

o f S we a s s o c i a t e t h e element

o f T ( i . e . t h e c e l l o f T t h a t l i e s i n t h e same

column). We denote by Q(S,T) t h e L a t i n square o b t a i n e d by r e p l a c i n g each e n t r y s o f Q w i t h t h e element a S a T ( s ) . O b v i o u s l y we have: (a) i f

U

i s a transversal o f Q then

U

i s also a transversal o f

Q(S,T);

( b ) i f R i s a L a t i n square which i s o r t h o g o n a l t o Q t h e n R i s a l s o o r t h o g o n a l t o Q(S,T). Let

Q

be a L a t i n square and l e t h be a symbol; we denote by

o b t a i n e d by r e p l a c i n g each e n t r y

s

THEOREM. For an even p o s i t i v e i n t e g e r k l e t l e t T1, T2 be two common t r a n s v e r s a l s o f A

Q

(h,s).

A, B be a p a i r o f DDOLS(k) and and B I f T1 and T2 have no

common c e l l w i t h each o t h e r and w i t h each d i a g o n a l exists a pair o f

Qh t h e copy o f

o f Q w i t h the ordered p a i r

.

D1

and

D2

DDOLS(3k).

P r o o f . Consider t h e two o r t h o g o n a l L a t i n squares o f o r d e r

3k

, then

there

C. Pellegrino and P. Luncellotti

332

O f course 8 possesses a tranSversa1 on t h e f r o n t d i a g o n a l , w h i l e t h e back diagonal i s a transversal o f B S t a r t i n g f r o m A and B we f o r m t h e f o l l o w i n g L a t i n squares o f o r d e r 3k

.

From ( a ) and ( b ) i t f o l l o w s i m m e d i a t e l y t h a t t h e square

i

on t h e f r o n t d i a g o n a l w h i l e we have :

Aij

( c ) each subsquare having (d)

A”

,D,

T,,T,

and

of

,D,

s t i l l has a t r a n s v e r s a l

has a t r a n s v e r s a l on t h e back d i a g o n a l . I n a d d i t i o n

Bij

and

6

of

i s a doubly d i a g o n a l L a t i n square

as p a i r w i s e d i s j o i n t t r a n s v e r s a l s ;

are orthogonal.

Since t h e square i s obtained from subsquares, we have f o r j=1,2, ...,k :

Hj

the set

o f the entries o f the

D,

transversals

, T,

A,,

of

H’. o f t h e e n t r i e s o f t h e J

D, o f

A,,

T, o f

A,,

and

7i by s u i t a b l y renaming symbols i n t h e j - t h column o f

which l i e on t h e

A,, and D, o f A, coincides w h i t t h e s e t ( k t j ) - t h column o f l y i n g on t h e t r a n s v e r s a l s of

a

D, o f

A,,

;

the set

K o f t h e e n t r i e s o f t h e ( k t j ) - t h column o f which l i e on t h e j t r a n s v e r s a l s D, o f A,,, T, o f A,, and D, o f A,, c o i n c i d e s w h i t t h e s e t

Kj

o f t h e e n t r i e s o f t h e ( 2 k t j ) - t h column o f

D, o f each

, T,

A,, j=1,2,

of

...,k

A,,

and D, o f

exchange i n

A,,

?i the

’li

elements o f

same row; s i m i l a r l y we exchange t h e elements o f Ki same row o f

A”

(property

d i s t i n c t c e l l s ) : from

A^

(e)

(c) and

l y i n g on t h e t r a n s v e r s a l s

. H

j and

implies t h a t the elemints o f (f)

matrix i s a L a t i n square. F u r t h e r e a s i l y shows.

and

K:

ij

Hj

appearing on

appearing on t h e and

K! occupy

J

i t f o l l o w s immediately t h a t t h e r e s u l t i n g

8

i s d o u b l y d i a g o n a l as t h e c o n s t r u c t i o n

333

Doubly Diagonal Orthogonal Latin Squares Observing t h a t

has p r o p e r t i e s which a r e analogous t o ( e ) and ( f ) we can

exchange elements i n

c

B

as we d i d i n d e r i v i n g

A”

from

A

d o u b l y d i a g o n a l L a t i n square ( d ) . Hence

8

and

8

which i s o r t h o g o n a l t o

are p a i r o f

and t h u s o b t a i n

R

a

because o f p r o p e r t y

DDOLS(3k).

EXAMPLE. S i n c e f o r each r s 2 t h e r e e x i s t s a p a i r o f DDOLS(2r) s a t i s f y i n g t h e h y p o t h e s i s o f t h e p r e v i o u s Theorem, we have t h a t f o r each r 3 2 we can c o n s t r u c t a p a i r o f DDOLS(3-2r). ACKNOWLEDGEMENTS. Work done w i t h i n t h e sphere og GNSAGA o f CNR, p a r t i a l l y s u p p o r t e d by MPI. REFERENCES J.Denes and A.D.Keedwel1, New York, 1974).

L a t i n squares and t h e i r A p p l i c a t i o n s (Academic Press

W.D.Wallis and L.Zhu, Four p a i r w i s e o r t h o g o n a l d i a g o n a l L a t i n squares o f s i d e 12, U t i l . Math. 21 (1982) 205-207. K . H e i n r i c h and A . J . H i l t o n , Doub1.y d i a g o n a l o r t h o g o n a l L a t i n squares, D i s c r . Math. 46 (1983) 173-182.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 335-338 0 Elsevier Science Publishers B.V. (North-Holland)

335

ON THE MAXIMAL NUMBER OF MUTUALLY ORTHOGONAL F-SQUARES

Consolato PELLEGRINO and N i c o l i n a A. MALARA D i p a r t imento d i Ma tema ti ca V i a Campi, 213/B 41 100 MOOENA ( ITALY)

I n t h i s paper we pi-ov t h a t t h e upper bound, g i v e n by Mandeli and Lee and Federer 131 f o r t h e number t o f orthogonal squares, a l s o h o l d s f o r F1(n;xl), F2(n;;x2), ... , Ft(n;ht) x 1 ,ml 1 9 t h e number t o f o r t h o g o n a l F1(n;x1,1,x1,2 F*("12,1 ,A*,2'* *. J ~ , ~ ~ , Ft(n;ht,l,xt,2 L ,A t,mt 1 squares.

,...,

...

1

-

,...

DEFINITIONS AND PRELIMINARY RESULTS

Hedayat and Seiden, i n c o n n e c t i o n w i t h some r e s u l t s b y o t h e r a u t h o r s , g i v e i n 111 a g e n e r a l i z a t i o n o f t h e concept o f l a t i n square: t h e c o n d i t i o n t h a t each element appear e x a c t l y once i n each row and i n each column i s s u b s t i t u t e d b y t h e c o n d i t i o n t h a t each element appear one and t h e same f i x e d r l u m b e r o f t i m e s i n each row and i n each column. They c a l l such squares frequency-square o r s h o r t l y F-squares. More p r e c i s e l y t h e y g i v e t h e f o l l o w i n g d e f i n i t i o n : DEFINITION 1.

...,am 1 .

Let

L

We say t h a t

write briefly

A

k.':J 1

F =

m a t r i x d e f i n e d on a

i f f o r each

times

Ak

nxn

F-square o f t y p e

,..., A,,,),

F(n;xl,x2

appears p r e c i s e l y

i s an

F

a

b

,..., A),

(n;xl,x2

k=1,2,.

A = Ial,a2,

m-set

..,m

and we

t h e element

ak

of

i n each row and i n each column o f

ik 2 1)

F

.

A ~ = X ~. .== A. = A t h e n m i s determined u n i q u e l y by n and h , m hence we s i m p l y w r i t e F(n;A) Note t h a t an F ( n ; l ) square i s s i m p l y a l a t i n I t i s easy t o prove t h a t an F(n;Al,x 2,...,hm) square of o r d e r n square e x i s t s m

In particular i f

.

.

i f and o n l y i f & X i = n

I n [l; Hedayat and Seiden a l s o e x t e n d t o F-squares t h e c o n c e p t o f o r t h o g o n a l i t y o f l a t i n squares t h r o u g h t h e f o l l o w i n g d e f i n i t i o n s : DEFINITION 2.

Given an F(n;xl,A 2 2 ' . . . , A

and

,...,p s )

F2(n;p1,p2

i s orthogonal t o the p a i r each

(ui,v.)

J=1,2,

J

F 2 , and w r i t e of

UxV

Ai

Let

F-square o f t y p e

be a

(n;xi ,1

i s a set o f

i p j , i,j=1,2

on a F1

appears

r

)

square on a

s-set

r-set

U = {u,,u2,.,.,ur}

,...,v S I ,

V = {vl,v2

we say t h a t

1.F2

, i f upon s u p e r i m p o s i t i o n o f

xipj

times, f o r each

i=1,2,

F,

F1 on and f o r

...,r

5

...,s .

OEFINITION 3.

, ,.,Ft

square

,..., t.

t

m.-set,

, A ~,2,.

1

.

i=1,2

"i ,mi )

,...,t.

on t h e s e t

Fi

be on

We say t h a t

F,,F2,

F o r each

Ai

mutually (pairwise) orthogonal

.

i, l e t

F-squares i f

FiLFj

,

C. Pellegrino and N. A . Malara

336

Hedayat, Raghavarao and Seiden proved t h a t the maximal number o f mutually I n [2] orthogonal F(n;x) squares i s (n-1) 2/(m-1) , where m=n/A. I n [3] Mandeli, Lee and Federer proved t h a t the maximal number t o f mutually orthogonal , Ft(n;xt) squares (where f o r each i = l y 2a...,t Fi i s F1(n;;X1), F2(n;;x2), defined on a mi-set and n=himi) s a t i s f i e s the i n e q u a l i t y

...

2.

ON THE MAXIMAL NUMBER OF MUTUALLY ORTHOGONAL F-SQUARES

I n analogy t o THEOREM. Let

we prove the f o l l o w i n g

[3]

...,h Z a m 2 ) , ... ,

F l ( ~ ; ~ l , l y h l , 2 a . . . a ; x l , m l ), F2(n;;x2,1,h2,2y

Ft(n;;xtal,xt,2,...,A

)

be

t

mutually orthogonal

tamt

i = l a 2 ,...,t Fi i s defined on the mi-set number t s a t i s f i e s t h e i n e q u a l i t y t

‘&mi

-t5

Ai

F-squares, where f o r each mi and n = xi,j Then t h e

&

.

.

(n-1) 2

1=

Proof. From Fh(n;;xh,lyhh,2y...yx ) we d e f i n e a n2 xmh m a t r i x Mh = [a:ja,], hamh where ah -1 i f the k - t h symbol o f Ah occurs i n the c e l l ( i , j ) ( i a j = l , 2 , ij,kn ) o f Fh and 0 otherwise. L e t M = [MlIM21 . I M t ] . By the property o f the

...,

..

(n-1) 2t 1 and

F-squares, the number o f l i n e a r l y independent rows i n M i s a t most so we o b t a i n

Now, we can w r i t e the product o f t h e transpose o f

M’M

=

L2Jm xm L~ 2 1

L2N2

M with

...

M i n t h i s manner:

2 Jm2xmtLt

:~ I

where

Li =

’

[ U ~ , ~ J( i = l y 2 y , . . a t )

,...,

i s a diagonal m a t r i x o f order

mi w i t h

urlr= i

f o r each r=1,2 mi, Ni = [nk,s] i s a diagonal m a t r i x o f order mi i nr,r = n f o r each r=1,2, ...ami i s a m a t r i x o f s i z e mixm and Jmixmj j ( i ,j=1,2,. ,t) w i t h the element 1 everywhere.

with

..

Maximal Number of Mutually Orthogonal F-Squares

337

Let

Om2xml

*..

L2

- J1

...

Lt

where

i s the matrix o f size

Omixm

j

everywhere.

As

A.

.#O,

-

JmtxmlL1

The e i g e n v a l u e s o f t

t- (mi-1) GT

and

t-1

are

M

.

(iyj=lyZ

1 . l

i s i n v e r t i b l e and t h e m a t r i x

A

1 ,J

as t h e m a t r i x

,..., t )

m.xm.

tn,

Jmtxm2L2

n

and

0

*

*

w i t h t h e element

M'M

*

0

has t h e same rank

Nt

j! .

with respective m u l t i p l i c i t i e s

1

,

Then t

1 +

(mi-1)

= rank(M) = rank(M'M) =

t rank(#) Hence

5 m i n { (n-1)

t

mi

-

t

1=

When

...=

XiYl=Xiy2=

and Federer

[3]

.

1.

i,m.

...

i

(i=1,2,

..., t )

. we have t h e r e s u l t b y Mandeli, Lee

Furthermore, t h e p r e v i o u s theorem suggests t h a t we c a l l a s e t

of m u t u a l l y o r t h o g o n a l

A~,,,~),

=A 1

< (n-1) 2

F-squares

Ft(n;;h t,l,~t,2,,.,y~t,mt)

F1(n;;xl ,l,;xl

y2

,...,A,,,,~),

a complete s e t i f

F2(n;;x2,1yx2,2y...y

338

C. Pellegrino and N . A . Malara

where

n=hiYl+xiy2t

...+x. ,mi 1

(i=l,2,.

.., t ) .

ACKNOWLEDGEMENTS. Work done w i t h i n t h e sphere o f GNSAGA o f CNR, p a r t i a l l y supported by M P I

.

REFERENCES (1

1

A.Hedayat, E.Seiden, F-squares and o r t h o g o n a l F-squares design: a g e n e r a l L z a t i o n o f l a t i n square and o r t h o g o n a l l a t i n squares design; Ann. Math. S t a t i s t . 41 (1970) 2035-2044.

121 A.Hedayat,

D.Raghavarao, E.Seiden, F u r t h e r c o n t r i b u t i o n s t o t h e t h e o r y of F-squares design, Ann. S t a t i s t . 3 (1975) 712-716. W.T.Federer, On t h e c o n s t r u c t i o n o f o r t h o g o n a l Fsquares o f o r d e r n f r o m an o r t h o g o n a l a r r a y (n,k,s,2) and an OL(s,t) s e t , J. S t a t i s t . Plann. I n f e r e n c e 5 (1981) 267-272.

131 J.P.Mandoli ,F.C.H.Leey

Annals of Discrete Mathematics 30 (1986) 339-346 0 Elsevier Science Publishers B.V. (North-Holland)

339

CARTESIAN PRODUCTS OF GRAPHS AND THEIR CROSSING NUMBERS Giustina Pica

+

D i p a r t i m e n t o d i Matematica e A p p l i c a z i o n i U n i v e r s i t a d i N a p o l i , Naples, I t a l y Tomat P i s a n s k i

++

Oddelek za Matematik0,Univerza v L j u b l j a n i L j u b l j a n a , Yugoslavia A l d o G.S.Ventre

+

I s t i t u t o d i Matematica,Facolta d i A r c h i t e t t u r a U n i v e r s i t a d i N a p o l i , Naples, I t a l y

Kainen and White have determined e x a c t c r o s s i n g numbers o f some i n f i n i t e f a m i l i e s o f graphs. T h e i r process uses r e p e a t e d C a r t e s i a n p r o d u c t s o f r e g u l a r graphs. I t i s shown how t h i s process can be s u b s t a n t i a l l y g e n e r a l i z e d y i e l d i n g e x a c t c r o s s i n g numbers and bounds f o r v a r i o u s f a m i l i e s o f graphs.

INTRODUCTION

I n t h i s paper graph embeddings and i m n e r s i o n s a r e s t u d i e d . I n o r d e r t o keep i t s h o r t we a d o p t s t a n d a r d d e f i n i t i o n s o f t o p o l o g i c a l graph t h e o r y t h a t can be found, say i n [ 2,3,4,5,13] . U s u a l l y o n l y normal imnersions o f graphs i n t o s u r f a c e s a r e considered, i . e . imnersions i n which no two edges c r o s s more than once and no edge crosses i t s e l f . I n p a r t i c u l a r , t h i s means t h a t two edges t h a t a r e a d i a c e n t do n o t c r o s s . We r e q u i r e i n a d d i t i o n t h e i m n e r s i o n t o be a 2 - c e l l immersion which means t h a t t h e complement o f t h e immersed graph i s a d i s j o i n t u n i o n o f open d i s k s ( 2 c e l l s ) and t h a t t h e r e e x i s t s a s e t o f edges t h a t can be removed f r o m t h e immersed graph i n o r d e r t o o b t a i n a 2 - c e l l embedding o f i t s spanning subgraph i n t o t h e same s u r f a c e . The connected components o f t h e complement o f t h e immersion a r e c a l l e d faces. I n a 2 - c e l l i m n e r s i o n o r embedding a l l f a c e s a r e open d i s k s . A f a c e i s s a i d t o be p a r t i a l i f i t has a t l e a s t one c r o s s i n g p o i n t on i t s boundary o t h e r w i s e i t i s s a i d t o be t o t a l . We w i l l make use o f t h e d e f i n i t i o n o f an ( s k)-embedding o f 1 1 1 t h a t we r e p e a t here f o r convenience ( s e e a l s o [ l o ] and 1125).

[

A 2 - c e l l embedding o f a graph G i n t o a s u r f a c e S i s s a i d t o be an (s,k)-embedding i f we can p a r t i t i o n t h e s e t o f f a c e s o f t h e embedding i n t o s+l s e t s F1yF2y,..,FS,R i n such a way t h a t t h e boundary o f each s e t Fi, l(i(s, i . e . t h e u n i o n o f boundar i e s of faces b e l o n g i n g t o F i s an even 2 - f a c t o r o f G, i . e . a spanning subgraph i’ o f G c o n s i s t i n g o f c y c l e s o f even l e n g t h s ; f u r t h e r m o r e , k o u t o f t h e s 2 - f a c t o r s c o n s i s t o f q u a d r i l a t e r a l s o n l y and a l l f a c e s o f R ( i f t h e r e a r e any) a r e q u a d r i l a t e r a l s . R i s c a l l e d t h e s e t of r e s i d u a l f a c e s and may be empty. I f k=s we a r e dea-

340

G. Pica. T. Pisanski and A. C.S. Ventre

l i n g w i t h q u a d r i l a t e r a l embedding. I f G has no t r i a n g l e s t h e embedding i s a l s o m i nimal, y i e l d i n g t h e genus o r n o n o r i e n t a b l e genus o f G (depending on t h e o r i e n t a b i l i t y t y p e o f S), see [Ill L e t G I have an (s,k)-embedding i n t o S ' and l e t G2 have an (s,k)-embedding i n t o S " . We say t h a t t h e two (s,k)-embeddings agree i f t h e r e e x i s t s a b i j e c t i o n between t h e v e r t e x s e t s o f G and G w h i c h induces a b i j e c t i o n 1 2 o f a l l s sets o f nonresidual faces.

.

Example 1 . P a r t ( a ) o f F i g u r e 1 shows an (1,O)-embedding o f K -2K2 i n t o t h e sphere. The o u t e r f a c e i s hexagonal and t h e r e a r e two r e ~ i d u a 1 ~ ' ~ f a c e Ps a. r t ( b ) of F i g u r e 1 shows an (l,O)-embedding of K i n t o t h e p r o j e c t i v e plane. There i s one hexagonal f a c e and t h r e e r e s i d u a l 3 9 3 f a c e s . The two ( 1 ,D)-embeddings agree, which i s shown by an a p p r o p r i a t e numbering o f v e r t i c e s i n b o t h graphs.

(b)

Figure 1 An i m n e r s i o n o f a g r a p h G i n t o a s u r f a c e S i s s a i d t o be an (s,k,c,e)-immersion if i t i s a 2 - c e l l immersion w i t h c c r o s s i n g p o i n t s and i t . i s p o s s i b l e t o o b t a i n a n (s,k)-embedding o f a spanning subgraph H o f G i n t o S by removal o f e edges, and by removal o f any e-1 edges t h e r e remain some c r o s s i n g p o i n t s ( e i s m i n i m a l ) . H i s s a i d t o be a reduced graph o f t h e (s,k,c,e)-immersion o f G. L e t G have an (s,k,c,e) i n t o S ' . We say t h a t t h e two immer-immersion i n t o S and an (s,k,c',e')-immersion s i o n s agree, i f t h e c o r r e s p o n d i n g (s,k)-embeddings o f reduced graphs agree. The f o l l o w i n g examples h e l p e x p l a i n t h e above d e f i n i t i o n s . Example 2. F i g u r e 2 r e p r e s e n t s p l a n a r (1,0,3,2)-immersion of K , The reduced graph and i t s (l,O)-embedding i s d e p i c t e d o n , F i g u r e l ( a ) . Note 3 y 3 t h a t t h e immersions o f K on F i g u r e s 2 and l ( b 1 agree. 3,3 The f o l l o w i n g two examples were f i r s t used by Kainen [6,7] and Kainen and White[9]. Example 3. P a r t ( a ) o f F i g u r e 3 shows an (1,1,4,4)-immersion of K i n t o the sphere. I f t h e edges 1-6, 2-7, 3-8, and 4-5 a r e removed a ( 3 , 3 ) - e m f j d d i n g w h i c h i s o f course a l s o an (1,l)-embedding of t h e 3-cube graph Q i n t o t h e sphere r e 3 s u l t s ; see F i g u r e 3(b). P a r t ( c ) o f F i g u r e 3 r e p r e s e n t s t h e well-known genus i n t o t h e t o r u s which i s a (4,4)-embedding. embedding o f K 494

34 I

Cartesiati Products of Graphs 2

1

6

3

4

5 Figure 2

N o t e t h a t i m m e r s i o n s on F i g u r e s 3 ( a ) and 3 ( c ) a g r e e as (1,1,4,4)i m m e r s i o n s , as t h e y have f a c e s 1-2-3-4 a n d 5-6-7-8 i n comnon.

(C)

Figure 3

and (l,l,O,O)-

342

G. Pica, T. Pisanski and A.G.S. Ventre

Example 4. F i g u r e 4 ( a ) r e p r e s e n t s a (2,2,8m - 8,4)-immersion o f t h e C a r t e s i a n product C x C 4 i n t o t h e sphere f o r t h e case m = 3. P a r t s ( b ) and ( c ) o f F i g u r e 2m analogous t o p a r t s ( b ) and ( c ) o f F i g u r e 3. Namely, by removing 4 are a p p r o p r i a t e f o u r edges A-B, C-0, E-F, and G-H we o b t a i n a p l a n a r , r e s dual ( 3 , 3 ) embedding o f PPm x C4 (which i s o f course a l s o a ( 2 , 2 ) - and even ( 1 , 1 -embedding) as d e p i c t e d by Figure 4(b). F i n a l l y , Figure 4(c) represents the f a m i l i a r t o r o i d a l (4,4)-embedding o f C 2m x C4. Note t h a t ( a ) and ( c ) agree as 1,1,8m - 8,4) - and ( 1 , I ,O,O)- immersions (and n o t as (2,2,p,q)immersions f o r any P and 4). R e c e n t l y Beineke and Ringeisen have shown [ I ] t h a t c r ( C x C ) = 2m. T h i s means m 4 t h a t t h e immersion o f F i g u r e 4 ( a ) i s f a r from o p t i m a l .

Cartesian Products of Graphs

343

Figure 4 CONSTRUCTION OF IMMERSIONS When d e a l i n g w i t h c r o s s i n g numbers on s u r f a c e s t h e f o l l o w i n g two c o m b i n a t o r i a l i n v a r i a n t s a r e handy. d (G) = q k Jk(G) = q

-

g(p g(p

-

2(1 - k))/(g 2 + k)/(g

-

-

2)

2)

Here p denotes t h e number o f v e r t i c e s , q denotes t h e number o f edges, and g denot e s t h e g i r t h o f G.In b o t h cases k i s a n o n n e g a t i v e i n t e g e r r e p r e s e n t i n g i n t h e f i r s t case t h e ( o r i e n t a b l e ) genus and i n t h e second case t h e n o n o r i e n t a b l e genus o f some s u r f a c e . They a r e sometimes c a l l e d E u l e r d e f i c i e n c i e s as t h e y r e f e r t o t h e graph and t h e s u r f a c e , and o n l y t h e E u l e r c h a r a c t e r i s t i c o f t h e s u r f a c e i s i n v o l ved. They were i n t r o d u c e d b y Kainen. The o r i e n t a b l e v e r s i o n was i n t r o d u c e d i n [ 6 ] w h i l e t h e n o n o r i e n t a b l e one was d e f i n e d i n [ 8 ] and l a t e r used by Kainen and White [ 9 1 E u l e r d e f i c i e n c y t e l l s us t h e number o f s u p e r f l u o u s edges which o b s t r u c t t h e embedding o f a graph i n t o t h e s u r f a c e . The P o l l c w i n g lemma shows how E u l e r def i c i e n c i e s serve as l o w e r bounds f o r c r o s s i n g numbers c r ( G ) and Cr (G). k k Lemma 5.

.

crk(G)>dk(G)

and

Zk(G)>dk(G).

.

F o r p r o o f o f t h e o r i e n t a b l e case see [ 6 ] The n o n o r i e n t a b l e case i s e s s e n t i a l l y To o b t a i n an upper bound f o r t h e c r o s s i n g number we t h e same; see a l s o [8,9] need t h e f o l l o w i n g lemna.

.

Lemma 6, L e t G be a connected graph w i t h p v e r t i c e s and q edges. I f G a d m i t s a n o r i e n t a b l e (s,s,c,e)-immersion, t h e n c r (G),
3 44

G.Pica, T.Pisanski and A.G.S. Ventre

P r o o f . By removal of e edges we o b t a i n i n b o t h cases a q u a d r i l a t e r a l embedding o f t h e reduced subgraph H. T h e r e f o r e t h e s u b s c r i p t k ( h ) corresponds t o t h e genus ( n o n o r i e n t a b l e genus) o f t h e s u r f a c e i n which t h e immersion o f G and t h e embedding of H takes p l a c e . Since t h e r e a r e c c r o s s i n g p o i n t s t h e statement o f Lemma f o l l o w s . The f o l l o w i n g lemma w i l l h e l p us t o f i n d graphs w i t h (s,s,c,e)-immersions, namely t h e C a r t e s i a n p r o d u c t s o f o t h e r graphs. These w i l l be t h e graphs f o r w h i c h Lemmas 5 and 6 w i l l a p p l y y i e l d i n g l o w e r and upper bounds f o r t h e c r o s s i n g numbers. Lemma 7. L e t G and H be two connected graphs and 0 1 , s>,k)O two i n t e g e r s . L e t t h e r e e x i s t a p r o p e r edge c o l o r i n g o f H u s i n g a t most s c o l o r s such t h a t d o f t h e c o l o r s used determine d 1 - f a c t o r s o f H. L e t t h e r e be f o r each v e r t e x v o f H an (s,k,c e )-immersion o f G i n t o some s u r f a c e S(v). Furthermore, l e t a l l t h e i r m e r and E = x e Then t h e C a r t e s i a n p r o d u c t G x H s i o n s ” ’agree. D e f i n e C admits an (s+d, min(k+Zd ,s+d)y C,E)-immerxion i n t o some s u r f a c e T. (Here T i s o r i e n t a b l e i f and o n l y i f a l l s u r f a c e s S ( v ) a r e o r i e n t a b l e and H i s b i p a r t i t e . )

=xc

.

The p r o o f i s e s s e n t i a l l y t h a t o f Lemma 2.3 o f [II] and i s o m i t t e d . I n t u i t i v e l y , t h e c r o s s i n g p o i n t s do n o t i n t e r f e r e w i t h t h e argument used i n t h e p r o o f , as t h e y o c c u r o n l y i n t h e r e s i d u a l f a c e s and a l l embeddings o f G agree on a l l n o n r e s i d u a l faces. S t a r t i n g now w i t h graphs such as those o f Examples 1,2,3 and 4 and u s i n g repeatedl y Lemma 7 i t i s p o s s i b l e t o o b t a i n composite graphs t h a t s a t i s f y Lemma 6. I n some cases t h e upper bound o f Lemma 6 and t h e l o w e r bound o f Lemna 5 c o i n c i d e t o g i v e e x a c t c r o s s i n g number; see [9] f o r examples! CONSEQUENCES

I n t h i s s e c t i o n we a p p l y our lemmas t o o b t a i n bound f o r c r o s s i n g numbers and e x a c t c r o s s i n g numbers i n some s p e c i a l cases. i n t o a surface o f Theorem 8. I f a connected graph G admits an (s,s,c,c)-immersion genus k t h e n c r (GI = c if t h e s u r f a c e i s o r i e n t a b l e o r 5 (G) = c i f i t i s nonk h o r i e n t a b l e , where k= 1-p/Z+(q-e)/4 and h=Z-p+( q-e) /2. P r o o f . Combining Lemnas 5 and 6 we o b t a i n t h e i n e q u a l i t i e s e(cr (G)
Cartesian Products of Graphs

345

Note t h a t t h e r e s u l t o f Theorem 6 i n case o f G = K was o b t a i n e d a l r e a d y by K a i 1 nen and White [ 9 ] . Theorem 10. L e t G be an a r b i t r a r y connected b i p a r t i t e graph and l e t k),2 be an i n t e g e r . For each i n t e g e r n > d ( G ) - 1 and f o r each m such t h a t 0,<M2n t h e r e i s 4mGrg-m(C21. x Qnt2 where g i s t h e genus o f CZk x

x G)

<

8(k

-

l)m

,

x G.

P r o o f . S t a r t i n g w i t h immersions as i n F i g u r e 4(a,c) we c o u l d e a s i l y p r o v e b y i n d u c t i o n on n t h a t C 2k x Qn+2 a d m i t s an o r i e n t a b l e ( n t l ,n+l,8(k-1 )m,4m)-immersion p r o o f o f Theorem 9 we observe t h a t a l s o C and t h e n as i n t h e 2k 'n+2 a d m i t s an immersion o f t h e same type. By t h e o r i e n t a b l e p a r t of Lemmas 5 and 6 t h e r e s u l t f o l lows. Note t h a t b y t a k i n g k = 2 i n Theorem 10 we o b t a i n t h e i n e q u a l i t y : 4m\ A ( G ) and f o r each m such t h a t 0(m<2n t h e r e i s 2m(Eg-m ( K 3,3 x Qn x G)( 3m, where g denotes t h e n o n o r i e n t a b l e genus o f K3,3 x Qn x G. P r o o f . The p r o o f i s e s s e n t i a l l y t h e same as t h a t o f Theorem 9 t h a t i s by i n d u c t i o n on n. The b a s i s o f i n d u c t i o n i s p r o v i d e d by F i g u r e l ( b ) and F i g u r e 2. A s m
I ] L.W.Beineke and R.D.Ringeisen,

[ 21 [ 31

[ 41 [ 53 [6]

[ 71

On t h e c r o s s i n g numbers o f p r o d u c t s o f c y c l e s and graphs o f o r d e r f o u r , J.Graph Theory 4 (1980) 145-155. R.K.Guy, C r o s s i n g numbers o f graphs, i n "Graph Theory and A p p l i c a t i o n s " , L e c t . Notes i n Mathematics 303 (ed. Y.Alavi, e t a l . ) , S p r i n g e r - V e r l a g , B e r l i n , Heid e l b e r g , New York, 1972, 553-569. R.K.Guy and T.A.Jenkyns, The t o r o i d a l c r o s s i n g number o f K J.Combinatoria1 n ,my Theory 6 (1969) 235-250. M.Jungerman, The n o n - o r i e n t a b l e genus o f t h e n-cube, P a c i f i c J.Math. 76 (1978) 443-451. P.C.Kainen, Embeddings and o r i e n t a t i o n s o f graphs, i n " C o m b i n a t o r i a l S t r u c t u r e s and t h e i r A p p l i c a t i o n s " , Gordon and Breach, New York, 1970,193-196. P.C.Kainen, A l o w e r bound f o r c r o s s i n g numbers o f graphs w i t h a p p l i c a t i o n s t o and Q(d), J.Combinatoria1 Theory B 12 (1972) 287-298. Kn*Kp,q P.C.Kainen, On t h e s t a b l e c r o s s i n g numbers o f cubes, Proc. Amer.Math.Soc. 36 (1972) 55-62.

[8] P.C.Kainen, Some r e c e n t r e s u l t s i n t o p o l o g i c a l graph t h e o r y , i n "Graphs and Combinatorics", Lect.Notes i n Mathematics 406 (ed. R.A.Bari and F.Harary) ,

G.Pica, T. Pisanski and A.G.S. Ventre

346

Springer-Verlag, B e r l i n , Heidelberg, New York, 1973, 76-108. C 91 P.C. Kainen and A.T. White, On s t a b l e c r o s s i n g numbers, 3 . Graph Theory 2 (1978) 181-187. [ l o ] T. Pisanski, Genus of Cartesian products o f r e g u l a r b i p a r t i t e graphs, J. Graph Theory 4 (1980) 31-42. C111 T. Pisanski, Nonorientable genus o f Cartesian products o f r e g u l a r graphs, J Graph Theory 6 ( 1 982) 391 -402. C121 G. Pica, T. Pisanski and A.G.S. Ventre, The genera o f amalgamations o f cube graphs, Glasnik Mat. 19 (39) (1984) 21-26. C131 A.T. White, Graphs, groups and surfaces, North-Holland, Amsterdam, London, 1984.

.

t

Work performed under t h e auspices of CNR-GNSAGA.

t+

The author was p a r t i a l l y supported by 6. K i d r i E Fund, Slovenia, Yugoslavia.

Annals o f Discrete Mathematics 30 (1986) 347-354 @ Elsevier Science Publishers B.V. (North-Holland)

AND

OVOIDS

347

CAPS

Giuseppe

1. -

INTRODUCTION

Let

(S,YJ

Y a

family of

IN

T a l l i n i (Roma)

be a l i n e a r s p a c e , t h a t is a s e t parts

in

SPACES

PLANAR

whose elements we c a l l p o i n t s ,

S

, whose elements we c a l l l i n e s , such t h a t any

S

l i n e h a s a t l e a s t two p o i n t s and two d i s t i n c t p o i n t s are c o n t a i n e d i n j u s t one

(S,p) i s a s u b s e t

l i n e . A subspace i n x f y

S‘

in

, t h e l i n e j o i n i n g them belongs t o

S’

i n t e r s e c t i o n of s u b s p a c e s i n

.

x,yES‘ ,

such t h a t f o r any

S

Obviously t h e s e t t h e o r e t i c a l

i s a s u b s p a c e , s o t h a t t h e f a m i l y o f sub-

(S,yj

s p a c e s is a c l o s u r e system.

9

Suppose a f a m i l y every JZ E points

there

planar

o n l y one element

is

,

9

t h e elements of

of

.

.p

are called

. Straightforward

planes

examples n 23 ,

s p a c e s a r e : t h e a f f i n e o r p r o j e c t i v e s p a c e s o f dimension

n 2 3 , [HI

any s u b s e t

H

= PG(n,q)

, w i t h r e s p e c t t o t h e i n t e r s e c t i o n s of

H

l i n e s and p l a n e s having t h r e e independent p o i n t s of

9 ‘ of

t h e f o l l o w i n g : c o n s i d e r a whatever f a m i l y

, n

2 4

H

.

and

PG(n,q)

9 the

w i t h its s e c a n t

A f u r t h e r example i s

d-dimensional

each o f them n o t b e l o n g i n g t o any

f a m i l y of l i n e s i n

PG(n,q)

. The

PG(n,q) ,

in

, two by two meeting i n a t most one l i n e , l e t

family of planes i n S = PG(n,q)

is c a l l e d

(S,&?,p)

The t r i p l e

w i t h r e s p e c t t o t h e i r l i n e s and t h e i r p l a n e s ;

PG(n,q)

19 1 2 2 ,

e x i s t s such t h a t

(S,Y)

c o n t a i n s t h r e e independent p o i n t s and through t h r e e independent

planar space of

of subspaces i n

subspaces i n

9’’ be SdE

triple

the

9‘. Set

(S,y$%p?q’?

is a p l a n a r s p a c e . Let

(S,x9) be a p l a n a r s p a c e . We d e n o t e

t h r e e by t h r e e n o t c o l l i n e a r . We c a l l (1.1)

Given any p o i n t lines t o

51

PE

through

a

cap H

ovoid

in

in (S,L?,!?)

a s e t of points

(S,z%

a cap

9

such t h a t :

t h e s e t t h e o r e t i c a l union o f t h e t a n g e n t P

is a s u b s p a c e Z p

&I

(S,a

such t h a t

G. Tallini

348 every plane through

space In

the

Z

P

P

not i n

meets Z i n a l i n e . The P P is c a l l e d t a n g e n t s p a c e t o through P 5

a

f o l l o w i n g we

h a v e t h e same s i z e

suppose

k23

,

that

the

lines

.

the planar

of

and t h e p l a n e s h a v e t h e same s i z e

(S,Y,y)

space v

,

t h a t is:

(1.2)

(S,a

Then

p’denotes If

R

is a S t e i n e r s y s t e m

the s e t of lines in

,

S(2,k,V)

IS1 = V

JT , (JT,

and for any

is a S t e i n e r system

d e n o t e s t h e number o f l i n e s t h r o u g h a p o i n t o f

t h e number of l i n e s t h r o u g h a p o i n t o f

R

(1.3)

,

= (V-l)/(k-l)

, it

(JT,%)

r

€9, if

J-C

S(Z,k,v)

r

and

(S,Y)

,

is

is:

= (v-l)/(k-l)

,

moreover : (1.4) If

s

= V(V-l)/k(k-1)

lyd

,

= v(V-l)/k(k-l)

i s t h e number o f p l a n e s t h r o u g h a l i n e i n s = (V-k)/(v-k)

(1.5)

.

( S ? g q ) , we e a s i l y o b t a i n

(R-l)/(r-l)

=

C o u n t i n g i n two d i f f e r e n t ways t h e number of p a i r s c o n s i s t i n g of a p l a n e J-C and a l i n e t h r o u g h i t , w e h a v e :

/.!??I la

1 4s ,

=

t h a t i s (see ( 1 . 4 ) , ( 1 . 3 ) ,

(1.5)):

1 9 1

(1.6) If

= V(V-l)s/v(v-l)

= VHs/vr = V R ( R - l ) / v r ( r - l )

l.ypl i s t h e number o f p l a n e s t h r o u g h a p o i n t

P

,

.

c o u n t i n g i n two d i f f e r e n t

ways t h e number of p a i r s e a c h of them c o n s i s t i n g o f a p l a n e t h r o u g h l i n e on i t n o t t h r o u g h

P

,

l ~ p l ( [ y J - C=l - r1YI-R ) ,

we obtain:

P

and a

t h a t is (see

(1.4),(1.3),(1.5)):

Iqp1= ( V - l ) ( V - k ) / ( v - l ) ( v - k )

(1.7)

If

=

(s,Y’,~)i s a p l a n a r s p a c e s a t i s f y i n g

Theorem 1.

-

If

in a

= PG(3,q) odd, an o v o id i n

(S,y,.!??)

Rs/r = R(R-l)/r(r-l)

(1.2),

an ovoid

we p r o v e :

a

e x i s t s , i t is

is an e l l i p t i c quadric i f PG(3,q)

, if

q

.

is e v e n .

q = k-1

(S,y,a=

&

Ovoids and Caps in Planar Spaces

- Let

Theorem 2 .

be a

H

r

If -

(S,SP) ,

h-cap o f a p l a n a r s p a c e

, r

h<.R+l

(1.8)

349

r

is even, o r if

Then:

odd.

i s odd a n d

1n76Hl

# r+l

f o r any

E.9, we h a v e : (1.9)

,

hlR-s+l the equality holds i f f

i f f (S,xfl = Theorem 3.

-

n >1

i s a n o v o i d , t h a t i s ( s e e Theorem 1)

H

.

PG(3,q)

(S,Zy)

If i n a p l a n a r s p a c e

,

H

divides

r-m

e q u a l i t i e s holding i f f

(Sz,.q)

o f a prime i n

@

W e remark t h a t ,

PG(d,q)

r

if

, Moreover

s-1

n = q

is o d d , a

,

PG(d,q)

=

(0,n)

m = (r-l)/n

with respect to l i n e s e x i s t s , then

a n i n t e g e r and

of type

,

m u s t be

,

n
the

i s t h e cowplement

H

.

(R+l)-cap

in

i s a s e t of t y p e

(S,y,p)

(0,2), s o by Theorem 3 , we h a v e :

-

Theorem 4 .

If -

- OVOIDS

Let P

(1.8) e q u a l i t y h o l d s ,

Sz

at

(S,Y,*q) s

is

3t (7

either

through

@

and t h e c a p

.

(S,y,% s a t i s f y i n g

r-1

to , tl , tr

P # Q ) , s o it meets

l i n e s through

and s o exterior,

InnfiI

P =

or t a n g e n t ,

I

PQ):

[

I

( 1 . 2 ) . If

P,QE

does n o t belong t o t h e tangent

PQ

Q E 3t and

t h e number o f p l a n e s t h r o u g h

(2.1) Let

n

P (because

It follows t h a t t h e

a r e secant of

PG(d,2)

= 2

(S,6qty)

IN

, any p l a n e

to

f MCD(s-l,(r+l)/Z)

( S , y + V ) = PG(d,2)

then

be a n ovoid i n a p l a n a r space

# Q

rp

.

s-1

is t h e complement o f a p r i m e i n

H

2.

i n (1.8) t h e e q u a l i t y n e v e r h o l d s i f

is o d d ,

does not divide

(r+1)/2 in

r

=

in

.

r

JT

in a line P , d i f f e r e n t from C '

a,

space t

t

P

,

P '

I t follows t h a t every plane of

or

r-secant

to

fi

and

(being

s(r-2)+2 , i . e . by ( 1 . 5 ) :

= R-s+l

.

b e t h e numbers of e x t e r i o r , t a n g e n t a n d

r-secant planes t o

G. Tallini

350

s1 respectively. We have (see (1.6),(2.1)5(1.7)):

(2.2ll is obvious. (2.2) and (2.2) follow by computing in two different ways 2

3

a

the pairs consisting of a point of

fi

and a plane through it and two points of

and a plane through them. By (2.2) we have (see (1.5)):

(2.3)

=

(R-s+l)(R-s)s/r(r-l)

t

=

t 0

=

(R-s+l)s(s-l)/r(r-l) 2 (s/r)[(V/v)R + s - (R+l)s]

1

Being

t 20 0

,

2

vcs

V

By (1.3) it is

By (1.5) we have

(k-l)R+l

=

s-1

;

-

, R-s

(R-r)/(r-l)

=

2

,

(r-1) /(R-r)> 0

PG(n,q) n-1

I

,

n 23

,

= (rR-2Rtl)/(r-l)

= r-2

and

a

2

+ (r-1) /(R-r)

k > , r , so

by (2.6) it is

9 is a projective plane of order

a Galois space

.

whence we have

R-s):

k - 1 2 ER(r-2)+l]/(R-r)

(2.6)

+ VR2 0

s(R+1)]

v = (k-l)r+l , whence we obtain:

(see also (2.5) and dividing it by

in

.

by (2.3) we have: 3

(2.4)

Being

, ,

t

q = k-1

is a

k

=

r

,

and every plane JG

.

It follows that (S,zrzq.% is n-1 (q t1)-cap in PG(n,q) (being

n-2

= R-s+l =

qi + 1 = q

qi -

i=O

n-1

+ 1). It is known (see [5])

that in

i=O

n-1 PG(n,q) , n > 4 , ( q +l)-caps don't exist, whence 2 (q tl)-cap in PG(3,q) So Theorem 1 i s proved.

n

=

3

and

is a

.

3.

-

Let

CAPS

H

IN

be a

(S,z.q)

h-cap in a planar space

n E 9 , n = IJGAH(L2

,

$7 n H

is a

n-arc in

( S , y , p ) satisfying (1.21. Given

. We

easily prove that:

351

Ovoids and Caps in Planar Spaces n s r + 1 ,

(3.1)

H

.

P,Q E H , P # Q

Let

a

n = r+l

(3.2)

in at most

H

has no tangent lines

By (3.1) each of the and

h < R + 1

,

(3

.

odd

planes through

s

P

points different from

r-1

r

3

, whence h

=

meet

PQ

I t 1 1 1 (r-l)s+2,

that is (see (1.5)): (3.3) the equality holding iff:

=

~ 9I , JdnHI22

J 7 E

,

I n n H I = r+l

so that, by (3.2): h = R+l

(3.4) @

H

0

has no tangent lines

H

F)

is of type ( O , r + l ) with respect to planes

Let us now suppose:

we remark rha-c, by ( 3 . 2 ) , (3.5) is fulfilled, if r

P f Q , by (3.5) and (3.11, each of the at most

points different from

r-2

P

s

is even. F o r any

planes through

and

(3

h l R-s+l

,

, whence

h

meets

PQ =

P,QEH , H

in

IH( S ( r - 2 ) s + 2

,

that is (see (1.5)): (3.6)

the equality holds iff

u n ~ @ [nnH122 , a

lnnql = r

,

that is: (3.7)

h

=

R-s+l e H

If (3.5) holds and a

=

s

,

there are just s

through

P , not in

fi

tangents to

H

z p , in a line. So

is an ovoid, by ( Z . l ) ,

h So Theorem 2 is proved.

=

/ H I = R-s+l

R-stl

F)

through

Jd

PQ

,J 7 nH

is

P , A s the number of planes through PO at

tangents by (3.7) is a subspace z

of such

=

s

with respect to planes.

, for any plane

R-s+l

r-arc having only one tangent at

is

H

h

is of class [O,l,r]

H

P H

. The

P of

( S a

set theoretical union meeting every plane

is an ovoid. Conversely, if

and (3.5) holds (see sect. Z ) , i.e.:

is an ovoid

.

G. Tallini

352 4. -

SETS

Let

OF

(0,n)

TYPE

s a t i s f y i n g (1.2). Let

S'

r'

be

(lS'l-l)/(k-l)

,

P

e n t from

t h e number S'

If w e s e t

( S S

lines

through

through

P

meets

SinH

JslnHI =

rl(n-1)

S' = S

,

or

VJGEY,

(4.3) is of t y p e

1

=

, H

(O,r(n-l)+l)

1nnHI

,

r e d u c e s t o a p o i n t , so i t is n o t

that

n = k-1

if

prime

n E pis

every

points differ-

of

(S,Y)

, H

.A

,

1

,

r(n-l)tl

n = k

. Therefore:

.

is t h e complement o f a s u b s p a c e

meeting any l i n e n o t b e lo n g in g t o

subspace

If

.

1

lo

=

2Snlk-1

(S,Y)

.

S'

with respect t o planes.

(4.4) We remark

+

+

( H I = R(n-1)

H

n-1

in

and

S ' = J'C r e s p e c t i v e l y i n (4.11, w e h a v e :

(4.2)

whence

a point in

0

S'nH f

such t h a t

of

(S,y,,fl

whence:

(4.1)

n

(s,2',.9)

IN

LINES

be a subspace o f

P E S ' n H , every l i n e i n

If

TO

RESPECT

be a s e t of type ( 0 , n ) with r e s p e c t t o l i n e s i n a planar space

H

=

WITH

i n a point.

S'

(S,y,.!?)

planar space

S'

in

We d e n o t e s u c h a proper , i f

is c a l l e d

t h e l i n e a r c l o s u r e o f w h a t e v e r t r i p l e t of i t s i n d e p e n d e n t p o i n t s .

We p r o v e : Theorem 5. -

(S,y,.!?)

type

exists,

(O,k-l)l

then

i s t h e complement o f a p r i m e

(S,2?,9)

in

=

PG(d,q)

.

H

PG(d,q)

of

&

H

I t follows t h a t a

p r o p e r p l a n a r s p a c e is a Galois s p a c e i f f i t c o n t a i n s a p r i m e . Proof point. the

in

S' Let

line 3t

= S-H

is a s u b s p a c e o f

P,Q€S'

PQ

meets

is p r o j e c t i v e ,

,

T E H

and

being

i.e.

. The

plane

proper.

(S,2',9)

PQ = S'nn

meeting any l i n e n o t i n

(S,a

Each o f

i n a point, so

(S,y,Y, =

PG(d,q)

joining

J'C

,

r = k with

the

S'

in a

meets

S'

in

l i n e s through

T

P,Q,T

r

and e v e r y p l a n e i n q = k-1

.

(S,zfl

Thus t h e t h e o r e m

is p r o v e d . Let

n

b e a n i n t e g e r s a t i s f y i n g ( 4 . 4 ) and

(see (4.3))

/nnHI

= r(n-l)+l

.

If

3t

P € n -H

E 9 such t h a t

,

3t

n H # 0 , i.e.

t h e number o f l i n e s t h r o i i g h

P

353

Ovoids and Caps in Planar Spaces in 3c ,

n-secant of 3cnH

m = (r-l)/n

hence

must

number of planes through (4.2),(1.5) and being

therefore It is

r-m

is

be an integer. If

1

meeting

s-1

n < r - m I r-1

,

whence

k = r

In fact, if

r-m
space

PG(d,q)

is

is an exterior line of

,

H

and

whence the contradiction

n 1 2 ) we should k> r

n = r-1 , but by (4.41,

so that every plane is projective. Then and

H , the

v = /Hl/(r(n-l)+l) , that is (by

(being m = (r-l)/n

the equalities hold iff

,

H

e

.

n+l > r ; by (4.4), k 2 n + l

have

u = (3cnHI/n , so that:

r-1 = mn) we have:

divides

.

r-m>n

,

is a set of type

.

So it is

r>k2n+l

,

(S,Y,YJ is a Galois

(O,k-lIl , that is the complement of

a prime. Thus Theorem 3 is completely proved.

REFERENCES

r_l]

F. Buekenhout, Une caract6risation des espaces affins bas6e s u r la notion de droite, Math. Z. 111 (19691, 367-371.

[ Z ]

F. Buekenhout et R . Deherder, Espaces linCaires finis B u l l . SOC. Math. Belg. 23 (1971), 348-359.

[ 3 ]

M.Hall, Jr., Automorphisms of Steiner triple systems, IBM J. Res. Develop. 4 (1960), 460-472 ( = Amer. Math. SOC. Proc. Symp. Pure Math. 6 (1962),47-66).

[ 4 ]

H. Hanani, On the number of lines and planes determined by Technion. Israel Inst. Tech, Sci. Publ. 6 (1954/5), 58-63.

Is]

G. Tallini,

Sulle

Q plans isomorphes,

d

points,

k-calotte di uno spazio lineare finito, Ann. Mat. (4)

42 (1956), 119-164.

[6j

G. Tallini, La categoria degli spazi di i-ette, 1st. Mat. Univ. L'Aquila, (1979/80), 1-25.

[ 7 ]

L. Teirlinck, On linear spaces in which every plane is either projective or affine, Geometriae Dedicata, 4 (1975), 39-44.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 355-362 0 Elsevier Science Publishers B.V. (North-Holland)

355

(k,n;f)-ARCS AND CAPS I N FINITE PROJECTIVE SPACES B. J . Wilson

Chelsea C o l l e g e U n i v e r s i t y of London 552 K i n g ' s Road London SWlO OUA

I n 5 1 some theorems which g e n e r a l i s e r e s u l t s o f d ' A g o s t i n i are g i v e n . Some p a r t i c u l a r examples o f ( k , n ; f ) - a r c s a r e d i s c u s s e d i n 52 a n d t w o i n f i n i t e classes are d e s c r i b e d i n 13. I n 14 t h e e x t e n s i o n o f t h e r e s u l t s of 5 1 t o h i g h e r dimensions is n o t e d . E.

1 . I n t h e p a p e r s of B a r n a b e i [ 2 ] and d ' A g o s t i n i [ 4 ] [ 5 ] a n a c c o u n t h a s been g i v e n of some r e s u l t s c o n c e r n i n g w e i g h t e d ( k , n ) a r c s i n f i n i t e and p a r t i c u l a r l y G a l o i s p l a n e s . These o b j e c t s a r e a l s o c a l l e d ( k , n ; f ) - a r c s i n [4] and [ 5 ] . I n t h i s n o t e w e a r e concerned w i t h e x t e n d i n g t h e work i n [ 5 ] . However w e f i r s t mention t h a t t h e i d e a of a w e i g h t e d ( k , n ) - a r c was o r i g i n a l l y proposed by M. T a l l i n i - S c a f a t i [16] and t h a t t h e theme of t h a t p a p e r , namely t h e embedding of t h e a r c i n a n a l g e b r a i c c u r v e was c o n t i n u e d by Keedwell i n 191 and [ l o ] .

We s h a l l work w i t h t h e d e f i n i t i o n of a w e i g h t e d a r c g i v e n i n [ 5 ] acknowledging t h a t t h e d e f i n i t i o n g i v e n i n [ I 6 1 i s e q u i v a l e n t and h a s p r i o r i t y . Thus w e a r e c o n c e r n e d w i t h a s e t K of k > 0 p o i n t s i n P G ( 2 , q ) t o e a c h p o i n t P of which i s a s s i g n e d a n a t u r a l number f ( P ) c a l l e d i t s w e i g h t and such t h a t t h e t o t a l w e i g h t of t h e p o i n t s on any l i n e d o e s n o t exceed a g i v e n n a t u r a l number n , i . e . f o r e a c h l i n e R of PG(2,q) w e have C f (P) 5 n PER

A l i n e h a v i n g t o t a l w e i g h t i is c a l l e d a n i - s e c a n t o f K . Points n o t i n c l u d e d i n K are a s s i g n e d t h e w e i g h t z e r o . F o l l o w i n g t h e

n o t a t i o n of

[5] w e l e t

w = max f ( P ) PEK and u s e L. t o d e n o t e number of p o i n t s o f w e i g h t j f o r j 3

=

O,l,...,w.

F o l l o w i n g t h e n o t a t i o n and t e r m i n o l o g y o r i g i n a t i n g i n [ I 3 1 l e t t i d e n o t e t h e number of i - s e c a n t s of K f o r which i =O,l,...,n. W e call t h e t i t h e c h a r a c t e r s o f K a n d , i f e x a c t l y u of them a r e non-zero w e say t h a t the a r c K has u characters. I f t h e v a l u e s of i f o r which t i i s non-zero a r e m l < < n t h e n K i s s a i d t o b e of t y p e ( m ~ , r n 2 , . . . , n ) . I n [ 1 3 ] , [ 1 4 ] , [71 and s u b s e q u e n t l y i n l a t e r p a p e r s , one of which i s [ I S ] c o n s i d e r a b l e a t t e n t i o n h a s been g i v e n t o ( k , n ) a r c s h a v i n g e x a c t l y two c h a r a c t e r s . In p a r t i c u l a r t h e connection between such a r c s and H e r m i t i a n c u r v e s h a s been e x p l o r e d and g e n e r a l i s a t i o n s i n t o h i g h e r d i m e n s i o n s d i s c u s s e d i n [ 1 5 ] . A good bibliography i s included i n [16].

...

B.J. Wilson

350 I t i s proved i n [ 5 ] t h a t f o r a 0 < m < n , it i s n e c e s s a r y t h a t

( k , n ; f ) - a r c K of t y p e ( m , n ) , when

w S n - m

(i)

and q 5 0 mod (n - m)

.

(ii)

Let W = C f(P) = C f ( P ) , and t h e n i t may e a s i l y b e s e e n [51 PEK P E E (2,q)

that m(qt 1) 5 W

5 (n-w)q+n.

(iii)

A r c s f o r which e q u a l i t y h o l d s on t h e l e f t a r e c a l l e d minimal and a r c s f o r which e q u a l i t y h o l d s on t h e r i g h t a r e c a l l e d maximal. The case of m = n - 2 w a s d i s c u s s e d a t l e n g t h i n [ 5 ] . By (ii)w e must I n order t o have a n a r c have g = 2h and t h e n ( i )r e q u i r e s t h a t w 5 2 . which i s n o t s i m p l y a ( k , n ) - a r c w e t h u s must have w = 2 so t h a t (iii) gives

( n - 2) ( q +1 ) 5 W 2 (n- 2) ( q +1 ) + 2.

(iv)

W * ( n - 2 ) ( q + 1 ) + 1 and t h e o t h e r two p o s s i b l e v a l u e s of w a r e d i s c u s s e d i n [ 5 ] . Such a r c s have p o i n t s I t may e a s i l y be shown t h a t

having p o s s i b l e w e i g h t s 0 , l and 2; w i t h t h i s i n mind w e c a n s t a t e t h e f o l l o w i n g r e s u l t s which a r e g e n e r a l i s a t i o n s of theorems i n [ 5 ] and which c a n be proved by s i m i l a r methods. THEOREM 1.

Let K be a (k,n;fl-arc of type (m,n) w i t h n > m > 0 of minimal weight FI=m(q+ 1 ) having some p o i n t s of weight w = n - m , some p o i n t s of weight a for exactly one value of a s a t i s f y i n g both 1 2 u S w - 1 and ( w , d = I and a t l e a s t one point of weight 0. l a ) Suppose there i s exactly one p o i n t of weight 0 . Then a = w - 1 and the p o i n t s of weight w form a l w q + w - q - 1,wl-arc of which the ( w - 1)-secants are concurrent i n the s i n g l e points of weight 0. ( b ) Suppose that 1 , > 1 . Then K c o n s i s t s of a q / w c o l l i n e a r p o i n t s each of weight w and the q 2 p o i n t s not co2lincar w i t h them, each of weight a. Further, n=aq+w. THEOREM 2.

Let K be a (k,n;f)-arc of type h , n l w i t h n > m > 0 of maximal weight W = ( n - w ) l q + 1 ) + W having some p o i n t s of weight 0 , some p o i n t s of weight a. f o r exactly one value of a s a t i s f y i n g both 1 s a ~ w 1- and ( w , d = 1 and a t l e a s t one point of weight w . l a ) Suppose there i s exactly one point of weight w. Then a = 1 and the p o i n t s of weight 0 form a I w q + w - q - l,w)-arc of which the ( w - l ) - s e c a n t s are concurrent i n t h e single point of weight w. Suppose k w > l . Then K c o n s i s t s of a q / w + 1 c o l l i n e a r p o i n t s each of weight Ibl w and t h e q2 poznts not c o l t i n e a r w i t h them, each of weight a. Further n = a q + w .

-

I t h a s been shown [ l ] t h a t t h e e x i s t e n c e of a ( n o q +n o q - 1 , n o ) - a r c w i t h n , > 2 r e q u i r e s t h a t q - 0 (mod n o ) and t h a t i n t h a t c a s e t h e a r c p o s s e s s e s e x a c t l y q + 1 ( n o - 1 ) - s e c a n t s which a r e c o n c u r r e n t i n a p o i n t N c a l l e d t h e n u c l e u s of t h e a r c . The a d d i t i o n of t h e p o i n t N t o t h e a r c then gives a (n,,q+n,-q,n,)-arc, known as a maximal

357

( k *ri;f)-Arcs und Cups in Finite Projective Spaces

(k,n,)-arc. For even v a l u e s of q such a r c s have been c o n s t r u c t e d i n G a l o i s p l a n e s by Denniston 161. I t h a s a l s o been shown t h a t n o s u c h a r c s c a n b e c o n s t r u c t e d i n G a l o i s p l a n e s f o r n o = 3 by Cossu f o r q = 9 [3] and Thas [ 1 9 ] . Hence t h e theorems 1 and 2 are o f i n t e r e s t o n l y if n-m>3. 2 . I t f o l l o w s from t h e r e s u l t s of § 1 t h a t f o r t h e c a s e n - m = 3 w e need t o c o n s i d e r f u r t h e r i n a d i s c u s s i o n of maximal and minimal (k,n;f)-arcs the cases

and

R, > 0;

II, > 0; 1, > 0;

In particular we discuss (v).

(1,

(vi)

> 0.

I n t h a t case

( n - 3 ) ( q +1 ) 5 W 5 ( n - 2 ) q t n . F o l l o w i n g t h e n o t a t i o n of

(vii)

[ 5 ] w e u s e t h e symbol v!

t o denote the

number of i - s e c a n t s which p a s s t h r o u g h a p o i n t of w e i g h t j . t h e arguments i n [ 5 ] t h e v a l u e s of v:

and

vA-~,

j = 0,1,2

Using

are fixed

i n d e p e n d e n t l y of t h e p o i n t u n d e r c o n s i d e r a t i o n . F o r W minimal, i . e . W = ( n - 3 ) ( q + 1 ) w e have i n p a r t i c u l a r vo = q + l n-3

vo = 0 n (viii)

I t f o l l o w s i m m e d i a t e l y t h a t no p o i n t o f w e i g h t 0 l i e s on a n n-secant. W e now a t t e m p t t o c o n s t r u c t examples of ( k , n ; f ) - a r c s which s a t i s f y t h e s e c r i t e r i a , i . e . ( v ) , n - m = 3 , and W = ( n - 3 ) ( q + 1). Easy c o u n t i n g arguments g i v e

Solving ( i x ) gives

tn = ( n - 3 ) q / 3 tn-3

= (3q2+6q-nq+3)/3

Now l e t a b e a n - s e c a n t on which t h e r e are no p o i n t s of w e i g h t 0 so w e suppose t h a t on a a r e a p o i n t s o f w e i g h t 1 and 6 p o i n t s of w e i g h t 2. Counting p o i n t s o f u and w e i g h t s o f p o i n t s on a g i v e s a + B = q + l (xi) c c + B = n solving (xi) gives a = 2 ( q + 1) - n = n- (q+l)

(xii)

B.J . Wilson

358

C o u n t i n g i n c i d e n c e s between p o i n t s of w e i g h t 2 and n - s e c a n t s g i v e s

Hence, u s i n g ( v i i i ) , ( x ) and ( x i i ) w e have

II,

= (n-3)(n-q-

1)/2

(xiii)

S i m i l a r l y c o u n t i n g i n c i d e n c e s between p o i n t s of w e i g h t 1 and n-secants gives R,vk =

tnCl

whence, u s i n g ( v i i i ) , ( x ) and ( x i i ) w e have

L1

=

( n - 3) ( 2 q + 2 - n )

(xiv)

From ( x i i i ) and ( x i v ) , c o u n t i n g t h e p o i n t s i n t h e p l a n e w e o b t a i n 2q2+ (11-3n)q+n2-6n+11-2Ro= 0 It i s t h u s n e c e s s a r y t h a t (n-q),

-

(xv)

(48-16R0)

s h o u l d be a s q u a r e . W h i l s t it i s n o t t h e o n l y c a s e f o r i n v e s t i g a t i o n a n o b v i o u s v a l u e t o t r y i s L o = 3 . The r e s u l t i n g s o l u t i o n s f o r ( x v ) a r e t h e n n = 2 q + 1 and n = q + 5. By c o u n t i n g t h e p o i n t s of a n ( n - 3 ) - s e c a n t c o n t a i n i n g t h r e e c o l l i n e a r p o i n t s of w e i g h t 0 i t may e a s i l y be s e e n t h a t f o r n = 2 q + 1 i t i s i m p o s s i b l e f o r t h e t h r e e p o i n t s of w e i g h t 0 t o be c o l l i n e a r . A s i m p l e example may be found i n P G ( 2 , 3 ) . Assign t h e weight 0 t o and ( O , O , I ) , t h e w e i g h t 1 t o t h e p o i n t s t h e p o i n t s ~1,0,0),(0,1,0) ~ 2 , 1 , 1 ~ , ~ 1 , 2 , 1 ) , ( 1 , 1 , and 2 ) ( l , l , l ) and t h e w e i g h t 2 t o a l l o t h e r points. T h i s y i e l d s a ( 1 0 , 7 ; f ) - a r c of t y p e (4,7). For t h i s c a s e of R, = 3 w i t h n = 2 q + 1 w e may o b t a i n from ( x )

Thus by ( x i i i ) t h e n - s e c a n t s form t h e d u a l of a

The e x i s t e n c e of such a n a r c would, i f q > 3, v i o l a t e t h e L u n e l l i - S c e c o n j e c t u r e [ I l l t h a t f o r a ( k , n , ) - a r c w i t h q i 0 (mod n ) it i s n e c e s s a r y t h a t k 5 ( n o l ) q + 1 . However i t was shown by H i l l and Mason [ 8 ] t h a t c o u n t e r e x a m p l e s t o t h i s c o n j e c t u r e c a n be found f o r a n i n f i n i t e number of v a l u e s of q .

-

We now c o n s i d e r t h e c a s e i n which ( v ) h o l d s , w i t h t o = 3 , W minimal and n = q + 5 . I f t h e t h r e e p o i n t s of w e i g h t 0 a r e c o l l i n e a r t h e n i t may be shown t h a t t h e p o i n t s of w e i g h t 2 form a c o m p l e t e ( 2 q + 3 , 4 ) a r c of t h e t y p e ( 1 , 2 , 4 ) . Examples of such a c o n f i g u r a t i o n have been found i n P G ( 2 , 3 ' ) . I f w e d e f i n e GF(3') by t h e r e l a t i o n a'= 2 a + 1 o v e r GF(3) t h e n one such example i s c o n s t r u c t e d as f o l l o w s :

359

Ik,n;fl-Arcs and Cops in Finite Projective Spaces The t h r e e p o i n t s ( 0 , l , a 3 ) ,( 0 , l , a 6 ) I ( 0 , l , a 7 ) h a v e w e i g h t 0 and t h e t w e n t y two p o i n t s

Q, ( 0 , 0 , 1 ) Q l ( O t 1iO) Q, ( 0 , 1 ,a:)

Q3(0ilra 1 (1, a l l 1 ( 1 ,a,a)7 (l,a,a 1

(1 , a 2, a 3 ) ( 1 ,a;,a;) (1,a la 1 (1 ( 1 ,a 3 , a 4 1 (1,a3

(1,a5,a1

(1,aSIa:) ( ~ , a ~ ~ a ( 1 , a 6 ,1 ) ( 1 , a 6 ,a) ( 1 ,a67,a7) (1,a r l ) ( 1 ,a7,a) (I ,a7,a7)

have w e i g h t 2 . The r e m a i n i n g p o i n t s of P G ( 2 , 3 ) a r e a s s i g n e d w e i g h t 1 g i v i n g a n ( 8 8 , 1 4 ; f ) - a r c K O of t y p e ( 1 1 , 1 4 ) w i t h t h e p o i n t s of w e i g h t 2 f o r m i n g a c o m p l e t e ( 2 2 , 4 ) - a r c of t y p e ( 1 , 2 , 4 ) . L e t v be t h e l i n e , x , = O , of c o l l i n e a r i t y of t h e t h r e e p o i n t s of w e i g h t 0 . On v a r e f o u r p o i n t s R,,R,,R,,R, of weight 1 and f o u r p o i n t s Qa,QlrQ2,Q3, w i t h c o o r d i n a t e s a s i n d i c a t e d a b o v e , of w e i g h t 2. F u r t h e r , t h e r e a r e p r e c i s e l y t h r e e p o i n t s P 1 , P 2 , P J of w e i g h t 1 which d o n o t l i e on v and which a r e j o i n e d t o t h e p o i n t s R i o n l y by 14-secants. T h e i r l i n e of c o l l i n e a r i t y p a s s e s t h r o u g h Q , and t h e n i n e l i n e s PiQ, a r e 1 1 - s e c a n t s meeting i n t h r e e s a t t h e s i x p o i n t s P l , P 2 , P 3 , Q l , Q 2 , Q 3 and o t h e r w i s e o n l y i n p a i r s . This configuration i s , i n P G ( 2 , 3 2 ) , d e t e r m i n e d c o m p l e t e l y by f i v e of t h e p o i n t s *.-

rQ,.

3. The ( 1 O l 7 ; f ) - a r c of t y p e ( 4 , 7 ) i n PG(2,3’) which w a s c o n s t r u c t e d i n 5 2 i s a p a r t i c u l a r c a s e of two o t h e r w i s e d i s t i n c t i n f i n i t e c l a s s e s . F i r s t l y w e n o t e t h a t t h e p o i n t s of w e i g h t 1 form a $-arc, t h i s b e i n g t h e i r r e d u c i b l e c o n i c

xi

t

x: + x ;

= 0.

I n P G ( 2 , q ) , w i t h q odd, l e t C be an i r r e d u c i b l e c o n i c . There a r e e x a c t l y q t 1 p o i n t s on C a t e a c h of which t h e r e i s a u n i q u e t a n g e n t l i n e t o C. The p o i n t s of t h e p l a n e which a r e n o t on C may t h e n b e p a r t i t i o n e d i n t o t h e d i s j o i n t classes of q ( q + 1 ) / 2 e x t e r i o r p o i n t s , t h r o u g h e a c h o f which p a s s e x a c t l y t w o t a n g e n t s t o C and q ( q 2 ) / 2 i n t e r i o r p o i n t s , t h r o u g h e a c h of which t h e r e a r e n o t t a n g e n t s t o C . A s s i g n i n g w e i g h t 0 t o e a c h i n t e r i o r p o i n t , w e i g h t 1 t o e a c h p o i n t of C and w e i g h t 2 t o e a c h e x t e r i o r p o i n t g i v e s a minimal ( ( q 2 + 39 t 2 ) / 2 , 2 q ; f ) - a r c of t y p e ( q + 1 , 2 q + 1 ) . By a s s i g n i n g d i f f e r e n t w e i g h t s t o t h e p o i n t s of t h i s c o n f i g u r a t i o n o t h e r ( k , n ; f ) - a r c s may b e o b t a i n e d . F o r example a s s i g n i n g t h e w e i g h t 0 t o e a c h i n t e r i o r p o i n t o f C , t h e w e i g h t 1 t o e a c h e x t e r i o r p o i n t of C and t h e w e i g h t ( q + 1 ) / 2 t o e a c h p o i n t of c a maximal ( ( q 2 + 3 q t 2 ) / 2 , ( 3 q + 1 ) / 2 ; f ) - a r c of t y p e (q t 1,2q + 1 ) is obtained.

-

A s e c o n d i n f i n i t e c l a s s of

( k , n ; f ) - a r c s i s s u g g e s t e d by r e g a r d i n g t h e t r i a n g l e of p o i n t s o f w e i g h t 0 i n t h e ( 1 0 , 7 ; f ) - a r c of 12 a s a s u b p l a n e . G e n e r a l l y l e t n o b e a s u b p l a n e of o r d e r q o of a ( n o t n e c e s s a r i l y G a l o i s ) f i n i t e p r o j e c t i v e p l a n e n of o r d e r q w i t h q > q ; + q , . I n t h i s c a s e t h e r e a r e some l i n e s of n which d o n o t c o n t a i n a n y p o i n t of n o . A s s i g n w e i g h t 0 t o p o i n t s of n o , w e i g h t u t o p o i n t s of TT which a r e n o t on l i n e s of n o and w e i g h t v t o t h e r e m a i n i n g p o i n t s where u / v = ( q - q:) / ( q q , q i ) i n i t s l o w e s t terms Then t h e r e i s formed a minimal ( ( q- 9 0 1 ( g + 9, + 1 ) I + 9,+ l ) u + ( q - q i - q , ) v ; f ) - a r c

-

-

(4

of t y p e ( ( q + q o ) u , ( q : + q o l+) u + ( q - q i - q , ) v ) .

B.J. Wilson

3 60

As in the case of the previous example reassignment of other weights to the sets of points involved leads to further (k,n;f)arcs. 4 . The definition of a (k,n;f)-arc given in § I may be extended to that of a (k,n;f)-cap [5] by substituting PG(r,q) for PG(2,q) with r > 2 . In [ 5 ] it was shown that (k,n;f)-caps of type (n- 2,n) , with r 2 3 do not exist. This proof required results listed by Segre [12] p 166 concerning the non-existence of certain k-caps in PG(r,q) with r 2 3.

If we use the notation Qr to denote the number of points in PG(r,q) then the results in [I21 showed that the number of points on a k-cap cannot be Qr-l. For a (k,n;f)-cap of type ( m - n) with O < m < n the minimal weight is mQy-l. However it may be shown using analogous arguments to those indicated above that a (k,n;f)-cap of minimal weight mQr-l and otherwise satisfying the conditions of theorem 1 cannot exist. A similar result can be obtained for maximal arcs. REFERENCES Barlotti, A., Su {k;n}-archi di un piano lineare finito, Boll. Un. Mat. Ital. 1 1 (1956) 553-556. Barnabei, M., On arcs with weighted points, Journal of Statistical Planning and Inference, 3 (19791, 279-286. Cossu, A., Su alcune proprieta dei {k;n}-archi di un piano proiettivo sopra un corpo finito, Rend. Mat. e Appl. 20 ( 1 9 6 1 ) , 271-277. d'Agostini, E., Alcune osservazioni sui (k,n;f)-archidi un piano finito, Atti dell' Accademia della Scienze di Bologna, Rendiconti, Serie XIII, 6 (19791, 211-218. d'Agostini, E., Sulla caratterizzazione delle (k,n;f)-calotte di tipo (n-2,n), Atti Sem. Mat. Fis. Univ. Modena, XXIX, (1980), 263-275. Denniston, R.H.F., Some maximal arcs in finite projective planes, J. Combinatorial Theory 6 (1969), 317-319. Halder, H.R., h e r Kurven vom Typ (m;n) und Beispiele total m-regularer (k,n)-Kurven, J. Geometry 8, (19761, 163-170. Hill, R. and Mason, J., On (k,n)-arcs and the falsity of the Lunelli-Sce Conjecture, London Math. Soc. Lecture Note Series 49 (1981), 153-169. Keedwell, A.D., When is a (k,n)-arc of PG(2,q) embeddable in a unique algebraic plane curve of order n?, Rend. Mat. (Roma) Serie VI, 12 (19791,397-410. [lo] Keedwell, A.D., Comment on "When is a (k,n)-arc of PG(2 embeddable in a unique algebraic plane curve of order n?1 : ) , Rend. Mat. (Roma) Serie VII, 2 (19821, 371-376.

36 1

( k , n;fl-Arcs and Caps in Finite Projective Spaces L u n e l l i , L. a n d S c e , M . , Considerazione arithmetiche e v i s u l t a t i s p e r i m e n t a l i s u i { K ; n l q - a r c h i , 1st. Lombard0 Accad. S c i . Rend. A 98 (1964), 3-52.

S e g r e , B . , I n t r o d u c t i o n t o G a l o i s Geometries, A t t i . Accad. Naz. L i n c e i Mem. 8 (1967), 133-236. T a l l i n i S c a f a t i , M . , { k , n } - a r c h i d i un p i a n o g r a f i c o f i n i t o c o n p a r t i c o l a r e r i g u a r d o a q u e l l i c o n due c a r a t t e r i (Nota I ) , A t t i . Accad. Naz. L i n c e i Rend. 40 (1966), 812-818. T a l l i n i S c a f a t i , M . , { k , n ) - a r c h i d i un p i a n o g r a f i c o f i n i t o c o n p a r t i c o l a r e r i g u a r d o a q u e l l i c o n due c a r a t t e r i (Nota 111, A t t i . Accad. Naz. L i n c e i Rend. 4 0 (19661, 1020-1025. T a l l i n i S c a f a t i , M . , C a t t e r i z z a z i o n e g r a f i c a d e l l e forme Rend. Mat. e Appl. 26 (19671, 273-303. h e r m i t i a n e d i un S r , q . T a l l i n i S c a f a t i , M . , G r a p h i c C u r v e s on a Galois p l a n e , A t t i d e l convegno d i Geometria C o m b i n a t o r i a e s u e A p p l i c a z i o n i P e r u g i a

(1971), 413-419.

T a l l i n i S c a f a t i , M., k - i n s i e m i d i t i p 0 (m,n) d i uno s p a z i o a f f i n e A r l q , Rend. M a t . ( R o m a ) S e r i e V I I , 1 (1981), 63-80. T a l l i n i S c a f a t i , M., d-Dimensional t w o - c h a r a c t e r k - s e t s a f f i n e s p a c e A G ( r , q ) , J . Geometry 22 (19841, 75-82.

-

i n an

T h a s , J.A. , Some r e s u l t s c o n c e r n i n g ( q + 1 ) (n-1) 1 , n ) - a r c s and { ( q + 1 ) ( n - 1 ) + l , n } - a r c s i n f i n i t e p r o j e c t i v e p l a n e s of o r d e r q, J . C o m b i n a t o r i a l Theory A 19 (19751, 228-232.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 363-372 0 Elsevier Science Publishers B.V. (North-Holland)

363

N. Zagaglia Salvi Diparthnto di Matematica Politecnico di Milano, Milano, Italy

Let C be a circulant (0,l)-matrix and let us arrange the elements of the first row of C regularly on a circle. If there exists a diameter of the circle with respect to which 1 ' s are synanetric, we call C reflective. In this papr we prove some properties of the reflective circulant ( 0 , l ) -matrices and of certain corresponding cam binatorial structures.

INIXOWrnION

A matrix C of order n is called circulant if C P = P C, where P represents the permutation ( 1 2 n 1.

.. .

Let C be a circulant (Ofl)-mtrixand let us arrange the e l m t s of the first row regularly on a circle, so that they are on the vertices of a regular polygon. If there exists a diameter of the circle with respect to which 1's are symnetric, we call C reflective. In this paper we prove some properties of the reflective circulant (O,l)-mtricesand of certain corresponding carbinatorial structures. In particular, % is proved that a circulant (O,l)-mtrixC of order n satisfies the equation C P = CT, 0s h 2 n-1, if and only if it is reflective. Moreover we determine the number of such C for every h. It is proved in certain cases the conjecture of the non-existence of circulant Hadamard matrices and, therefore, of the non-existence of certain Barker sequences. We also give a sufficient condition that the autcmrphism group of a directed graph is C the cyclic group of order n. n' Finally we determine a characterization for the tournaments with reflective circulant adjacency matrix. For the notations, I and J denote, as usual, the unit and all-one matrices: the matrix C denote the transpose of C. T

... , cn-13

I. L e t c be a circulant matrix. If [co, cl, it follows [2] that the eigenvalues of c are n-1 x = c c,bJjr

r

j=o

is the first nm of C,

(1)

3 2ai where 0 5 r 5 n-1 and w = exp( 1.

n

Consider the circulant matrix A

=

C P. The first row of A is obtained frcm the

N . Zagaglia Salvi

364

first row of C by shifting it cyclically one position to the right. n-1 where 06 r
Jrr

.

PFOKSITtCN 1.1 - !he eigenvalues Ar and vr , 0 5 r 5 n-I , of the cir ant mah "3ir trices C and CP , 0 2 h 5 n-I , as in (1), satisfy the relations v =A (w ) r r

.

- The eigenvalues A trices c and D, as in (11, satisfy

PROKSITION 1.2

and p

0 5

r 5 n-I , of

the circulans ma-

r' h e relations p =A if and d r r

y if D =

cT.

...

2 Roof.If A= diag( 1, w, w , , wn-I 1, where w=exp(-),2ri then there exists an n unitary mtrh U such that P = U A U*. If [c0, c , c.n-1 1 and [do, dl, d:n 11 are the first rows of C and D, n-1 we have C = .C c.P3 = U r Ux where r= ;i!,cjA7 and D = U( nf'd,AJ)U*.

...,

'

...,

3=0 3

7=0 7

n-I ' If p equals 'i for all r, it f o l l m jgodjA7 = TIE L e r s e i5 easily proved. Q DEFINITICN 1 . 3

exp (

I- n

- Let wr, up two nth roots of unity , 0 5

. If d

2~i

7 and hence D =

=

I r-pI , we

T'

r,p 2 n-I, and w =

say that such roots are at distance d and we write

dist( wr, wP)=d. Clearly, then, dist( Gr, Gp)=d.

-

h THMlREM 1.4 A circulant (0,l)-mtrix C satisfies the equation CP =C T' 0 5 h 5 n-I, if and only if it is reflective. proof. k t C be a reflective circulant (O,l)-matrix of order n; let A r' 0 I r 5 n-I, be the eigenvalues of C as in (1). p2 PS Suppose that A l = w + w +...+a , 0 5 p < p 2 <
...

...,

P,+l. Pi Since C is reflective, it follms that also for the s nth roots of unity w , 1 5 i I s, there exists a dimter 2 of the mit circle with respect to which such m t s are symnetric. a a 1 2 If 2 contains at most o m of the above roots, consider an ordering H = I w ,u , a , w of such roots obtained by traversing the unit circle counterclockwise, so that the diameter 2 is encountered only once. In case a does not contain any o! ci"i ' "i+l s-(i-l) s-i )=dist( w roots, s is even and dist( w , w ,w ), I ~ i s 5 - 1 . 2

...

5%

In case 2 containrone root, s is odd and that r w t is w a ai a i+1)=&st( w's-(i-I) s-i s-1 ?hen dist( w , w , w ),ISis--. 2

If g contains two roots, we ignore one of these, then we proceed as before. a

So the two roots on g are wax

and w

.

365

Cornbinatorial Structures

-

ci

...,

a

-011 2 ,s It is easy to see that the roots w , w , w are syrmmetric for the diameter b, where b is the reflection of g in the real axis.

There are two cases:

a

i 1) the diameter a contains at m s t one of the roots w , lSiSs. a +h a s -1 k t h be the minirmnn positive integer such that w = w

.

cis "s-1 the symnetry of the roots with respect to g,we obtain: dist( w , w )= a a a a a +h -a2 a +h s- 1 s-2 dist( w w 'I= dist( .1' It follows that w = w , 0 a al+h a - 3 S h hrw , ..., w = ij andhence hlw = h Thus A ( w ) =A ISrk-1. 1r r t h Consider the matrix D = 8 Let p- be the eigenvalues of D, as in ( 1 ) . By Prop. By

',

.

-

whir

the a v e statements, pr = Ar. ( so thai, 1.1, we have p = r r By Prop. 1.2 D=C a. T' i 2) The diameter 5 contains two of the roots w , l
.

-

I

e

h

Suppose, m, that the circulant (O,l)-matrix C satisfies the equation 8 =CT, 06hSn-1; we prove that C is reflective. h If [co, cl, c is the first rcw of C, then the first rows of 8 and

...,

3

n-1 cT are respectively [c-h, cl-h, c J and [cot cn-l, ...,cJ n-lh F r m the relation C P = C we obtain the equalities c = c where O5i5n-1 TI i-h n-it and the indices are mod n, so that C is ref1ective.a

.

...,

Recall that, if A = [a.

,I and B are two matrices of order m and n respectively,

13

then the direct product of A with B is a matrix of order rm defined by ,-

a B a B A X B =

21

...

22

...

Let C be a reflective circulant (0,l)-matrix. If there are two diameters of the circle with respect to wich the 1's are symnetric, then there are t m integers k and k kl k such that C P = C , i = 1,2. h 2' 2' I& C satisfies such equations and 1% <;f
-A

h circulant (O,l)-matrix C of order n satisfies C P

=

C, IhSn-I,

N . Zagaglia Salvi

366

i f and only i f it i s C = 0 or C = J f o r (h,n)=l and C = J x D f o r ( h , n ) = t > 1 , q n where q = - and D is circulant (0,l)-matrix of order t. t h Proof. The f i r s t rrm of 8 is obtained f r a n the f i r s t row of C by s h i f t i n g it cyclically h p x i t i o n s to the right. h is the f i r s t row of C, f r a n 8 =C we obtain If [cl, C2' . . . I CJ

where the indices are mod n. W e have t w cases to consider:

a) (h,n)=l b) ( h , n ) > l .

...,

Case a ) . I f ( h , n ) = l , then the n e n t r i e s of H = { I , l+h, l t ( n - l ) h } are a l l 1 d i s t i n c t and H is equal to the set {1,2,. ,n} 1 Hence in the sequence (2) there are a l l the entries of the f i r s t row of C and it is or C = 0 or C = J.

.. .

Case b). Let (h,n)=t>l w i t h n=tq. I n HI only q elements of t h e set K

..., I + (q-I)hl

are d i s t i n c t . It i s easy t o prove that a l s o the elements of the set K =I i , i + h , . i i j. i 6 [l,t] are d i s t i n c t and, mreover, K . is d i s j o i n t fran K

1

={ l , l + h ,

.., i+(q-1) h l ,

f

1 1' c repeated q times determines the f i r s t I t follows that the sequence c c2 t row of c. hen, i f D i s the c i r c u l a n t matrix whose f i r s t r m is [cl c2 ct] , we have

...

...

C = J

xD.

q Conversely, i f it is

c

= J

q

x

D, then it is easy t o prove that

ah=

C.

II

Remark t h a t , i f a circulant (0,l)-matrix C of order n=hq s a t i s f i e s the r e l a t i o n c Ph = C, then there are also s a t i s f i e d the relations c ph" = C, s=1,2,. ,q. k let C be a circulant (0,l)-matrix such that C P = CT, k€[O,n-l].

..

r

r

I f k is even, say k=2r, then we obtain CP = ( CP

IT;

r

denoted CP =E, it follaws

E = E

T'

If k i s odd, k=2q+l, 920, then we obtain (CPq)P=(mq)Trwhere, denoted CPq=E,

ET .

it i s EP=

'IIIEORFM 1.6- A n-square c i r c u l a n t (0,l)-matrix C = J

x

D, where D has order t

and n=tqr is r e f l e c t i v e i f and only i f D is reflectiye. Proof. Let C = J X D, of order n=tq, t , q > l , be a circulant (O,l)-IMTx; then D is ( 0,1) - c i r c u l d t , of order t, and by Theorem 1.5 it is s a t i s f i e d CP =C. If C is reflective, there exists an integer r such that, denoted d = E , we have E = %or E P =

ET.

L

Since it is El?= = E, by Theorem 1.5, there exists a circulant matrix F of order t such t h a t E = J x F. 9

Combinatorial Structures

367

.-

It is easy to prove that E = E iff F = F and E P = E iff F P = F where? T T T TI represents the permutation ( 1 2 t). r fir Since C P = E, then it is D P = F and C is reflective iff D is ref1ective.a

...

DEFINITICN 1.7 - A set D = { d if the circulant is reflective. d,, d2,

of k integers mod n is reflective to w h i c h the 1's are in the positions

...,%,

- If the set of integers mod n D = {dlld then it is reflective the set t~ = {tdl, td2, ...,tc$

THMlRFM 1.8

, ..., $2 is reflective, , for every t prime to n.

Proof. Let D = {d } be a reflective set and let C be the circulant (0,l)-matrix with ch the 1 ' s are in the positions If we arrange the e l m t s of the first rcw of C rqlarly on a exists a diameter g of the circle with respect to which the 1's are symnetric. L e t H = { a , a2, . ., ak } an ordering of the elements of D obtained by traversing the unit circle counterclockwise, so that the diameter 2 is encountered only once. Let ~1 be the last element of H before traversing a ( a can belong to a). r r Let c1 ' the syrrpnetric of a. as regards t o g , iC[l,r]. i

,

.

Consider the set tH ={ k l l ta2,

..., tak 1 .

Because t is prime to n, the ele-

ments of t€l are distinct and tcci f toi' (mod n) I but perhaps for i=r. ~ r e m e r as , we have a - a , =a ' ki' kr' , where td dif#e&ces

-

-a '

, then tar - h.= t(a -cr.)=t(ai' -a ' I = r i r &e mod n and i C [ltr-11.

Then tcci', l$isr, is synanetric of tcr. as regards to the diameter that reduces to half the distance between tclr and to:'; hence tD is reflective.0

2. Let C be a reflective circulant (0,l)-mtrix. is the first rcw of C, fran the relation ah= If [col c1, ...,c n-1 hc [O,n-d , by the above remarks we obtain

1

cTI

c = c i-h n-i

(3)

where Oliln-I and the indices are rrcd n. The relation (3) is an equality between two different elements except for i-h :n-i ( mod n ) ; in that case there exists an integer k 5 0 such that 2i = h + kn.

(4)

Let n and h odd, n=2t+l, h=2r+l and r't; then we have 2i=2r+l+k(2t+l). For k=l we obtain i=r+t+l and clearly this is the d y possible value.

If n is d d and h even, n=2t+l and h=2r, we have 2i=2r+k(2t+l)and i=r is the only possible value. If n is even and h odd, n=2t and h=2r+l,we have 2i=2r+l+k(2t);this equation is clearly impossible.

N.Zagaglia Salvi

368

For n and h even, n=2t and h=2r, the equation 2i=2r+k(2t)has two solutions: i=r and i=r+t, corresponding to k=O and k=l. THEXlFEM 2.1

- Let h,n be two integers , OSh5n-1. When n is odd, there are

h P = , C, while , when n n+A for h wen. such matrices for h cdd and 2

circulant (0,l)-matrices C that satisfy the equation

?

is even, there are

n%

2

C

Proof. Let n be odd. By the above remarks, from (3) we obtain and one identity. So in the first row of C exactly n-1 + 1 = 2 arbitray. As every e l m t can be 0 or 1, there are 2

n-1

- equalities n+? elements are 2

"+Acirculant ( 0 , l ) -matrices that

h

satisfy the equation C P = CT, for n odd. n n-2 If n is even, the relations (3) correspond to 5 equalities for h odd and to 2 equalities and two identities for h even.

n+32such matrices

% distinct mtrices when h

is odd and 2

Hence, there are 2 when h is even.0

3. Recall that a difference set D = {d1 l d2,

...,$I

is a k-set of integers

rnod v such that every a f 0

(mod v) can be exspressed in exactly X ways in the form d d. E a (mOa v) , where d. and d . are in D. We suppose O<X
-

PROPOSITICN 3.1

- If C is a circulant incidence matrix of a (v,k,A)-configura-

h tion, it never satisfies the equation 8 = CT. Praf.If row of

[corcl,

cT is

...,cn-1]

is the first r m of a circulant matrix C, the first

..., cl]

, i.e.

[cofcn-l,

c. has been replaced by c - ~ ,where the

indices are mod n. p be the positions of the elements equal to 1 on the first row k t p l , p2,

...,

S

of C. If C is an incidence matrix of a (v,k,X)-canfiguration,then fp , p ,...,psi 1 2 is a difference set. Since -1 is not a multiplier for a difference set [37 , then there exists no such that { p,+h, p2+h, ...,pS+h) = integer hL [O,n-I] -pl r -p2, -ps) r

...,

i.e.

c d ~ e snot satisfy c ph

=

cT.a

Recall that a (v,k,X)-mtrix is a (O,l)-mtriY A of order v which satisfies A%=

(k-X)I+XJ

where 0 < X < k < v-1. It follows that detA

v- 1 =

k(k-X

)

2

.

Combinatorial Structures

369

matrix of + I ' s and -1's such that H H p I . It has been conjectured [6] that a Hadarnard mtrix cannot be circulant for v > 4 . Remrk that we can define the notion of reflective circulant Hadmard matrices, analagous to the corresponding notion for the (0,l)-matrices.

A Hadamird matrix H of order v is a

PR(IPOS1TION 3.2- A Hadarmrd matrix of order v cannot be reflective circulant o r coincident block circulant , for v > 4.

Proof. If H is a --aiiective c i r c u l m t o r coiiiiident block circulant Hadamrd f i s a reflective o coincident block circulant $ v ~ r v and ) A = f v ~ 1f i By m - 0 ~ . 3.1 a reflective circulant matrix cannot be the incidence m a t r i x of a (v,k,X )-configuration, then cannot be a (v,k,X )-matrix. Moreaver, since it is clear that the determinant of a coincident block circulant matrix is zero, such a matrix cannot be a (v,k,X )-matrix. [I

matrix, then K = -( H + J 2 (v,k,A )-matrix, w i t h k =

\

.

z(

The hrop. 3.2 is a p a r t i a l answer to the problem, i n d i g i t a l c d c a t i o n s , of the existence of f i n i t e sequences of 1 and -1 {b.Iv w i t h the property that

t h e i r aperiodic auto-correlatim- coefficients

1 1

V

c . = C b.b 3 i.1 1 i + j ' 1 2 j Lv-1, should be 1 ,O,-I. It is w e l l knm that there is a one-to-one correspondence between such s e q u e n ~ sand circulant Hadamrd matrices [I ,pg. 981 W e note that the results of the Prop. 3.1 and the Prop. 3.2 were implicitely proved i n [ 3 ]

.

.

4. L e t G be a directed graph with n vertices;we car. suppose that its adjacency matrix is not symnetric. We call cyclic a graph w i t h circulant adjacency matrix. For a cyclic graph the mapping i + i + l , and its powers, are clearly autanorphim by definition. Hence, these graphs always have the cyclic group of order n, C , a s a subgroup of t h e i r autcmorphisn group. In f5lthe problem is posed h w t o give a procedure for determining the aukmorphism group of a cyclic graph. The follmdng Theorem 4.2 gives a subset of the graphs for which the autcmorphisn group i s exactly C n' LDW?I 4.1-

If C

f

0,J is a circulant matrix of order prime n,then det C

...

7

*

0.

is the f i r s t row of C, the eigenvalues of such Proof. I f [co , c , ,c n- 1 m t r i x are as i n = c c .w = 0. If det c = 0, there exists an e i envalue A Hence F( w)=O, where F ( x ) =1 c , x ' ~with iq n, I s a rational plyncxnial of 7 degree
11).

40d

...

''

J.0

N. Zagaglia Salvi

370

- Let G be a directed cyclic graph having a prim number n of vertices and reflective circulant adjacency matrix. Then the automrphim group of G is precisely C . n

l%3m4.2

Proof. Let G be a directed graph with a prime number n of vertices. If A is the adjacency matrix of G and Q is the permutation matrix Corresponding to an automrphim of G, we have (5)

4,AQ=A.

If A is reflective circulant, there exists a kt [I ,n-11 for h i c h A Pk =

men we have k

\.

k

4 y A P Q P T = A .

By LemM 4.1, (5) and (6) we obtain

k

P

k

Q P T = Q .

It follows prkuprk = Q, r c [I . I , Since n is prime, The integers rk, rt[l,n], mod n are distinct; then there exists jt[l,n]such that jk- 1 (md n). Hence we have P Q P = Q, i. e. Q is T circulant.0 In [5] it is proved that if a directed graph with a prim n m h r of vertices has C as autawrphism group, then the adjacency matrix A of G has distinct n eigenvalues. COROLIARY 4.3- If A is the reflective circulant adjacency matrix of a directed & r of vertices, then the eigenvalues of A are distinct. graph G with a p r h n

If R is the mtrix corresponding to the pewtation i+ n-i, we have the followinq

- Let G be a directed graph with a prime number of vertices and reflective circulant adjacency matrix. Then G is self-converse an has exactly n ismrphistns with the converse, corresponding to the matrices RP , hC[l ,nJ . COFOLTARY 4.4

4,

Proof. Let G be a directed graph with a prime nunher of vertices and reflective circulant adjacency matrix. holds, R Corresponds to an isanorphisn of G w'th t& conver4 . 2 , every autmrphim of G corresponds to Pti , h Cll ,n] ;

f ? . y ; e k m

then every matrix RP corresponds to an isamorphim of G with G'. It is well k n m that the number of i m r p h i s n s of a self-converse graph G w i t h the converse GI is equal tc the nunber of autanorptusms of G.U

5. Recall that a tournwent T of order n is a directed graph in which every pair of vertices is joined by exactly one arc. A tournament matrix A is the adjacency matrix of a tournament T and satisfies A + % = J - I .

37 1

Combinatorial Structures

LetR the reflective circulant tournament matrix, of odd order n, that satisfiesRP = A T' It is easy to prove that the I Is of the first row ofR are in even positions, the 0's in odd positions. THEy)REM is C

5.1

- The autmrphisn group of the tournament 2 corresponding to P

n'

Proof. Let Q be the permutation matrix corresponding to an autmrphisn cf then we have Q

and, being A P

p Q =4

;

(8)

=AT, it follows

QTA P Q PT

=a.

(9)

R nonsingular [ 9 ] Since is regular, it follaws ~ n t is obtain P Q P = Q, i. e. Q is circulant. T

a

183 the follawing theorems are proved. THFxlRFM 5 . 2 - If a tournament matrix A satisfies A Q = i+

;

by (3) and ( 9 ) w

In

where Q is a permu-

tation matrix, then Q corresponds to a n-cycle. PROPOSITIW 5 . 3 - Every tournament matrix A for wfiich AQ, where Q corresponds to a n-cycle, is also a tournament matrix, is penrutationally similar t o R

.

Now, we have the following

-

THMRFM 5.4 A reflective circulant toumamntmatrix is reflective circulant if and only it is pennutationally similar t o A

.

Proof. Let A be a reflective circulant tournament matrix of order n. Since the row sums of A are constant, it follaws that the t o u r n m t corresponding to A is regular; hence n is odd. Bec use A is reflective circulant, there exists an integer kC[l ,n-1] for which A P =%. k By Theorem 5.2 P corresponds to a n-cycle; then, by Proposition 4 . 3 A is pernutationally similar toR. The converse is easily pr0~e~1.O

r:

- The automorphisn groq of a tournament with reflective circulant adjacency matrix is C . n

COROLWIY 5 . 5

Proof. The proof follaws imnediately fran Theorems 5.1 and 5 . 4 . CDY5.6 - The determinant of a reflective circulant tournament matrix of n-1 order n is

-.2

Proof. Since the matrix of cdd order n

R satisfies the relations

+

It

=

N . Zagaglia Salvi

312

-

J - I andRP =AT,it follm A (I + P)= J I; then we have detR.det(I+P)= det(J - I). It is immediate to prove that det(1 t P ) = 2 and det(J - I)=n-I for n odd. BY Proposition 5.3, the prmf is capletai.CJ

[I]

Bamrt, L.D., Cyclic Diffemce Sets (Lscture Notes in Mathematics,

n. 182, Springer Verlag, 1971). [2]

Biggs, N., Algebraic Graph Theory (Cambridge University Press,1974).

A note on multipliers of difference sets, J. of bs. Nat. B. of Standards, ~01.69 B (1965) 87-89.

131 Brualdi, R.A.,

[4]

Davis, P.J., Circulant matrices

[5]

Elspas, B. and Turner, J., Graphs with circulant adjacency matrices, J. Cunbinatorial Theory 9(1970) 297-307.

[6]

(

A Wiley-Interscience Publication, 1979)

.

Ryser, H.J., Canbinatorial Mathematics ( Cams Math. Monograph, N. 74, New York, 1963).

[7]

Turner, J., Point-Symwtric Graphs with a Prim N d r of Points, J. Cmbinatorial Theory 3 (1967) 136-145.

[8]

Zagaglia Salvi, N., Sulle matrici tome0 associate a matrici di pennutazione, Note dihtematica, vol. I1 (1982) 177-188. Zagaglia Salvi, N., Alcune proprieta delle matrici torneo regolari, in: Atti del Convegno "Geanetria Cconbinatoria e di incidenza: fondamenti e applicazioni" La Mendola, 4-11 Luglio 1982 ( Editr. Vita e Pensiero) 635643.

Annals of Discrete Mathematics 30 (1986) 373-382 0 Elsevier Science Publishers B.V. (North-Holland)

373

OVALS I N STEINER T R I P L E SYSTEMS

Herbert Z e i t l e r

Mathematisches I n s t i t u t , U n i v e r s i t g t Bayreuth D-8580

B a y r e u t h , West Germany

T h i s a r t i c l e i s a r e p o r t o n some j o i n t w o r k s w i t h

H. Lenz, B e r l i n [ 2 , 7 , 8 ] .

Using t h e n o t i o n o f a

r e g u l a r o v a l , S t e i n e r t r i p l e systems of o r d e r v = 9

o r 13 + 12n w i t h n

E

INo are c o n s t r u c t e d .

INTRODUCTION L e t V w i t h I V I = v > 3 be a f i n i t e s e t a n d B w i t h I B I = b a s e t o f 3 - s u b s e t s of V. The e l e m e n t s of V a r e c a l l e d p o i n t s , t h e e l e m e n t s o f B l i n e s or blocks.

I f any 2-subset o f V i s contained i n e x a c t l y one

l i n e , then t h e incidence s t r u c t u r e (V,B,E)

i s called a S t e i n e r

t r i p l e s y s t e m of o r d e r v , b r i e f l y S T S ( v ) . Each p o i n t o f a n S T S ( v ) 1 1 l i e s o n e x a c t l y r = ~ ( v - 1 )l i n e s a n d t h e r e a r e e x a c t l y b = - v ( v - I ) 6 l i n e s . The c o n d i t i o n v = 7 o r 9 + 6 n , n E INo i s n e c e s s a r y and s u f f i c i e n t f o r t h e e x i s t e n c e of a n S T S ( v ) . The s e t of t h e s e " S t e i n e r numbers" v w i l l b e d e n o t e d by STS. Any non empty s u b s e t H c V i n a S T S ( v ) i s c a l l e d a h y p e r o v a 2 i f f e a c h l i n e of S T S ( v ) h a s e x a c t l y e i t h e r t w o p o i n t s ( s e c a n t ) o r no p o i n t ( e r t e r n a 2 l i n e ) i n common w i t h H. Thus w e o b t a i n I H I = l + r . = V x H t o g e t h e r w i t h a l l e x t e r n a l l i n e s of H forms

The complement

a s u b s y s t e m S T S ( r ) a n d , v i c e v e r s a , t h e complement of a s u b s y s t e m STS ( r ) i s a h y p e r o v a l i n STS ( v ) P r e c i s e l y f o r a n y v = 7 o r 1 5 + 1211,

.

n E INo

t h e r e e x i s t s a n STS ( v ) w i t h a t l e a s t o n e h y p e r o v a l

[21,

14I

[7]. The s e t of a l l t h e s e s p e c i a l S t e i n e r numbers w i l l be d e n o t e d by HSTS. The r e m a i n i n g S t e i n e r numbers i n RSTS = STS\HSTS

o r 1 3 + 12n, n

E

INo.

are v = 9

,

374

H. Zeitler

1 . OVALS

1 . 1 DEFINITIONS Any non empty s u b s e t 0 c V i n a S T S ( v ) w i l l b e c a l l e d a n o v a l i f f there e x i s t s exactly one tangent a t each point of 0 ( t h i s tangent

meets 0 i n e x a c t l y o n e p o i n t , t h e s o - c a l l e d p o i n t o f c o n t a c t ) a n d e a c h l i n e of S T S ( v ) h a s a t most t w o p o i n t s i n common w i t h 0 . Somet i m e s t h e complement 5 = V \ O t o g e t h e r w i t h a l l t h e e x t e r n a l l i n e s

t o 0 w i l l b e r e f e r r e d t o a s t h e “ c o u n t e r o v a l ” . The p o i n t s o f 0 w i l l be c a l l e d t h e on-points,

t h e points on t h e tangents to 0 with

t h e points of contact deleted w i l l be c a l l e d t h e ex-points

and a l l

t h e remaining p o i n t s t h e in-points. 1 . 2 ENUMERATION THEOREMS 1 . 2 . 1 CLASSES OF POINTS

Each o v a l has e x a c t l y r = -1( v - l ) p o i n t s and t h e c o r r e s p o n d i n g 2 1 ) oints. counter oval exactly r + 1 = ~ ( v + l p Proof:

Each p o i n t o n t h e o v a l l i e s o n r l i n e s , a n d p r e c i s e l y o n e

of them i s a t a n g e n t . 1 . 2 . 2 CLASSES O F LINES

W i t h r e s p e c t t o a n o v a l 0 t h e r e e x i s t e x a c t l y r = ~1 ( v - 1 ) t a n g e n t s , 1 1 exactly = s ( v - l ) ( v - 3 ) s e c a n t s and e x a c t l y - r ( r - I ) =’(v-I) (v-3) 6 24 external l i n e s .

(;)

Proof:

An e a s y c o u n t i n g a r g u m e n t .

1 . 3 THEOREM

The number of t a n g e n t s t h r o u g h a n e x - p o i n t o f a n o v a l - 0 i s e v e n o r odd, a c c o r d i n g t o r b e i n g even o r odd. Proof:

L e t s b e t h e number of s e c a n t s , t t h e number o f t a n g e n t s

t h r o u g h a n e x - p o i n t o f 0 . By 1 . 2 . 1 ,

t + 2 s = r, and t h e s t a t e m e n t

follows.

1 . 4 THEOREM

For a n y v

E

HSTS t h e r e e x i s t s a n S T S ( v ) w i t h a t l e a s t o n e o v a l .

375

Ovals in Steincr Triple Systems

Proof:

When v E HSTS t h e r e e x i s t s a n S T S ( v ) w i t h a t l e a s t o n e

h y p e r o v a l . Any r p o i n t s of t h i s h y p e r o v a l form a n o v a l . T h e s e o v a l s

are s p e c i a l ones, because a l l t h e t a n g e n t s m e e t i n e x a c t l y one p o i n t , t h e k n o t of t h e o v a l . 1 . 5 DEFINITION

O v a l s a l l whose t a n g e n t s a r e c o n c u r r e n t will b e c a l l e d k n o t o v a l s . I f through each ex-point

of a n o v a l t h e r e p a s s e x a c t l y two t a n g e n t s ,

then t h e oval w i l l be c a l l e d a regular oval. 1 . 6 THEOREM

In a n S T S ( v ) w i t h v E HSTS t h e r e a r e no r e g u l a r o v a l s , i n a n S T S ( v ) w i t h v t RSTS t h e r e e x i s t no k n o t o v a l s . Proof:

I f v E HSTS, t h e n r = 3 o r 7 + 6n, n

E INo,

a n d r i s o d d . By

1.3, r e g u l a r o v a l s c a n n o t e x i s t i n s u c h a n S T S ( v ) . I f i n a n S T S ( v ) w i t h v E RSTS a k n o t o v a l e x i s t e d , t h e n a d d i n g t h e k n o t t o t h e p o i n t s of t h e o v a l w e would o b t a i n a h y p e r o v a l ; t h e r e f o r e , v E HSTS, a contradiction. 1 . 7 THEOREM

For any r e g u l a r o v a L 0 i n at? S T S ( v ) t h e r e e x i s t s e x a c t l y o n e i n point. Proof:

L e t e be t h e number o f e x - p o i n t s a n d i t h e number o f i n -

p o i n t s of 0 . The e x - p o i n t s a n d i n - p o i n t s t o g e t h e r a r e t h e p o i n t s o n the counter oval

6. T h e r e f o r e ,

e+i =

=

r + '1. By 1.5 e a c h ex-

p o i n t l i e s on e x a c t l y two t a n g e n t s a n d o n e a c h of t h e r t a n g e n t s to 0 t h e r e are e x a c t l y t w o ex-points.

Consequently, e = r and i = 1 .

1 . 8 THEOREM

For a r e g u l a r o v a l t h e n u m b e r s of l i n e s i n t h e d i f f e r e n t c l a s s e s t h r o u g h p o i n t s of d i f f e r e n t c Z a s s e s a r e a s f o l l o w s :

H. Zeitler

316

secants

tangents

in-point

1 1 ? ( r - 2 ) = T(v-5) 1 1 =p - 1 )

0

on- po i n t

r - 1 = ?1( v - 3 )

1

ex- po i n t

2

Tr

external lines 1 1 ~(r-2= ) T(v-5) 1 1 = x(v-1)

zr

0

i

The proof f o l l o w s immediately from t h e p r e c e d i n g theorems.

2 . THE M A I N THEOREM For a n y v E RSTS t h e r e e x i s t s a n STS(v) w i t h at l e a s t o n e r e g u l a r

oval. Now t h e S t e i n e r numbers v

E

HSTS and v E RSTS w i l l b e c l a s s i f i e d i n

a g e o m e t r i c a l way u s i n g t h e e x i s t e n c e o f S T S ( v ) ' s w i t h k n o t o v a l s and w i t h r e g u l a r o v a l s , r e s p e c t i v e l y . D i s r e g a r d i n g t h i s d i s t i n c t i o n between o v a l s w e c a n summarize: F o r any S t e i n e r number v E STS t h e r e e x i s t s a n STS(v) w i t h a t l e a s t one o v a l . I n [ 2 ] t h i s theorem i s proved by c o n t r a d i c t i o n . W e now g i v e a d i r e c t proof by c o n s t r u c t i o n [ 8 I .

Throughout t h e proof by a n o v a l w e always mean a r e g u l a r o v a l . Taking i n t o a c c o u n t theorem 1 . 6 , w e s t a r t w i t h a S t e i n e r number v E RSTS and c o n s t r u c t a n STS(v) w i t h a t l e a s t one r e g u l a r o v a l . 2 . 1 THE POINTS

1 With v E RSTS, 101 = r = -2( v - l ) = 4 o r 6 + 6n, n E INo, and 1 161 = r + 1 = z ( v + l ) = 5 o r 7 + 6n, n E INo. T h e r e f o r e , t h e number o f p o i n t s o n t h e o v a l 0 is e v e n and t h e number o f p o i n t s o n t h e c o u n t e r o v a l i s odd. W e c o n s i d e r two r e g u l a r r-gons w i t h t h e same c e n t r e M whose e d g e s

have d i f f e r e n t l e n g t h s . F u r t h e r m o r e , t h e r-gons a r e r o t a t e d w i t h

. The

...

v e r t i c e s Po,P, , ,Pr-, of t h e v e r t i c e s Qo,Q,,...,Qr-l of t h e o u t e r polygon as e x - p o i n t s and f i n a l l y t h e p o i n t M i s t a k e n a s t h e i n - p o i n t of t h e o v a l 0 . W e c o u n t t h e v e r t i c e s of each polygon c o u n t e r c l o c k w i s e . F i g u r e 1 shows t h i s i n t e r p r e t a t i o n f o r r = 1 6 . r e s p e c t t o e a c h o t h e r by a n a n g l e

f

t h e i n n e r polygon a r e t a k e n a s o n - p o i n t s ,

Ovals in Steiner Triple Systems

377

0

M

Figure 1 I t t u r n s o u t u s e f u l t o r e p r e s e n t t h e numbers r modulo 1 2 , Conse-

q u e n t l y , r = 1 2 or 4 or 10 o r 6 + 12n. n E INo. I n t h i s p a p e r w e r e s t r i c t o u r s e l f t o t h e case r = 4 + 12n. The p r o o f s i n t h e o t h e r cases may be done i n a s i m i l a r way. 2 . 2 CONSTRUCTION OF SPECIAL 3-SUBSETS

F i r s t o f a l l w e c o n s t r u c t s p e c i a l p o i n t sets h a v i n g no c o n n e c t i o n with an oval. L e t a r e g u l a r r-gon be g i v e n w i t h v e r t i c e s O,l,

...,r-1

and c e n t r e M .

( a ) F i r s t c l a s s of s u b s e t s I 1 There a r e = T(v-l) d i a m e t e r s i n t h e g i v e n p o l y g o n . The two end-

H , Zeitler

378

p o i n t s o f each d i a m e t e r t o g e t h e r w i t h t h e c e n t r e M f o r m a 3-subset o f t h e f i r s t class. ( b ) Second c l a s s o f s u b s e t s L e t i , j w i t h i <j b e t w o v e r t i c e s o f t h e g i v e n p o l y g o n . Then t h e

minimum o f j

- i,

r

+i-

j

is c a l l e d t h e " d i f f e r e n c e " of t h e t w o ver-

t i c e s . By t h i s d e f i n i t i o n , t h e f o l l o w i n g d i f f e r e n c e s a r e p o s s i b l e : 1 1 1 1 , 2 , . . . , p - 1 = a ( " - 5 ) . (The d i f f e r e n c e ?r i s a l r e a d y e l i m i n a t e d w i t h t h e f i r s t c l a s s ! ) Now w e form o r d e r e d " d i f f e r e n c e t r i p l e s " ( d l , d 2 . d 3 ) i n s u c h a way t h a t e a c h o f t h e m e n t i o n e d d i f f e r e n c e s occ u r s a t m o s t o n c e and d l + d 2 = d3. Which d i f f e r e n c e s c a n o c c u r i n t h e t r i p l e s ? How many d i f f e r e n c e t r i p l e s of t h i s kind are p o s s i b l e ? I n [ I ] ,

[21,

[31,

q u e s t i o n s a r e a n s w e r e d . Here w e g i v e t h e r e s u l t o n l y

[ 6 1 these

-

w i t h o u t any

proof.

r = 4 t 12n, n E IN (1,

5 n + l , 5n+21 , ( 2 ,

3n,

3n+2),

(31

5n,

3n-1,

3n+3),

(2n-1,

4n+2, 6 n + l )

5n+3), (4,

,

(2n, 2n+l, 4 n + l ) .

1 The d i f f e r e n c e 3 n + 1 = -r i s missing, any o t h e r d i f f e r e n c e o c c u r s 4 e x a c t l y once a s r e q u i r e d . A l t o g e t h e r w e h a v e 2n d i f f e r e n c e t r i p l e s .

The l e a s t v a l u e f o r r , i . e . r = 4 1 d o e s n o t p r o v i d e d i f f e r e n c e t r i p l e s . We o b t a i n t h e a f f i n e p l a n e A G ( 2 , 3 ) . I t i s e a s y t o show t h a t i n t h i s system t h e r e e x i s t r e g u l a r ovals ( f o r i n s t a n c e t h e s e t o f p o i n t s (x,y) with x,y

E

GF(3) and x 2 - y 2 = 1 ) . I n t h e f o l l o w i n g w e

o m i t t h i s s p e c i a l case. 2.3 THE EXTERNAL L I N E S Now t h e p o i n t s 0 , 1 , . . . , r - 1

i n 2 . 2 a r e t h e ex-points

QoIQ1f...fQ,-l

and M i s t h e i n - p o i n t o f a n o v a l 0 . Next, t h e e x t e r n a l l i n e s t o 0 a r e c o n s t r u c t e d u s i n g t h e d i f f e r e n c e t r i p l e s g i v e n i n 2.2. F o r better t y p i n g w e f r e q u e n t l y w r i t e ( P , x ) a n d (Q,x) i n s t e a d o f Px a n d Qx. (3)

The e x t e r n a l l i n e s c o n t a i n i n g M

The s u b s e t s o f t h e f i r s t c l a s s d e t e r m i n e d by t h e p o l y g o n d i a m e t e r s

Ovals in Steiner Triple Systems

319

a r e t h e e x t e r n a l l i n e s o f 0 c o n t a i n i n g M . We c a n w r i t e {M

,

(Q, x )

,

(Q,x+

1 with x t

2 t h e number x modulo r .

{O, 1,.

. ., r - I

1. W e a l w a y s u n d e r s t a n d

( b ) The r e m a i n i n g e x t e r n a l l i n e s

( Q , x ) , ( Q , x + 2 ) , ( Q , x+3n+2 ( Q , x ) , ( Q , x + 4 ) , (Q1x+3n+3

( Q , x ) , ( Q f x + 2 n ), ( Q l x + 4 n + l x t {0,1,

..., r - I }

1 I n t h i s way w e o b t a i n a l t o g e t h e r r . 2 n = - - ( v - I ) ( v - 9 ) a d d i t i o n a l 324 s u b s e t s o f p o i n t s . These l i n e s are e x t e r n a l l i n e s o f 0,too. Together 1 1 w i t h t h e -r = - ( v - l ) e x t e r n a l l i n e s c o n t a i n i n g M w e h a v e g o t 2 4 1 1 1 ~ ( v - 1 )+ ? ; r ( v - I ) ( v - 9 ) = ~ ( v - 1 ()v - 3 ) e x t e r n a l l i n e s . By 1 . 2 . 2 , t h i s

is t h e set of a l l e x t e r n a l l i n e s . In t h e difference t r i p l e s 2 . 2 the difference 3n+ 1 is missing. The a s s o c i a t e d p a i r of e x - p o i n t s

( ( Q , x ), (Q,x+3n+l)) w i t h

w i l l b e i n v e s t i g a t e d l a t e r on i n connection with

x E {O,I,...,r-l} the tangents. 2 . 4 THE SECANTS

I n f i g u r e 2 w e have t h e s e c a n t s o f a n o v a l contair.inq t h e ex-point

-

f o r r = 16.

B

0

*

0

0

0

.,

M

, , , , , , , ,.

....0............... .........-0

----_ --- -- -------z

-0

01,-

o--

-0

0-

-

--+

-Q

-Q

o Q Figure 2

D

A

Q

H. Zeitler

3 80

Secants containing a n ex-point

( a ) First c l a s s o f p o l y g o n c h o r d s W e s t a r t w i t h t h e p o l y g o n v e r t i c e s A and B ( f r o m B t o A c l o c k w i s e ! ) 1 having t h e d i f f e r e n c e Tr. I n t h e set of polygon chords o r t h o g o n a l 1 t o t h e diameter through B we choose t h o s e T ( r - 4 ) c h o r d s which a r e n e a r e s t t o B ( f u l l l i n e s ) . Then t h e n e x t p o l y g o n c h o r d o f t h i s k i n d i s a p o l y q o n d i a m e t e r CA. The e n d p o i n t s o f t h e p o l y g o n c h o r d s

c h o o s e n i n t h i s way h a v e t h e d i f f e r e n c e s 2 , 4 , .

. .,-!-r-2, 2

respectively.

( b ) Second c l a s s o f p o l y g o n c h o r d s The p o l y g o n v e r t e x which f o l l o w s A g o i n g c l o c k w i s e from B t o A i s D . 1 I n t h e s e t o f p o l y g o n c h o r d s p a r a l l e l t o CD w e c h o o s e t h o s e -r 4 c h o r d s which a r e f a r t h e s t from B ( d o t t e d l i n e s ) . The e n d p o i n t s o f 1 t h e s e p o l y g o n c h o r d s h a v e d i f f e r e n c e s l 1 3 ~ . . . r ~ r - rle~s p e c t i v e l y . One o f t h e s e c h o r d s i s a n e d g e o f t h e p o l y g o n . The e x - p o i n t

lying

close t o t h i s edge i s Q. The e n d p o i n t s o f e a c h p o l y g o n c h o r d , b o t h i n ( a ) a n d i n ( b ) , t o g e t h -

e r w i t h Q form a n o v a l s e c a n t . I n t h i s way w e o b t a i n a l t o g e t h e r 1 1 = ;(r-2) -4( r - 4 ) + s e c a n t s c o n t a i n i n g Q. By 1 . 8 w e h a v e f o u n d a l l s e c a n t s of t h i s kind.

ar

T h e r e i s no s e c a n t c o n t a i n i n g Q and t h e t w o o n - p o i n t s A and B a s

w e l l . T h e r e f o r e , t h e s e t w o p o i n t s must b e t h e p o i n t s o f c o n t a c t o n t h e t w o t a n g e n t s t h r o u g h Q . By o u r c o n s t r u c t i o n t h e d i f f e r e n c e o f A 1 and B i s e x a c t l y -r. 4 I n t h e same way it i s p o s s i b l e t o c o n s t r u c t t h e s e c a n t s t h r o u g h a n y ex-point

(Q,x)

.

Now w e c a n e x p l i c i t l y w r i t e down a l l t h e s e c a n t s c o n t a i n i n g a n e x point (Qrx):

1 1 ( Q l x ) , ( P , x + ~ r - l ) , ( P r x 2+ - r + l ) } .

Furthermore,

1 (A,x) = ( P r x + , r ) ,

1 3 ( B , x ) = ( P , x + ~ r ) ,( C , X ) = ( P , x + - r ) . 4

We f u r t h e r see t h a t t h e t a n g e n t c o n t a i n i n g ( Q l x ) a n d ( Q , x + + )

has

t h e p o i n t o f c o n t a c t ( B , x ) . The d i f f e r e n c e o f t h e s e t w o e x - p o i n t s i s

1 Tr .

Again, x E { O r l t . . . , r - l } .

38 1

Ovals in Steiner Triple Systems Secants containing t h e in-point The e n d p o i n t s o f a n y d i a m e t e r o f t h e i n n e r p o l y g o n form a n oval 1 s e c a n t t o g e t h e r w i t h M . I n t h i s way w e o b t a i n e x a c t l y secants

zr

c o n t a i n i n g M. By 1 . 8 w e h a v e f o u n d a l l t h e s e c a n t s o f t h i s k i n d : 1 namely, t h e y a r e { M , ( P , x ) , ( P , X + ~1 ,~ x) E { O , l , ...,r - 1 ) . Adding t h e numbers o f 1 1 o b t a i n r--(r-2) +-r = 2 2

oval s e c a n t s i n t h e t w o d i f f e r e n t classes w e 1 1 -r(r-1) = - ( v - l ) ( v - 3 ) . 2 8

2 . 5 THE TANGENTS When c o n s t r u c t i n g t h e e x t e r n a l l i n e s i n 2 . 3 some p a i r s o f e x - p o i n t s w e r e l e f t w i t h o u t a c o n n e c t i n g l i n e . The d i f f e r e n c e b e t w e e n t w o 1 p o i n t s o f s u c h a p a i r w a s d = -r 4 = 1 + 3n. The c o n s t r u c t i o n o f t h e o v a l s e c a n t s i n 2 . 4 was d o n e i n s u c h a way

t h a t t h e d i f f e r e n c e of t h e t w o e x - p o i n t s on each t a n g e n t g i v e s t h e number d . T h e r e f o r e , t h e oval t a n g e n t s a r e c o m p l e t e l y d e t e r m i n e d : 1 1 { ( Q , x ), ( Q , x + T r ) , ( P , x + z r ) } , x E { O t l r - . . f r - l } . Our c o n s t r u c t i o n i n case r = 4 + 12n w i t h n E IN i s c o m p l e t e d . CONCLUDING REMARKS

Long a g o P . Erdos a s k e d t h e f o l l o w i n g q u e s t i o n s : Which c a r d i n a l i t y i s t h e maximal o n e f o r a p o i n t s e t M i n a n S T S ( v ) c o n t a i n i n g no l i n e ? F o r w h i c h S t e i n e r numbers v may t h i s maximal case o c c u r ? A l l t h e s e q u e s t i o n s w e r e a l r e a d y answered i n [ 5 ] . H e r e w e relate

t h e s e p r o b l e m s t o h y p e r o v a l s a n d o v a l s . The r e q u i r e d maximal c a r d i n a l i t y o c c u r s i f each p o i n t of M l i e s on s e c a n t s o n l y . T h i s y i e l d s I M I i;

l + r . T h e r e f o r e , i n case o f maximal c a r d i n a l i t y M i s a h y p e r -

o v a l . P r e c i s e l y , f o r any v

€

HSTS t h e r e e x i s t s a n S T S ( v ) w i t h s u c h

a s e t o f maximal s i z e l + r . I f s u c h h y p e r o v a l s do n o t e x i s t , t h e n

1M[ 5 r . The u p p e r bound i s a c h i e v e d i f M i s a r e g u l a r o v a l . F o r a l l S t e i n e r numbers v

E

RSTS t h e r e e x i s t S T S ( v ) ' s w i t h s e t s o f t h e

r e q u i r e d maximal c a r d i n a l i t y r . A n o t h e r matter i s t h e i n v e s t i g a t i o n o f f i n i t e a f f i n e a n d p r o j e c t i v e s p a c e s w i t h r e g a r d t o ovals. W e m e n t i o n some r e s u l t s : I n P G ( d , 2 ) w i t h d > 2 t h e r e e x i s t e x a c t l y 2 ( 2d + l - 1 ) k n o t o v a l s a n d no r e g u l a r o v a l s . I n AG(2,3) t h e r e e x i s t e x a c t l y 54 r e g u l a r o v a l s a n d no k n o t o v a l s . I n AG(d, 3 ) w i t h d

2

3 t h e r e e x i s t neither knot

H. Zeitler

382 o v a l s n o r r e g u l a r ovals.

W e n o t i c e t h a t t h e r e a r e s t i l l many u n s o l v e d p r o b l e m s a b o u t o v a l s i n a n S T S ( v ) . (The t o t a l number of o v a l s i n a g i v e n STS (v) ; t h e number of non-isomorphic S T S ( v ) ' s w i t h o v a l s i f t h e S t e i n e r number i s g i v e n ; i n v e s t i g a t i o n o f automorphism g r o u p s ; )

...

F o r S t e i n e r s y s t e m s S ( k , v ) w i t h k Z 3 t h e r e a r e o n l y a few r e s u l t s concerning o v a l s , hyperovals and similar n o t i o n s [ 4 ] . Extensions o f t h e ErdBs p r o b l e m s t o g e n e r a l s y s t e m s of t h i s k i n d a r e unknown, too.

REFERENCES

H i l t o n , A.J.W., On S t e i n e r and s i m i l a r t r i p l e s y s t e m s , Math. Scand. 24 ( 1 9 6 9 1 2 0 8 - 2 1 6 L e n z , H . and Z e i t l e r , H . , A r c s a n d O v a l s i n S t e i n e r T r i p l e S y s t e m s , C o m b i n a t o r i a Z T h e o r y , L e c t u r e No t e a i n M a t h e m a t i c s 969 ( 1 9 8 2 ) 229-250 (Springer Verlag, Berlin, Heidelberg, N e w York) P e l t e s o n , R . , E i n e Losung d e r b e i d e n H e f f t e r s c h e n D i f f e r e n z e n probleme, C o m p o a i t i o Math. 6 1 1 9 3 9 ) 2 5 1 - 2 5 7 R e s m i n i , M. d e , On k - s e t s of t y p e (m,n) i n a S t e i n e r s y s t e m S ( Z , l , v ) , F i n i t e G e o m e t r i e s and D e s i g n s , L . M . S . L e c t u r e N o t e S e r i e s 49 (1981) 104-113 S a u e r , N . and Schonheim, J . , Maximal s u b s e t s o f a g i v e n s e t h a v i n g no t r i p l e i n common w i t h a S t e i n e r t r i p l e s y s t e m o n t h e s e t , Canad. M a t h . BUZZ. 1 2 ( 1 9 6 9 ) 7 7 7 - 7 7 8 Skolem, T . , Some r e m a r k s o n t h e t r i p l e s y s t e m s o f S t e i n e r , Math. S c a n d . 6 ( 1 9 5 8 ) 273-280 Z e i t l e r , H . a n d Lenz, H . , H y p e r o v a l e i n S t e i n e r - T r i p e l Systemen, M a t h . Sem. B e r . 3 2 ( 1 9 8 5 ) 1 9 - 4 9

Z e i t l e r , H . and Lenz, H . , R e g u l a r Ovals i n S t e i n e r T r i p l e S y s t e m s , JournaZ of C o m b i n a t o r i c s , I n f o r m a t i o n and S y s t e m Sciences, Delhi (to appear)

383

PARTICIPANTS

L.M.

V.

Abatangelo

Abatangelo

Bari Bari

S. Antonucci

Napoli

E.M.

Milano

Aragno M a r a u t a

R.

Artzy

Haifa

L.

Bader

Napoli

G.

Balconi

Pavia

A.

Rarlotti

Firenze

U.

Bartocci

Perugia

L.

Benbteau

Toulouse

W.

Benz

Hamburg

L.

Berardi

L I Aqui l a

C.

Bernasconi

Perugia

L.

Bertani

Heggio E m i l i a

A.

Beutelspacher

Mainz

A.

Bichara

Roma

M.

Biliotti

Lecce

P. R i o n d i

Napoli

P. B i s c a r i n i

Perugia

F.

Ronetti

Ferrara

A.

Bonisoli

Munchen

A.A. A.

Bruen

Caggegi

1. C a n d e l a

G.

C a n t a l u p i Tazzi

Western O n t a r i o Napol i

Bari Milano

Participants

384 R . C a p o d a g l i o d i Cocco

Bo 1ogna

M.

Bari

Capursi

B. C a s c i a r o

Bari

P.V.

Roma

Ceccherini

N.

Cera

Pescara

N.

Civolani

Milano

A.

Cossu

Bari

C.

C o t t i Ferrero

Parma

M. Crismale

Bari

F . De C l e r c k

Gent

E . Ded6

Milano

M.

De Finis

M.L. M. M.M.

D e Resmini

De S o e t e

Deza

C . D i Comite

Roma Roma Gent Paris

Bari

V.

Dicuonzo

Roma

L.

D i Terlizzi

Bari

B . DlOrgeyal

D i jon

J . Doyen

Bruxe 1l e s

F. Eugeni

L Aqui l a

G.

Faina

Pe r u g i a

L.

Faggiano

Bari

M.

Falcitelli

Bari

A.

Farinola

Bari

G.

Ferrero

Parma

0. F e r r i

L I Aqui l a

P. F i l i p

Miinchen

S.

Fiorini

Malta

M.

Funk

Potenza

M.

Gionfriddo

Ca t a n i a

M.

Grieco

Pavia

W. Heise A.

Herzer

Miinchen Mainz

Participarz ts R.

H i l l

Salford

J . Hirshfeld

Sussex

D.

Hughes

London

V.

Jha

Glasgow

H.

Kellerer

Miinchen

H. Karzel Keedwell

A.D. G.

Korchmaros

H.J.

Kroll

Miinchen Gui 1d f o r d Potenza Munchen

F . Kramer

Clu j-Napoca

H.

Kramer

Cluj-Napoca

P.

Lancellotti

Modena

B.

Larato

Bari

D.

Lenzi

Lecce

A.

Lizzio

Catania

G.

Lo F a r o

Messina

P.M.

Lo R e

Napol i

G.

Luisi

Bari

G.

Lunardon

Napoli

H.

Lunenburg

K a iserslautern

N.A.

Malara

Modena

M.

Marchi

Brescia

A.

Maturo

Pescara

F . Mazzocca

Case r t a

N.

Melone

Napoli

M.

Menghj-ni

Roma

G.

Menichetti

Bologna

G.

Micelli

Lecce

G.

Migliori

Roma

F. Milazzo

Catania

S. Milici

C a ta n ia

A.

Maniglia

Lecce

D.

Olanda

Napoli

A.

Palornbella

Bari

385

Participan Is

Pastore

A.M.

Bari

S.

Pellegrini

Bresc i a

C.

Pellegrino

Modena

G.

Pellegrino

Per u g ia

C. P e r e l l i Cippo

Brescia

M. P e r t i c h i n o

Bari

G. P i c a

Napoli

F. P i r a s

Cagl i a r i

Pornilio

A.I.

L.

Porcu

Rorna Milano

L. Puccio

Mess i n a

G.

C a t an i a

Quattrocchi

P. Quattrocchi

Modena

G.

Bari

Raguso

L. R e l l a

Bari

T . Roman

Buc a r e s t i

L.A.

Rosati

Firenze

H.G.

Samaga

Hamburg

M.

Scafati

Roma

R.

Scapellato

Parma

E. Schroder

Hamburg

R.

Schulz

Berlin

D.

Senato

Napoli

H.

Siernon

Ludwigsburg

C. Soma

Roma

R. S p a n i c c i a t i

Rorna

A.G. S p e r a

P a l e rmo

.

Stangarone

Bari

K.

Strarnbach

Erlangen

G.

Tallini

Roma

P..

Terrusi

Bari

R

J . Thas

Gent

M.

Ughi

Perugia

V. Vacirca

C a t a n i a.

Participants K.

Vedder

Gissen

A.

Venezia

Roma

F. Verroca

Bari

R.

Vincenti

P e rugi a

A.

Ventre

Napoli

H. Wefelscheid

Essen

B.

Wilson

London

N.

L a g a g l i a Salvi

M i 1a n o

H.

Zeitler

Bay r e u t h

E.

Zizioli

Brescia

R. Z u c c h e t t i

Pavia

387

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