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i s mapped ( a f t e r some c o m p u t a t i o n ) t o
A f t e r more c o m p u t a t i o n we see t h a t r’(j)
T
= C-a2j,a2j,2b2j,l]
.
T h i s completes a proof of t h e f o l l o w i n g :
3.1.
The se,t M o f
2 x 2
upper t r i a n q u l a r m a t r i c e s c o r r e s p o n d i n g t o F i s h e r ’ s
f l o c k f o r t h e s t a n d a r d cone K c o n s i s t s o f a l l t h e - m a t r i c e s o f t h e f o l l o w i n g two
types:
In 3.1 i t would be q u i t e c o n v e n i e n t t o have a f o r m u l a f o r bk in t e r m s o f ak alone. So g i v e n a v a l u e t (=ak) I t i s easy t o see t h a t bk = +ila:(z+l) 2- 4 z / ( l + z ) . such t h a t t 2 - 2/(l+a) i s a nonsquare i n F, t h e c o r r e s p o n d i n g bk i s
-+iJ t 2 ( z + 1 ) 2 - 4 z / ( l + z ) ,
l e a s t +t i s p a i r e d w i t h
b u t we do n o t see how t o d e t e r m i n e t h e s i g n o f bk. +iJ t 2 ( z + 1 ) 2 - 4 z / ( l + z ) .
At
S.E. Payne
364 The cone
r?
i s i n v a r i a n t under t h e p r o j e c t i v e c o l l i n e a t i o n T, so we may c o n s i d e r
what happens t o F i s h e r ' s f l o c k under T. TI
i s a p l a n e i n F i s h e r ' s f l o c k f o r K,
= [a,b,c,llT
PCTl-lP-l[
T r a n s f e r r i n g t o K,
a, b, c ,1IT.
0
-ab2 = a2 -l,a2j-l
But
= (a2j+2-a2j)/2b.)
-
j,
and f o r t h a t k we have
a r i s e s f r o m F i s h e r ' s f l o c k F f o r K (as i n 3.1) i f and o n l y a r i s e s f r o m t h e image o f F under t h e a c t i o n o f P[T]-'P-'.
How i s t h e s e t M of 3.1 a f f e c t e d by r e p l a c i n g z w i t h some o t h e r
NOW we ask:
generator z
= 2aa2j-a2j+lyb2j+l
T h i s proves the following.
1;
ift h e m a t r i x
we o b t a i n
if and o n l y if k = ( q - 1 ) / 2
1;
3 . 2 . The m a t r i x
i s replaced b y
'I
O
M u l t i p l y i n g t h i s m a t r i x t i m e s C-a2j,aa2j,2b2j,l]T,
a2j = aq-2j = a2k+l b 2 j = -bqmZj = -b2k+l.
71
So compute :
l o
(Here we use
then
we see t h a t i f
r of
gcd(r,q+l)
= 1,
so
r = 2j+1 f o r some j?
From E q u a t i o n 22, we have
So t 2 square.
- 4/N(l+z)
i s a square i n F i f and o n l y i f
This implies t h a t
nonsquare i n F )
to the set
t -+ t / a j
{t
E
F: t 2
We c o n t i n u e t o use t h e n o t a t i o n ak,bk
-
{t
4/N(l+z2j+l)
modulo (q+1)/2.
E
-
i s a nonsquare i n F).
i s replaced with 1
So i f t = aZi, t h e n t / a j = a2k A f t e r some c o m p u t a t i o n we see t h a t
= 1.
4/N(l+z2j+l) is a F: t 2 4 / N ( l + z ) i s a
as a g i v e n i n S e c t i o n 2 , and l e t
t h e c o r r e s p o n d i n g elements o f F o b t a i n e d when z gcd(Zj+l,q+l)
-
(t/aj)'
maps t h e s e t
:k,6k
be
z 2 j + l,
f o r some k determined
The Thas-Fisher Generalized Quadrangles
aZi
* = 'ja2k
iff
(,2k
p 3 (Zj+l)-'
Put
12j+1)+2i+j+l-l)
(mod q+l).
( 2 j + 1 ) + 2 i -j-l) =
(,-2k
365
o,
(24)
Then the l e f t f a c t o r i s zero i f and o n l y i f
And t h e r i g h t f a c t o r i s zero i f and o n l y i f
k
[i-j/Z]p
(mod (q+1)/2),
i f j i s even.
(26)
A s i m i l a r computation shows t h a t bZi
=
2
.
aj-bZk
i f f (z
2k (2J+1 )+2i+j+l+l
-
(,-2k ( 2j + l ) + 2 i -jq) =
I t now f o l l o w s t h a t
-6,,, bZi/aj
=
bZk,
j odd j even.
This e s s e n t i a l l y completes a p r o o f o f the f o l l o w i n g . L e t M,be the,s,et o f m a t r i c e s a r i s i n g from F i s h e r ' s f l o c k f o r K s t a r t i n g
3.3.
w i t h the generator z (as i n 3.1).
Let
obtained when z i s replaced w i t h i?j*',
4.
THE
ii be
the corresponding s e t o f m a t r i c e s
gcd(2j+l,q+l)
= 1.
Then
GQ OF THAS
Most o f the d e t a i l s needed t o prove t h e theorem o f t h i s s e c t i o n have a l r e a d y been given.
4.1.
For each odd prime power q o n l y one GQ a r i s e s from F i s h e r ' s f l o c k .
Proof: The idea i s t h a t n e i t h e r r e p l a c i n g t h e o r i g i n a l f l o c k w i t h i t s image under y i e l d s d i f f e r e n t GQ. I n view o f 3.2 gcd(r,q+l) = 1, T nor r e p l a c i n g z w i t h zr, and 3.3, t h e f o l l o w i n g r e s u l t s complete t h e p r o o f .
S.E. Payne
366
;1 1;
and define 0: K + K by 0: (a,c,O) + (aQ,c,BQ). Then 0 is Put Q = an automorphism of K for which 8: (apAaT ,a(A+AT)) -t~QPQ(QAQ)(o(4)T,aQ(QAQ+(QAQ)T)). Since Q
Q
=
1; -:I
, A
=
1;
is replaced with Q A Q =
For t EF, t # 0, define 0: K-tK by $(o,c,B)+ (a,tc,tR). @ is an automorphism of K that replaces A with tA.
I -:I.
It is easy to check that
REFERENCES
c11 Fisher, 3. C. and Thas, J. A., Flocks in PG(3,q), Math. Z. 169 (1979) 1-11. C21 Kantor, W. M., Generalized quadrangles associated with G2(q), dour. Combin. Theory (A) 29 (1980) 212-219. 2 C3l Kantor, W. M., Some generalized quadrangles with parameters (q ,q), Math. Z. 192 (1986) 45-50.
C41 Kantor, W. M., Generalized quadrangles and translation planes, Algebras, Groups and Geometries 3 (1985) 313-322. C51 Payne, S. E., Generalized quadrangles as group coset geometries, Congressus Numerantium 29 (1980) 717-734. C61 Payne, S. E., A garden of generalized quadrangles, Algebras, Groups and Geometries 3 (1985) 323-354. C71 Payne, S . E., A new infinite family of generalized quadrangles, Congressus Numerantium 49 (1985) 115-128.
C81 Payne, S. E. and Thas, J. A., Finite Generalized Quadrangles, Research Notes in Mathematics No. 110 (Pitman Pub. Inc., 1984). C91 Thas, J. A., Generalized quadrangles and flocks of cones, preprint.
Annals of Discrete Mathematics 3 7 (1988) 367-374 0 Elsevier Science Publishers B.V. (North-Holland)
367
ON GROUP SPACES DEFINED BY SEMIDIRECT PRODUCTS OF GROUPS J . Pfalzgraf
Fachbere i c h Mathemat i k U n i v e r s i t a t des Saarlandes D-6600 Saarbrikken Fed. Rep. Germany
0 . INTRODUCTION
The " c l a s s i c a l " examples of noncomutative geometric spaces such a s ray s p a c e s , c i r c l e spaces, d i s c spaces, n e a r f i e l spaces, spaces defined by t h e more general F-groups, a l l t u r n out t o be group spaces with r e s p e c t t o s p e c i a l semidirect products of groups. W e introduce t h e concept of a semidirect product space (SPS) induced by t h e " a f f i n e action" of a semidirect product of groups ( c f . 1 . 2 ) . I t is p o s s i b l e t o make e x p l i c i t constructions of spaces with prescribed properties. This p o s s i b i l i t y is indicated by examples using f r e e groups t o show t h a t t h e equivalence of t h e p r o p e r t i e s "semiaffine" and " t a c t i c a l " , i n [A21 Satz 2 . 1 , is t y p i c a l f o r f i n i t e spaces ( i t does not hold f o r i n f i n i t e spaces, i n g e n e r a l , c f . 3.6). As another a p p l i c a t i o n of SPS we present a geometric version of Witt's cancell a t i o n theorem f o r q u a d r a t i c forms over f i e l d s ( c h a r a c t e r i s t i c not 2) involving two geometric i n v a r i a n t s which can be defined f o r a q u a d r a t i c form ( c f . 4 . 5 ) . F i n a l l y w e mention t h a t with respect t o a r b i t r a r y group extensions 1 + N E G + 1 f u r t h e r n a t u r a l questions of geometric type a r i s e . +
1.
+
NOTATION
Noncommutative geometric spaces (X,U ,I1 ) a s introduced by J.Andr& ( c f . A1 1 [A4]) can be described by t h e i r p a r a l l e l s t r u c t u r e ( c f . [Pl I): each map <,>:X2 + R , defining a p a r a l l e l i s m on t h e p o i n t set X ( R i s t h e set of d i r e c _ t i o n_ s o r_i d e_ a l p o i n t s ) , leads t o a so-called LP1'-space (X,<,>,R), i . e . t h e axioms ( L l ) , ( L Z ) , ( P l ' ) hold, and v i c e versa (see [ P l l f o r more d e t a i l s ) . are defined by x 0 y : = x U y"[<x,y>} ; x U y := { X ) U {zi <x,z>=<x,y>} i s t h e set of proper p o i n t s of x 1 y ; <x,y> i s t h e i d e a l p o i n t ( d i r e c t i o n ) of the line.
x
0y
is parallel t o u
0v ,
iff
< x , y > = < u , v > .Further axioms of a
space X a r e expressed by corresponding equations i n t h a t "<,>-calculus" cf. [Pll 2.5.
,
Now l e t G x X + X be t h e a c t i o n of a group G on a p o i n t set X . Then t h e group space ( c f . [All,[A21), denoted by V(G,X), i s defined by t h e p a r a l l e l map ( c f . [PZI): <,>:X2 R , with R := G \ X 2 , < x , y > : = G(x,y) t h e o r b i t of ( x , y ) . (1.1) -f
G a c t s componentwise on each p a i r of X z . For l i n e s we o b t a i n x u y = { x } u Gx.y.
J. Pfalzgraf
368
V( ,
) i s a covariant functor from "Group Operations" t o "Noncommutative
Geometric Spaces". For more information about group spaces see [All
-
[A41 and
[PZ]. Our i n t e r e s t here is concerned with group spaces associated with semid i r e c t products of groups. 1.2
Definition
Let G , N be groups ( m u l t i p l i c a t i v e l y w r i t t e n ) , G a c t s on N: gx := T ~ ( x ), x E N , g E G. E := N M G with respect t o t h a t a c t i o n . This gives rise t o t h e " a f f i n e action" of E on N , defined a s E x N -L N with T:
G
+
Aut(N), denoted by
W e obtain t h e semidirect product [ a , g ] . x : = a.gx
,
[a,gl
E
E, x
E
N
,
leading t o t h e group space V(E,N). W e call
t h i s a semidirect product space (SPS f o r s h o r t ) . W e note here t h a t a l l t h e 1-point s t a b i l i z e r s Ex , x because we know t h e s t a b i l i z e r group of 1
Ex = [ x , l l E1[x,ll-l
=
E
E
N which i s
N , can be c a l c u l a t e d
El
=
(1 ,GI, hence
{ [ ~ . ~ x - ' , g l/ g E G}.
I t t u r n s out t h a t a l l t h e following " c l a s s i c a l " examples of noncommutative geometric spaces a r e s p e c i a l SPS.
2.
EXAMPLES
2.1
Ray Space SIRn
n- 1 S u { O } , <x,y> : = (y-x)/ IIy-d1 y i s t h e ray x + IR otherwise. Then f o r x # y , x
X
:= IRn
,R
:=
,
if x # y
and <x,y> = 0
u
20(y-x) . We have t h e following p r e s e n t a t i o n of the ray space SRn a s SPS: SIRn = V ( I R n ~ I R , O , I R n ) , with r e s p e c t t o t h e n a t u r a l a c t i o n of TR,o on Rn. 2 . 2 Affine Space AG(n,JR) AG(n,IR) is obtained from SIRn by i d e n t i f i c a t i o n of antipodal d i r e c t i o n s , t h u s
R :=
d-'" { O } and <,>
x # y and <x,y> 2.3
:=
is defined on X : = Rn by <x,y> := [t (y-x)/ i~y-xll}, 0 , x = y. Then AG(n,IR) = ~ ( R n X , I R * , I R n ) , IR* := IR \ { O } .
C i r c l e Space
x U y i s t h e c i r c l e with radius IIx-yII : = IR2 , R : = IR 20 , <x,y> := / / x - y / (; and i t s midpoint x a s basepoint of t h e l i n e . X can be described by
X
IR')
o r by V(IR2xlSO(2,1R),IRz), a l t e r n a t i v e l y . Thus we see t h a t t h e covariant functor , ) i s not f a i t h f u l .
V(IR2X,0(2,1R),
V(
There i s an obvious g e n e r a l i z a t i o n of t h e above d e f i n i t i o n of c i r c l e space t o IRn leading t o spheres a s l i n e s .
On Group Spaces Defined by Semidirect Products of Groups
<x,y>:=
m
f o r x = y. Then x u y = { x l } x IR
{ x ) (~ {y,}
"=I(,
,
i f x1 = y, and
IR), i f x1 # y , . I t holds ( X , < , > , R )
x 7
a
E
IR,a
=
XU y
369 =
V(IR2xrH, IR2) with
IR*} IGL(2,IR).
E
Of course, t h i s d e f i n i t i o n can be g en er al i zed ( c f . [ P11,[ P21). 2.5
Remark
A l o t of f u r t h e r examples can be co n s t r u ct ed i n t h i s way ( f o r example
using IRn %I G , G a subgroup of GL(n,IR), e t c . ) Evidently one can d e f i n e t h e above spaces over more ge ne ra l c o e f f i c i e n t domains. In t h i s context we mention t h e following observation.
Consider t h e d i s c space 2.4 over a r i n g A with u n i t 1 , inste a d of a f i e l d . Then, using t h e analogous d e f i n i t i o n s (A* denotes t h e group of i n v e r t i b l e elements ( u n i t s ) of A) we o b t a i n t h e following r e s u l t
(X,<,>,R)
(2.6)
=
\I(AZ+H,A2) i f , a n d only i f , A is a f i e l d .
Proof: I f A is a f i e l d , then t h e e q u a l i t y holds ( c f . [PZ]). Conversely assume t h a t A i s n o t a f i e l d . Then t h e r e e x i s t s c
E A, c # 0 , which is n o t a u n i t . Now consider t h e p o i n t s x = (O,O), y = (O,c), u = ( l , l ) , v = (1,O) i n X = A' and show t h a t t h e p a r a l l e l classes (AZ- H). (x,y) and P < X ,Y> { ( x ' , y ' ) I < x ' , y'> = < x , y > } are d i s t i n c t (hence t h e spaces a r e d i s t i n c t , having d i f f e r e n t p a r a l l e l i s m s ) : obviously ( u , v ) E P s i n c e
4
values y i e l d s t h e equation 2.7
0 = 1 + a c , hence c
E
A*, a c ontra dic tion.
F-Groups and Near f i el d Spaces
As mentioned i n [ A41 93, t h e n e a r f i e l d spaces ( t h e s e are re gula r desarguesian n e a r a f f i n e spaces and v i c e v er s a) are SPS. I t t u r n s out t h a t even t h e more g e n e r a l F-groups ( c f . [A31 ) lead t o SPS.
For convenience we r e c a l l t h e d e f i n i t i o n of a F-group and adopt t h e n o t a t i o n of [ M I ( t h i s work i s a w e l l w r i t t e n co n t i n u a tion of [A31 de a ling w ith se ve ra l f u r t h e r t o p i c s of l i n e a r al g eb r a over n e a r f i e l d s ) . ( 1 ) A p a i r (F,V) i s c a l l e d F-group ( c f . [A31 ,[MI) i f t h e following four c o n d i t i o n s hold: (F1) (V,+) i s a group and ( F , , ) a monoid.
(F2) (F3) (F4)
F o p e r a t e s on V, i . e . (aB) . x = a(f3.x) and 1 .x = x and t h e r e e x i s t 0,-1 E F such t h a t 0.x = 0 , (-11.x = -x and furthermore a (x+y)= ax+cly. F* = F { O } i s a ( m u l t i p l i c a t i v e ) group. Cancellation: ax = Bx implies x = 0 o r a = .
370
J. Pfalzgrrsf
These conditions imply t h e following: (V,+) i s a b e l i a n ;
a.Ov
=
Ov ; a(-x) = - a x ; ( a ( - 1 ) ) x = ( ( - 1 ) a ) x = -ax ;
F* a c t s l i k e a group of automorphisms on ( V , + ) . The geometric spaces A(F,V) = (V,U
(1)
,I]
) i n [A31 p.295 ( f o r more d e t a i l s
see J.Andr6, Affine Geometrien uber Fastkorpem. Mitt.Math.Sem.Giefien x ( 1 9 7 5 ) ) , a r e defined by L := {Fx + y 1 x E V 1 { 0 } , y E V } a s set of lines and x U y = x = x + F(y-x) and p a r a l l e l i s m x 0 y I( x ' U y ' , i f f F(y-x) In terms of t h e <,>-model we can d e s c r i b e A(F,V) as follows:
F(y-x)
<,>
+
: Vz
f
R
,
R := {F*.x
and < x , y > : = 0 , x Note:
=
I
x
[O}}
V \
E
{ O } ; Cx,y>:= F*(y-x)
,
F(y'-x').
=
if x # y
y.
obviously F*v
=
F*w, i f f [ t h e r e e x i s t s a f 0: v = a w l
,
iff
Fv
=
Fw.
Since F* a c t s like automorphisms on ( V , + ) we can d e f i n e t h e semidirect product V
A
y(VH F* ,V).
F* and t h e SPS
Now w e s t a t e
A(F,V) = y(VMF*,V)
(3) Proof:
I t i s r e a d i l y seen t h a t < x , y > = < u , v > i s equivalent t o ( V M F * ) ( x , y )
=
( V d F * ) ( u , v ) , s i n c e the l a t t e r equation means [ t h e r e e x i s t s [ w , ~ ]E V M F * : ( x , y ) = [ w , a l ( u , v ) l , i f f [ x = w + clu , y a(v-u)] , i f f
F*(y-x) = F*(v-u) , i f f
=
w
+
a v l , i f f [y-x = av - au
=
<x,y> =
Thus t h e p a r a l l e l s t r u c t u r e s of both spaces are e q u a l , hence ( 3 ) h o l d s . Remark
2.8
S(F,U),
The group spaces
F a n e a r f i e l d , U*
<
F*
,U
= U * u
[ O ] , a s mentioned
i n [A41 Prop.5.3 ( t h e y were o r i g i n a l l y s t u d i e d by K.P. Rofiler i n h i s Diploma r b e i t , Saarbriicken 1985) t u r n out t o be SPS, t o o . Namely with respect t o t h e semidirect product
FWU*.
With a f f i n e a c t i o n s of semidirect products one can make e x p l i c i t c o n s t r u c t i o n s of spaces with p r e s c r i b e d p r o p e r t i e s ( e x p l o i t i n g t h e knowledge of t h e 1-point s t a b i l i z e r s ) . This p o s s i b i l i t y w i l l be i n d i c a t e d i n t h e next s e c t i o n . 3.
A CONSTRUCTION WITH FREE GROUPS
L e t Fn = F ( a l ,..., an) be t h e f r e e group of rank n ( n 2 2 ) with b a s i s { a,,..., an}
and
'p
t h e automorphism defined by ' p ( a . ) = a i + l
,
1 5 i < n
, 'p(an)
=
a l , then
G : = { i d , ' p , . . . ,(pn-l} < Aut(Fn) is c y c i i c and we o b t a i n t h e SPS
, we r e c a l l from [PZ] 3.5
(3.1)
V(Fn=4G,Fn)
(3.2)
f o r 1 5 t 5 n-1 :
(3.3)
for x
[ 1 ,x,(px,. (3.4)
E
Fn\
. . ,,pn-lx] /x
u yl
'pt
{l} :
has no f i x e d p o i n t s # 1
1 U x = [ I } u ( Fn X I G ) ~ X = [ l } (1,GI.x ~
and t h i s implies t h e g e n e r a l case =
n+l
,
x f y
E
Fn
.
=
On Group Spaces Defined by Semidirect Products of Groups
371
We make the following 3.5
Remark
y ( F n w G,Fn) a r e examples of ( n + l ) - t a c t i c a l desarguesian skewaffine spaces with Pappos-condition ( c f . [PZ]). (1)
( 2 ) By the way we note t h a t these examples serve t o show t h e independence of the parallelogram-condition (Pgm). Indeed, the commutator c r i t e r i o n f o r (Pgm) does not h,,ld ( c f . [P2] 3.6). Satz 2 . 1 of [A21 shows t h a t a f i n i t e imprimitive skewaffine space i s k - t a c t i c a l (i.e. ( x
y(
k f o r a l l l i n e s x u y , x # y) i f , and only i f , it i s a
=
a f f i n e space ( i . e . the Def. 2 . 2
, conditions
flat conditions
(Fl),(F2) f o r subspaces hold, c f . [A21
(Sl),(SZ)).
In the desarguesian case such a space is a group space with respect t o an imprimitive Frobenius group and vice versa ([A21 Satz 4.1,4.2). Subsequently we show t h a t [A21 Satz 2 . 1 is a " f i n i t e space result", it does not remain t r u e , i n general, i f X is an i n f i n i t e point s e t . To see t h i s , we take the t a c t i c a l spaces V(Fn*G,Fn)
and show t h a t ( F l ) does not hold, hence they a r e
not semiaffine. 3.6
Example
F l a t condition ( F l ) says: then
if U
<
X i s a subspace of a space X and L $ U
I U n LI 2 1 .
For V(Fn%IG,Fn) we s h a l l now construct a subspace 1
E
U
<
Fn and x
4U
a line, such
that I(xu 1 ) n U l > 1 : ( 1 ) I f U < Fn is a subgroup on which G a c t s (by r e s t r i c t i o n ) , then we obtain the subgroup U X G < FnMG and the corresponding subspace ( U H G I . 1 , c f . [P21 1 . 7 . Obviously
(2)
For 1
# x
(UHGl.1 E
Fn
= U.
the 1-point s t a b i l i z e r is (F X G ) = {[x.gx-l,gl ! g c G), -1" x- 1 = {x} u {l,x'p(x ),xcp'(x ) , ...,X ' ~ " - ~ ( X - ' ) ] .
hence x u 1 = { x } (Fn>dG)x.l ~ With t h i s notation:
(3) Let U be the subgroup of Fn generated by the elements -1 n- 1 -1 ( s ) = anal . Then G a c t s on U and x := a l a2a3 ,...,'p On the other hand: 1 E U and s E (x 1) n U , hence I ( x U Therefore (Fl) does not hold f o r U and V(Fn'>OG,Fn) i s not
u
s : = ala;'
4 U.
, cp(s)
1) n U [ b 2. semiaffine.
Another application of semidirect product spaces i s presented i n t h e next s e c t ion.
=
J. Pfalzgraf
372
4.
TWO GIDETRIC INVARIANTS FOR QUADRATIC FORMS
F i r s t we f i x t h e g e n e r a l n o t a t i o n s . Let A be a commutative r i n g with 1 , M an A-module and q : M form ( c f . [ B ] ) .
-f
A
a quadratic
For a q u a d r a t i c module (M,q) we can d e f i n e t h e following
geometric space (X,<,> ,R) a s a geometric i n v a r i a n t : 9 q(x-y) f o r x # y and X := M , R : = A u {m} , m (4.1) A , < x 7 y > q:= < x , y > := m , x = y. 9 Let O(M,q) o r O(q), f o r s h o r t , denote t h e orthogonal group c o n s i s t i n g of a l l
4
isometries u : M 4.2
+
M ( i . e . u i s a l i n e a r isomorphism with
q(ux) = q ( x ) ) .
Remark
(1) Obviously O(q) i s t h e group of a l l d i l a t a t i o n s of t h e geometric space ( X , < , > q ) which are l i n e a r isomorphisms. (2) I f we had defined < x , y > : = q(x-y) f o r x , y E M , then it i s r e a d i l y 9 seen t h a t q is a n i s o t r o p i c i f , and only i f , (LO) holds f o r (X,<,> 1 . 9 With r e s p e c t t o t h e n a t u r a l a c t i o n of O(M,q) on M we o b t a i n t h e SPS (4.3) V(MMO(q),M) a s another geometric i n v a r i a n t of t h e q u a d r a t i c module (M,q).
Subsequently w e show f o r q u a d r a t i c forms over f i e l d s of c h a r a c t e r i s t i c not 2 t h a t Witt's c a n c e l l a t i o n theorem can be expressed by t h e e q u a l i t y r e l a t i o n o f both geometric i n v a r i a n t s f o r a l l q u a d r a t i c forms over t h a t f i e l d . In [PZ] 3 . 9 , 3.10 t h e following i s mentioned without proof. 4 . 4 Notations ( 1 ) Let A = K be a f i e l d of c h a r a c t e r i s t i c # 2 . X = M a f i n i t e dimensional v e c t o r space over K . We s h a l l r e s t r i c t our c o n s i d e r a t i o n s t o nondegenerate q u a d r a t i c forms q : X cancelled).
+
K ( n o t e t h a t r a d i c a l s of q u a d r a t i c forms always can be
e say "The c a n c e l l a t i o n theorem holds over K" i f t h e following i s t r u e (2) W f o r q u a d r a t i c forms q , p 1 , p 2 over K: q I p1 = q 1- p2 implies p 1 1 pz . 4.5 Theorem With t h e above n o t a t i o n s t h e following is equivalent (1) (2)
The c a n c e l l a t i o n theorem holds over K For a l l q u a d r a t i c forms (X,q) over K t h e geometric space ( X , < , > 9 is equal t o t h e semidirect product space v(X -24 O(q) , X I .
Proof: ( 1 ) + ( 2 ) : Let (X,q) be a q u a d r a t i c form. We s h a l l show t h a t t h e parallel classes of both geometric spaces are i d e n t i c a l . This i s t r i v i a l f o r
On Group Spaces Defined by Semidirect Products of Groups
3 73
diagonal elements ( x , x ) . Therefore l e t x # y, then obviously (X>aO(q)).(x,y) E = { ( u , v ) 1
, we s e t w:= u-v (hence w # 0) and <x,Y'q then q(w) = q ( z ) . Now we show the existence of a u E O(q) such t h a t
reverse inclusion: l e t (u,v)
z : = x-y , u ( z ) = w.
E
P
i ) q(w)=q(z)#0 : f o r a:= q(w) = q(z) the two nondegenerate subspaces (K.z,(a))
and (K.w,(a)) of X lead t o orthogonal decompositions of (X,q): (K.z I Xl,(a) 1 q l ) and (K.w I X2,(a) I q2) with q1,q2 nondegenerate. From the isometry (a) I q1 = ( a ) I q2 we obtain by the cancellation theorem an isometry 'p
: q1 = q 2 . Let uo : K.z
K.z
I
X1 = K.w
+ K.w be defined by uo(z):= w, then u := uo X 2 is an element of O(X,q) such t h a t u ( z ) = w.
I 'p
:
ii) q(w)=q(z)=0 : i t follows t h a t dim(X) > 1 and there a r e two hyperbolic
planes H1 = (K.z @ KI. z l , h l ) , H2 = (K.w 8 K.w',h2) belonging t o z,w respectively ( c f . [L] chI 3.2,3.4), with h l ( z ) = h l ( z l ) = 0, Bh ( z , z ' ) = 1 , Bhl being the 1 corresponding b i l i n e a r form of hl (similar with h 2 ) . In the case dim(X) = 2 , i . e . X = H 1 = H 2 there e x i s t s obviously the isometry u ( z ) : = w , u ( z ' ) : = w'. I f dim(X) > 2 , then (X,q) = H1 I (X1,ql) and we can apply lemma 4 . 2 of [B] chIII 54 t o H2 c (X,q), implying the existence of an isometry u ~ 0 ( q )such t h a t ~ ( z )= w (and ~ ( z ' )= w ' ) . Altogether we have proved the existence of a u E O(q) with a(z) = w and w e continue the proof: taking such an isometry u the following holds f o r [v- u y, u] (2)3(1):
E
X 9 O(q): [v-uy,u]. (x,y)= (v+u(x-y) ,v)=(u,v).
Let q,p1,p2 be (nondegenerate) quadratic forms, q = ( a l ) i
...I
(an),
a . # 0 , such t h a t q 1 p1 = q I p2 . By induction we may r e s t r i c t ourselves t o 3 the case n = 1 ( c f . a l s o the remark i n [L] ch1,thm 4 . 2 , s t e p 3 and 4 . 7 ) . Therefore we s h a l l prove f o r 0 # a E K: ( a ) I p1 = ( a ) I p2 implies p1 = p2. Now let ' p : (K.x,(a)) I (Xl,pl) = (K.x,(a)) I (X2,p2) be an isometry. We s e t , y := ~ ( x ) then , q(y) = a = q ( x ) , hence
p
a r e equal, therefore [ t , u ]
9 [t,o].(O,x) which implies t
'p
-1
u : K.x
Cf. a l s o 4.6
I
X2 = K.x
I
X1
E
X >QO(q) e x i s t s such t h a t (0,)')
=
= 0 and u(x) = y = 'p(x). For the resulting isometry we obtain 'p-'o(x) = x and f i n a l l y (X2,p2) = ( X l , p l ) .
[ B l c h I I I 5 4 , Cor. ( 4 . 3 ) .
Remark
The foregoing theorem suggests an a l t e r n a t i v e formulation of cancellation in terms of the two geometric invariants which might be convenient f o r more general s i t u a t i o n s .
3 74
J. Pfalzgraf
5. REMARKS ON GROUP EXTENSIONS
We close with some final remarks on group extensions (5.1) 1 -N-E-tG+l The classification of group extensions involves the first three cohomology functors Hi(G,-), i = 1,2,3 (cf. e.g. [Br] chIV). Let us restrict to the case N = (A,+) a G-module and (5.2) O+A+E+G+l an extension which gives rise to the given G-action (cf. [Brl loc.cit.1, then the second cohomology group HZ(G,A) describes the equivalence classes of extensions (5.2). Such an extension is split if, and only if, E is the semidirect product A-G. The classification of all conjugacy classes of splitting homomorphisms s:G E involves H 1(G,A) = Der(G,A)/Ider(G,A). From a geometric point of view it would be natural to generalize SPSs in passing to arbitrary group extensions. In the case of (5.2) we have the group spaces V(G,A), V(E,A), E acting by conjugation (this is compatible with the original action of G on A). J.Andrk suggested to express the splitting of an extension in geometric terms. This finallv leads to characterize geometrically when a group space is a SPS. Since in the characterization of split extensions derivations and inner derivations play an essential role, it might be very interesting to study these objects from the geometric point of view. Moreover, these observations together with results on simplicia1 structures of geometric spaces give a motivation to try to introduce (co-)homological methods for our geometric spaces. +
REFERENCES ( ,159-168 Andrk ,J.,Zur Geometrie der Frobeniusgruppen. Math.2 .=1977) AndrC ,J.,Eine geometrische Kennzeichnung imprimitiver Frobeniusgruppen. 1981) ,120-135 Abhandlungen Math.Sem.Univ.Hamburg Andr6,J. ,Lineare Algebra uber Fastkorpern.Math.Z.E(1974) ,295-313 Andr6,J.,Noncommutative geometry, nearrings and nearfields. Proceedings of a Conference on Nearrings and Nearfields, Tubingen 1985. Baeza,R.,Quadratic Forms Over Semilocal Rings. Springer Lecture Notes in Mathematics 655, Springer Verlag 1978 Brown,K.S.,Cohomology of Groups. Graduate Text in Math. 87, Springer Verlag 1982 Lam,T.Y.,The Algebraic Theory of Quadratic Forms. W.A.Benjmin, 1973 Wller,K.,Lineare Algebra uber Fastkorpern.Diplomarbeit,Universitat des Saarlandes, Saarbriicken 1984 Pfalzgraf,J.,On a model for noncommutative geometric spaces. J.Geometry g(1985) ,147-163 Pfalzgraf,J.,On geometries associated with group operations. Geometriae Dedicata 2J( 1986) ,193-203
x(
Annals of Discrete Mathematics 37 (1988) 375-376 0 Elsevier Science Publishers B.V. (North-Holland)
375
ON PERMUTATION PROPERTIES FOR FINITELY GENERATED SEMIGROUPS
Giuseppe P i r i l l o I.A.G.A. - I.A.M.I. C o n s i g l i o Nazionale d e l l e Ricerche
Viale Morgagni 67/A Firenze ( I t a l i a )
Summary. we p r e s e n t a f i n i t e l y g e n e r a t e d semigroup which does not have t h e p r o p e r t y
P
d e f i n e d by R e s t i v o and Reutenauer, b u t s a -
t i s f i e s a weaker permutation p r o p e r t y
11
I n t h i s p a p e r , we improve a r e s u l t of
1,
P*.
t o which we r e f e r f o r d e f i n i t i o n s of
p r o p e r t i e s P n , P*, P and P* concerning semigroups. n More p r e c i s e l y , we prove t h a t P i s s t r o n g e r t h a n P*, even upon r e s t r i c t i o n t o f i n i t e l y g e n e r a t e d semigroups.
I n f a c t , we have t h e f o l l o w i n g p r o p o s i t i o n .
P r o p o s i t i o n 1. There e x i s t s a f i n i t e l y g e n e r a t e d semigroup which has t h e property
-
5
and does n o t have t h e p r o p e r t y
P*
Proof. _ _ Let
a
semigroup
and
Let t h e s u b s e t
/
The s u b s e t
I
be two l e t t e r s and
( r e s p e c t i v e l y , f r e e monoid
A+
I = { W 6 A+
b
W
I
of
= U vu2 ;
of A+
A={a, b}
t h e a l p h a b e t of t h e
free
A*).
be d e f i n e d as follows
A+
1
P.
x i y
u l , u2 ~ A * ; v = b a b a ; x L y , l ; i L 1 1
i s an i d e a l of
and t h e Rees q u o t i e n t
A+
S = A+ / I
has t h e r e q u i r e d p r o p e r t i e s . I n f a c t , we can show:
1) S
has t h e p r o p e r t y P
2 ) f o r each 1. Let
n
1. 2 ,
u 1 ' u 2 , u ~,u 4 and u
i) If there e x i s t s a l e t t e r ces
i
and
j
t h e words
* 5;
S does n o t have t h e p r o p e r t y P
5
.
be a r b i t r a r y words of A+.
c €{a, b}
u
1
and
such t h a t f o r a s u i t a b l e p a i r of i n d i u
j
Without l o s s of g e n e r a l i t y , we can suppose ul. and
n
a r e powers
of
c , then
i = 1 and j = 2 .
u2. u3. u 4 . u5
US t h e words
G. Pirillo
376
u 2 . u l . u3. u 4 . u 5 coincide.
Hence, i n t h i s c a s e t h e r e s u l t f o l l o w s .
ii) I f a l e t t e r with t h e p r o p e r t y considered i n i ) does not e x i s t , t h e n f o u r
words among them c o n t a i n
ul, u2, u3, u 4 a
and
u
and
5
c o n t a i n t h e l e t t e r a and t h r e e o f
b.
But i n t h i s c a s e , from t h e d e f i n i t i o n of tations
f
and
g
of
{ l , 2, 3 , 4 , 5)
I , t h e r e a r e two d i f f e r e n t permu-
such t h a t
u f ( 1 ) . u f ( 2 ) . Uf ( 3 ) . uf ( 4 ) . U f (5) and
u g ( l ) . ug(2). ug(3)- u9(4)- ug(5) b e l o n g t o I , i . e . t h e y r e p r e s e n t t h e same e l e m e n t of t h e R e e s q u o t i e n t S = A+ /I.
2. C o n s i d e r t h e f o l l o w i n g n - t u p l e ab, ab
2
,
...,
abn
of words o f A+.
I t i s easy t o see t h a t
2 (ab) (ab )
.... ( a b n ) Z I
( a b f (I) ) ( a E ( 2 ))
.. .. ( a b f ( n ) )
and, f o r every f # i d ,
So, f o r e a c h n > 2 , S d o e s n o t have t h e p r o p e r t y
6
P
n
I
.
REFERENCE
1. G. PIRILLO, On p e r m u t a t i o n p r o p e r t i e s f o r s e m i g r o u p s , P r o c e e d i n g s of t h e I n t e r n a t i o n a l Conference on Group Theory, Bressanone - B r i x e n , May 1986 ( t o appear i n Springer Lecture N o t e s ) .
Annals of Discrete Mathematics 37 (1988) 377-384 0 Elsevier Science Publishers B.V. (North-Holland)
ON k-SETS OF TYPE (O,m,%)
377
IN S 21,
Rita PROCESI CIAMPI
-
q
WITH THREE EXTERIOR HYPERPLANES
Rosaria ROTA
Dipartimento di Matematica, UniversitP degli Studi di Roma "La Sapienza", Piazzale Aldo Moro 5, 00185 Roma, Italia
It is well known that the study of k-sets of type (m,n) in an affine space is equivalent to the study of kcsets of type (O,m,n) in a projective space such that their exterior lines are only those belonging to a fixed hyperplane. This consideration suggested us the problem of studying the k-sets of type (O,m,n) such that the exterior lines are exactly those belonging to two [l] or three hyperplanes. In this paper we present the results concerning the case of three exterior hyperplanes either belonging or not belonging to the same pencil. In the first case necessary conditions are stated for the existence of such k-sets. In the second case we obtain a non existence theorem for r25.
1. INTRODUCTION with three exterior hyperplanes S1 Let K be a k-set of type (0,m,n) in S r-1 ' r>9 i S2 , S:-l not belonging to the same pencil and let S t - 2 = Sp-ln$-l , i,j,
.
r-1
h E C 1,2,3} and not in S
; such subspaces intersect in a
i
^sr-3
,K
-
For an S'
r-2
containing S
r-3
with exactly one n S k 2 is a k'-set of type (0,m,n) in S ' r-1 r-2 exterior hyperplane. Furthermore for S" containing S and different from r-1 r-2 Si and S' , KnSi-l is a k"-set of type (0,m,n) in S"r-1 with two exterior r-1 r-1 hyperplanes. Therefore [l] : 1.1. There does not exist in S , r25,a k-set of type (O,m,%) r>4 exterior hyperplanes not belonging to the same pencil.
with three
-..IS1r-I' Sr-I' 2 S3 is a k-set of type (O,q-d,q-l,q). r,q r-1 If we now consider the case in which the three exterior hyperplanes belong to
We first observe that K
the same pencil, K
= S
this case, for any S '
=
S
ISr-]
' Sr-1
'Sr-l
r>4 containing the
r-1
. S
1 is of type (O,q-2,q).
r-2
Furthermore in
intersection of the three exterior
hyperplanes and different from them, K n S ' is a k'-set of type (O,m,n) with r-1 exactly one exterior hyperplane, thus, for r > 3 , [ 3 ] only three cases are possible for (m,n) : (0,I), (q-l,q) and ( ( q - i q l / Z , (q+dql/d) q odd square, therefore: O
Research supported by G.N.S.A.G.A.
-
C.N.R.
R. Procesi Ciampi and R . Rota
378
1.2. If a k-set of type (U,m,n)
with three exterior hyperplanes belonging to a
-
-
,r>3, then q must be an odd square and rn=fq-Jq)/Z,n=(q+Jq)/2.
pencil exists in S
TJ9
To study the cases r = 3 , r=4, we will need the results concerning the case r=2.
k-SETS OF TYPE (O,m,n)
2.
IN
TI
4
L e t K be a k-set of type (O,rn,n) in
with exactly three exterior lines not 4 belonging to a pencil. A s in L21 we can obtain for k the equation: TI
o
k ' - k L ~ + i q + ~in+m-lI i I+rnn~q'+q-~i =
(2.1)
We must observe thatK
n..{r r } is a k-set of type (O,q-Z,q-I), k=q'-2q+l q 17rz>3 and this is the only possible case for n-m=I . In fact for 7FQ-2, n=q-:, k = =
q'-2q+I
=
i s the only acceptable solution of ( 2 . 1 )
( 2 . 1 ) we obtain A = 4(n-1) [n(q+3)-q(q+l)/+I
if and only if n l q ( q + l ) / ( q + 3 )
,
ing t o a pencil and n-m=l, thenK =
in TI
Li
7~
1
J
,
3
for P.
in
with three exterior lines not belong-
9
{rl,r2,r3 I .
represent the numbers of um,un,uA,u',w n m ,wn,vm,vn
m-secants and n-secants respectively for: = r . n r . , R#P.=rinrh
. Otherwise, for m=n-1
wich is greater or equal to zero
and thus n=q-1. Therefore:
2.1. If K is a k-set o f type (O,m,n)
We shall now recall that [l]
,
,
Q E ~ -bY,rl,r2,r3},
4 i=1,2,3 , and for TEK
.
for REr. and RfP = h Since the values of such
numbers do not change in this case, we have:
, k must satisfy 1 2 . 2 ) 9 and m ( q t l ) ~ k ~ ? l i n ( n ( q - l ) , n ( q + l / - q;) furthermore m=s(n-m) , n = I s + l )(n-m) and 2.2. For the existence of a k-set of type (U,m,n)
n-m divides q and k
in
TI
.
2.3. If .Y is a k-set of type (O,m,n)
in
TI
ing to a pencil and K is prime then K =
with three exterior lines not belong-
4 7~
~
4
h1,r2,r3}
.
We must now observe that: u
=
0
implies
k = m(q+l) , k 1
2
= n ( q + l l - q , n = q ( q + l ) / ( q + 3 ) ; such value
of n is an integer only for q = 3, from which rn
w rn
=
0
implies
k
v
=
0
implies
q = 3
m
1
= n(q-I)
,n
, k 2 = miq+l)-q+2n , m = 2n-q 2
,m =
1 which implies vm
=
=
1
,n
=
. 1 (impossible).
From these and the analogues for the other values rl] we obtain:
2
On k-Sets of Type fO,m,nl 2 . 4 . Through any P#Ph, h=1,2,3
379
, at least one m-secant and one n-secant must
pass. Through Ph at least one n-secant must pass, but not necessarily one msecant. In fact w =O implies m=Zn-q, k = n(q-I), m 1 The equality m=2n-q
k
2
=
m(q+2).
, m=q-Za
can also be expressed as n=q-a
, a ( q , and
we have: 2.5. The values m=q-2a , n=q-a lead to integers solutions of ( 2 . 1 ) for any a dividing
4.
Furthermore from 2 . 4 .
the following is also true:
2.6. If K is a k-set of type (O,m,q-2)
then m=q-2 and K=II \ I r r r 1 1' 2' 3 q
.
Let now n=q-2 and then liu 2 3 . The first case u =I is easily excluded. For 9-2 9-2 u =2 then k =2q+mq-m-4 and k =q2-4q+Zm+2 , thus: 9-2 1 2 m'iq-11-m(2q~-7q+7)+q3-6q'+~oq-4 =o, of which the only possible integer solution is m=q-4 and u
2=2 give u s k=(q-4)(q+2/
9-
2.7. If a k-set of type (O,m,q-2)
. The values m=q-4, n=q-2 . Thus:
=(q-4)/2 which implies u 9-2 with u
k=(q-4)(q+2l.
=2 exists in II
4
q-2
, then q=8, m=4 and
.
For u and k =q2-5q+3m+4 With the same tecnique as =3 , then k =(m+3)(q-2) q-2 1 2 before, together with the condition w 20 , we obtain:
m
2.8. If a k-set of type (O,m,q-2)
with u =3 exists in II then only two cases 9-2 9 are possible: 1) m=q-4, k=(q-lI(q-Z) or 2) q=10 , m=q-4 , k=(q+/(q+Z/
.
The following proposition gives the only possible example f o r case 1) in 2.8. 2.9. If K is a k-set of type (O,q-4,q-2)
where
r
.
and k=(q-Zl(q-2)
thenK=n \{T,rl,r2,r3} 4 is an oval passing through the three intersecting points of r l,r2,r3 and
vice versa.
Proof. Let r"={Q, is an oval in
II
. 9
,...,Qq-?}
, Qi&KU{r1,r2,r3). We prove that r=r"u{P ,P ,P 1 1
Let us consider a line r different from r l,r2,r3
2
3
. If P . E ~ 1
then r is a (q-Z)-secant since w =0, thus r must contain exactly one Q.EI'' . q-4 J If P.dr and r is a (q-Z)-secant, then r is exterior to T ; otherwise rnr={Q.,Q.}. Thus
r
1
is an oval; the vice versa is obvious.
Let u s now consider some special cases. If.n-m=2
, thus 7'9
discriminant of (2.I) is A=4m2(q+3)-4m(q2-q41+(q+2)'
and q even, the
and A 2 0 if and only if
J
R.Procesi Ciampi and R.Rota
3 80
m
.
B=(q’-q-6+Jq4-3q3-18q2-4q+24)/2(q+3)
2.10. If.? is a k-set of type (O,m,n)
and
It is easy to verify that A C 2 , B>q+;thus: and n-m=2,
then m=q-4, n=q-2.
Similarly for n-m=3, q>8 : 2 . 1 1 . If K is a k-set of type (O,m,n)
and n-m=3 , then m=q+,
n=q-3.
In general for n-m=a we obtain A20 if and only if m
,
.
~=(q‘+q-aq-3a+Jq4+q3-2qZ,2q3-4a~q~-a~q~a~)/2(q+3)
For a=4 and q > 2 3 is A<4; thus: 2.12.
If K is a k-set of type (O,m,n)
and n-m=4, q > 2 3 , then m,B
.
The cases q=16, q=20 give respectively the pairs ( 4 , 8 ) or (8,lZ) and (12,16). At this point let u s consider the case in which the exterior lines belong to a pencil. We must observe that the (2.1) does not change in this case as the formulas for u , u ,u’,u’,v ,v , while it results: w =( n ( q - 2 ) - k) / ( n - m ) and w m n m n m n m =(k-m(q-Z))/(n-m) Proposition 2 . 4 . thus becomes:
=
.
2.13. Through any P#Ph
,
h=1,2,3
,
at least one m-secant and one n-secant must
pass. Through Ph at least one n-secant must pass, but not necessarily one msecant. In fact w =O implies k = n ( q - 2 ) , k =3n+m(q+l)-q, m 1 2
m=/3nq+n-q2+2q)/2q
.
Because of the different formula for w we also obtain a different limitation m for k ; namely m ( q + l ) ~ k ~ m i n ( n ( q - 2 ) , n ( q + l ) - q thus ), proposition 2.5. becomes: 2 . 1 4 . The values m=q-Za, n=q-a give kq(q-2)
for any a dividing q
as only possible solution of (2.1),
.
Following the previous results we obtain: 2.15. I f K is a k-set of type (O,m,q)
with three exterior lines belonging to a
pencil, then m=q-2 and K = T I ,{rl,r2,r3}. 9 2 . 1 6 . There does not exist any k-set of type (O,m,q-l)
with three exterior lines
belonging to a pencil. Proof. Let PET \&,rl,r2,r31
4
and PEr which i s a Iq-1)-secant;
then r n r n r = 1 2 3 k =q2-2q+m and thus
=1, from which we have k =(m+l)q-I, 1 2 q-1 m2q-m(2q2-4q+3)+lq3-3q2+2ql=0 with non acceptable solutions. Similarly:
=Ssr. This implies u
On k-Sets of Type (0,m.n)
381
There are no k-sets of type (O,m,q-2) with three exterior lines belonging
2.17.
to a pencil.
From these results and those obtained in the case of the exterior lines not belonging to a pencil we have: 2.18. I f a non trivial k-set of type (O,m,n) with three exterior lines belong-
ing to a pencil exists, then n-m>2. For n-m=3 must be m=q-6, n=q-3. there are no non trivial k-sets of type (O,m,n) 4 with three exterior lines belonging to a pencil. 2.19.
3.
I f q is a prime, in
i l
k-SETS OF TYPE (O,m,n) I N S
Let K be a k-set o f type (O,rn,n)
3,q
AND S
4,q
in S 7
'J4
with three exterior planes
i11,i12
,i13
not belonging to a pencil. I f we consider the points o f K distributed over the
.
.
planes of the pencil through an n-secant t intersecting the line rh= n l n n J , we obtain the following relation: IKI=k+Nk +(q-N)k2-/q, where k k are the s o l u 1 I' 2 tions of ( 2 . 1 1 , ;=I?[ , where I?=Kn;i,and ii=tUrh , moreover N is the number of those planes through t intersectingli in a kl-set. Similarly for an rn-secant in ii
, we obtain: IK [=i+Mk 1+(q-M)k2m q . Thus N-M=(n-mlq/& where
-
A is the discrimi-
h nant of (2.1). Since n-m divides n,m then N-M=q/JA',A'=A/(n-mlL and, as q=p , A ' = p Z 2 , l
.
The case A ' = l is easily excluded. We will also need the relation
, where
A
is the discriminant of ( 9 . 1 ) in [l].
results of 1 2 . and those obtained in
ill
Summarizing the
we get:
3.1. The necessary conditions for the existence in S
o f a k-set o f type 339 (U,m,n) with three exterior planes not belonging to a pencil are: n-rn divides
q,m,n,k
, 5<m
. Furthermore if n-m=3
then rn=q+, n=q-3. If q is
a prime there are no k-sets of type (O,m,n) with three exterior planes not belonging to a pencil in S
3,q
.
Moreover the number k=IKI must satisfy: (3.1)
(n+m-l)]+mn(q4+q3-q2-2q+l) k2-k[l+(q2+q+l)
=0
.
-
We now consider the following four cases: (a) q odd square, m=(q-/q)/2, n= h =(q+/T)/Z; ( 8 ) q=zh; (Y) q=p2, p # 2 ; ( 6 ) q=p p > 2 , h>2
-
(a) Solving ( 2 . 1 ) , ( 3 . 1 ) and ( 3 . 1 ) of [l],
i = ( q z & q d w ) / 2, k=(q3f_i;;JZq3+qL-3q+l)/2
Thus the following must be true: 2q-I=U2
we get:
, k '=(q+,qi-)/2
.
, 3q-2=b2 , 2q3+qL-3q+I=cL, which
R . Procesi Ciampi and R . Rota
382
proves to be impossible, then: -
-.
,q
3.2. In S
odd square, there are no k-sets of type ( O , ( q - J q l / 2 , f q + J ~ ) / 2 )
3,q with three exterior planes not belonging to a pencil.
(13)
Since A ' = ~ + 4 ~ ( s + l ) = 2 ~, ~and
a
is odd (see Ill),
it results:
h
3.3. In S , q=2 , there are no k-sets of type (O,m,n) 334 planes not belonging to a pencil.
with three exterior
az
(2~+1)+2IZs+l)~-2+p~-7, ,
(y) In this case a ' = ( p ' + l ) p L + ( 2 s + l ) 2 ( p ' + l ) - 2 p i p ' + l /
with z=1 or 2=2. The value z=1 is easily excluded; for 2=2 we obtain:
( 2 ~ + 1 / ~ ( p ~ + 3 ) - 2 p ( p (2s+1)+2p'-2 ~+l) t/pG-3pL+5)/(p3+3)
.
=
D
,
from which 2 s + ! = f p ( p L + 1 ) +
Since ( p ' - 1 ) L < p 6 - 3 p L ~ < ( p ' ) L , we get:
, q = p L , p # 2 , there are no k-sets of type (O,m,n) 3,q rior planes not belonging to a pencil.
3.4. In S
with three exte-
(6) Finally in this case the study of A ' leads to the condition: p -P 3h+P 2h-2a P 2h+P 2z+k<0, where l < z i h and n-m=pa.
4k-2a
+p
3h-2a
-
Thus:
3.5. If a k-set of type (U,m,n) pencil exists in S 3,9
,
with three exterior planes not belonging to a 22 q=ph, k # 2 , then a W 2 , 2,h-a In-m=pa, A'=p ) .
Since proposition 1.1. denies the existence of k-sets of the aforesaid type for
r>4 , we only have to consider r=4. In this case summarizing the previous results we obtain:
3.6. In
a k-set of type (U,m,n) with three hyperplanes not belonging to a 4,q h pencil can not exist if: q is prime, q=2 , q=p2, q odd square and m = ( q - / q ) / 2 , h n=(q+/q)/Z Furthermore if q=p , h#2 , a necessary condition for the existence 22 1. of such a k-set is: a < h / 2 , z>h-a fn-m=pa,A'=p S
.
We can now consider the case in which r=3 and the three exterior planes belong to a pencil. In this case k=lKl must satisfy:
k'-k
(3 . 2 )
[Z
+ I q'+qtl ) (n+m-l
)I tmn (q'+q
-4' - 2 q ) =U
.
Furthermore the general condition for the existence of such k-sets are the f o l lowing:
, q prime, the only possible k-set of type (O,m,n) with three ex3,q [r1,n2,n3}. Moreover if a non terior planes belonging to a pencil is K = S 3,q k trivial k-set of the aforesaid type exists in S , q=p , then n5q-3, n-m divides 394
3.7.
In S
'
On k-Sets of Type (0.m.n) m,n,q
3 83
.
and n-m#1,2
Since the proofs of 3 . 3 . , 3 . 4 . and 3.5. are based on equation ( 2 . 1 ) , the analogous propositions are still true when the three planes pass through a line r.
-
-.
If q is an odd square m=(q-Jq)/Z, +qJ2qz+q-2)/2
n=(q+Jq)/2,
from
and (q3+qJ2q2+q-2)/2=dq+(q-2) (q2-q)/2,
planes through r intersecting K in a fq2+q)/2-set
where
d
is the number of
(see r31). Therefore: 7q2-
.
from which q=(12d+23~J3Zd2+88d+1)/14 Moreover, if rln,
-q(12d+13)+4d2+8dt6=
0,
then n ( ) K is a &set
of type (O,(q-Jql/Z,fq+Jqi/d)
-
-
-__
belonging to a pencil, thus ;=(q2+JqJ3q-2)/2. 3 . 8 . If
(3.21 we have k=(q3+
with three exterior lines
From these observations follows:
I< is a k-set of type (O,(q-JSi/Z,fq+J~)/Z)
in S , q odd square, with 3,9 three exterior planes passing through a line r, the following integers must be perfect squares: 2q +q-2,
3q-2,
32d +88d+I, where d is the number of planes
through r intersecting K in a (q2+q,J/2-set. Finally, for r > 3 3.9.
, proposition 1.2.
can be extended as follows:
If K is a k-set of type (0,(q-&)/2,
planes belonging to a pencil in S for any t such that 2
Proof. For every S
.
PJ
,
4
fq+JG)/2) with three exterior hyper-
t-1 +q-2=aL t’
r23, q odd square, then Zq
s
I
in S non containing k =kf\S must satisfy the r,q r-2’ t $9 9 , whose discriminant must then be a equation: k2-k qt+qt-’/q(qt’’-2qt-’-q+2)=0 t,q
t
t
perfect square.
REFERENCES
111 r21
L31
Procesi, R. and Rota, R . , Sui k-insiemi di tipo (O,m,n) di uno spazio di Galois Sr,q, in print. Tallini Scafati, M., Calotte di tipo (m,n) in uno spazio di Galois Sr,q. Rend.Acc.Naz.Lincei (8) 53 1972, pp. 71-81. Tallini Scafati, M., I k-insiemi di tipa (m,n) di uno spazio affine Rend.Mat. (1) 1981, vol. 1, Serie VII, pp. 63-79.
This Page Intentionally Left Blank
385
Annals of Discrete Mathematics 37 (1988) 385-390 0 Elsevier Science Publishers B.V. (North-Holland)
AN ALGORITHM FOR L
S
-
COLOURATIONS
Luigia PUCCIO Dipartimento di Matematica 1-98100 Messina, Italy *
-
Universitg di Messina
An algorithm for L - colourations of undirected graphs is described. This algorithm gua:antees an optimal colouration and the correct value of the s-chromatic number for an arbitrary graph. Moreover, it can be used in case of directed graphs too.
1. INTRODUCTION Let G=(V,E) be an undirected graph and K a mapping of V into a set C whose elements are called colours. K is an L - colouration of G if for any x,y E V, x # y , K(x) = K(y) implies d(x,y) > s,swhere d(x,y) is the distance of y from x in G [lo, 1 7 1 , i.e. the minimum length of all chains from x to y. When s = 1 an L - colouration is the usual colouration. If K ( x ) # K(y) for all x,y E V, then 1 K is called an L m - colouration. Unless otherwise stated, by a graph we always mean a simple, i.e. without loops and multiple edges, connected and undirected graph. The s-chromatic number is defined as the minimum number of colours, y s ( G ) , such that G has an L - colouration. S
The determination of the s-chromatic number and some related parameters has been the subject of a good amount of research papers (see references). It is well known that classical colourations have many applications. The same is true for L - colourations. Since their definition implies the idea of distance betwEen vertices, a graph with an L - colouration can provide an S . interesting model to solve typical problems i n Operations Research, e.g. the location of emergency centres, or medians, and transportation and assignment problems. Let G=(V,E) be a graph; for any s E X , s > 1, and x E V, define Fs(x) = { Y E V : d(x,y) T (x) =
{y
gs(x) =
*
E V : d(x,y)
{X
:
>
s
}
< =
s + l },
V\c(x),
X CV, d i d < s+l } , G
Research supported by the Community of Mediterranean Universities.
L. Puccio
386 where X
G
is the subgraph of G whose vertices art. in X ;
d (G) =
max
1X1
x €gS
,
d (G) is called the s-density of G ; S
As(G) = obviously, d s ( G )
2
Y s ( G ) - ds(G) ;
0.
The most important result is that, for every s > 1 and any h E W , there exists a planar graph such that d (G) = h. Therefore, As(G) : s
sup (
>
1, G t .? ) =
00
,
where 9 is the set of all planar graphs. Fig. 1 shows such a planar graph; the black vertices yield the minimal configuration for s = 2 and k = 2h - I .
In [8] the problem is posed of finding the values for the following parameters: v (h) = min { n S
d s ( h ) = min { n
€
N
€
:
m (h) = min {n E W
N
:
3
G = (V,E) 3 '
3G
=
(V,E) 3'
:
3
G = (V,E)
IVI= n
=>
1 V 1 = n + d (G) 3' I E 1 = n =>
As(G) = h )
=> A
,
( G ) = h}
As(G) = h }
,
,
where s, h are positive integers. The values for those parameters are known just in very few cases. In general, upper bounds are known [13] which might be improved. The facts we just recalled seem to provide a good motivation for the research of an efficient algorithm for such L - colourations. Here we present an algorithm which guarantees both an o&imal L - colouration and the correct value of the s-chromatic number for an arbitrary graph. Furthermore, this algorithm applies also to directed graphs for which L - colourations can be S defined in a similar way [IS].
2.
DESCRIPTION OF THE ALGORITHM
There are many algorithms for the classical colouration. They are based on different procedure such as formulation as a zero-one programing [2] , sequential methods based on vertex ordering [ 141, backtrack methods [ 151 , optimal independent colourings [ 19 ] , formulation as a set covering problem [ 31. Our algorithm is an appropriate combination of the last two ones. In fact, its procedure is a formulation as a set covering problem, where the sets of the coverings are maximal independent 1-chromatic sets, called maximal 1-subgraphs. Recall the following definitions and theorem. An independent set of a graph G=(V,E) is a subset of V such that no two of its vertices are adjacent in G. An independent set is maximal when there is no independent set containing it. THEOREM [ 3 ] - If a graph is r-chromatic then it can be coloured with r (or
A n Algorithm for L,-colourations
387
L. Puccio
388
fewer) colours first colouring with one colour a maximal indepentent set S [GI, 1 Sl[G]>] and so on, until
next colouring with another colour a set S [
-
colourations as we next show. For a graph
G=(V,E) define the graph GS=(Vt,E') as follows. V' = V and for any x,y E V, ( x , y )E E' if and only if dG (x,y) < s + l . Thus in GS=(V',E') there is a link between x and every element in the set $(x).
PROPOSITION - A classical colouration of G'=(V?,E') G=(V,E), and conversely.
is an L - colouration of
Clearly, this result plays a fundamental role in the realization on our algorithm.. We try to find an optimal colouration of G S = ( V ' ,E' ) whit the help of the above mentioned theorem; consequentely, we get an optimal L S colouration for G = ( V , E ) and its s-chromatic number. 2.1. Steps of the algorithm
We just outline the procedure. The program of its implementation is written in FORTRAN 77 and was tested on several graphs. Such directed and undirected graphs were constructed by means of a random generator program [ b ] Details of the programs are available from the author.
.
STEP 1 - INPUT:
n = number of vertices of G=(V,E),
A = incidence matrix of G, s = kind of colouration. STEP 2:
Calculate the incidence matrix B of GS=(V',E'). By definition, GS has no loop. Then , from the classical formula to calculate the reachability matrix it follows
STEP 3: Find all maximal 1-subgraph of G. Consider T (x) €or every x E V=V'. Maximal independent sets are subsets of TS(xj u {x) They are found by depth-first search.
.
STEP 4: Store all maximal 1-subgraphs in an array Q in decreasing order of their numbers of vertices. Such an ordering turns out to be useful to find the optimal colouration. All maximal 1-subgraphs yield a covering of the vertex set V=V'. We have to find a partition with the minimum number of elements since we want an optimal colouration. Thus, the element number of the partition is the s-chromatic number 7, (G)
.
STEP 5:
Assign the first colour to the vertices of the first maximal 1subgraph in 9, i.e. the 1-subgraph with the maximum vertex number.
STEP 6: Consider the sequential next maximal 1-subgraph in the array 9.
An Algorithm for Ls-colourations
389
Three cases may occur: (a) All its vertices are coloured. Return to step 6. (b) No vertex is coloured. Assign to all vertices another colour. Return to step 6. (c) Some vertices are not coloured. Put them in a list L with pointer. Return to step 6. When all maximal 1-subgraphs are investigated, look at the list L. If L is empty, go to step 8. Colour vertices o f list L which are distinct and not coloured. There is a different colour for every not vanished pointer.
STEP 7: a) b)
STEP 8 - OUTPUT:
3.
List vertices of G with their own colours and the schromatic number Ys(G).
COMMENTS
We have the best implementation of the algorithm if the incidence matrices of G=(V,E) and GS=(V' ,E' ) , and the maximal 1-subgraphs storage array are logical. Obviously, the algorithm efficiency depends on the dimension of these matrices. However, if we use the algorithm to improve the first values of the parametres vs(h), ds(h), ms(h),then the incidence matrices of considered graphs are not too big. When the graph G is the model for a problem in operations research its incidence matrix can be very big but,usually, it is a sparse matrix. Therefore, we can use sparse matrix techniques. In the wrost cases, we can -S consider the complement G of GS. Then an L - colouration of G is obtained s -s assign the same colour to adjacent vertices of G If this is the case, we (G) instead o f T (G). consider the set
r
In [18] M
-
.
S
colourations are defined for directed graphs.
Let G=(V,E) be a directed graph and K a mapping of B into a set C of colours. - colouration o f G if for any x,y E V, x # y , K(x) = K(y) implies there is zo path from x to y with length less than s+l. 0 (G) is the schromatic number.
K is an M
In case of directed graphs the incidence matrices are not symmetric. Nonetheless, our algorithm can be used, with no changes, also for M S colourations.
REFERENCES [l] (21
[3]
[41
Antonucci, S., Generalizzazioni del concetto di cromatismo d'un grafo, Boll. Un. Mat. Ital., (5) 15-B (1978) 20-31. Berge, C. Graphes et hypergraphes, (Dunond, Paris, 1970). Christofides, N . , Graph Theory, an algorithmic approach, (Academic Press, 1975). Gallo, G., Pallottino, S., Ruggeri, C. and Storchi, G., Metodi ed algoritmi per la determinazione di cammini minimi, Monografie di Software
390
L. Puccio
Matematico N.29, I.A.C. (1984). Gionfriddo, M., Sulle colorazioni L d'un grafo finito, Boll. Un. Mat. S Ital., (5) 15-A (1978) 444-454. Gionfriddo, M., Su un problema relativo alle colorazioni L, d'un grafo planare e colorazioni L , Riv. Mat. Univ. Parma, (4) 6 (1980)151-160. [71 Gionfriddo, M., Alcuni risultati relativi alle colorazioni Ls d'un grafo, Riv. Mat. Univ. Parma, (4) 6 (1980) 125-133. Gionfriddo, M., Sul parametro A s d'un grafo Ls- colorabile e problemi i8 relativi, Riv. Mat. Univ. Parma, (4) 8 (1982) 1-7. Harary, F., Graph Theory, (Addison-Wesley, 1960). Kramer, F. and Kramer, H., Un problsme de colrolation des sommets d ' u n graphe, C.R. Acad. Science Paris Ser. A 2 8 (1969) 46-48. [ 11-1 IAzzio, A. and Milici, S., Determinazione di v3(i), v3(2), v4(;) in un grafo e condizioni sufficienti per A = 0 , Riv. Mat. Univ. Parma. Marino, M.C. and Puccio, I $ . , Su alcuni parametri associati a colorazioni L,, in un grafo finito non orientato, Le Matematiche, vol. XXXV, fasc. 1-11 (1980) 301-310. [13 Marino, M.C. and Puccio, L., Sul parametro d,(G) d'un grafo planare, Riv. Mat. Univ. Parma, (4) 9 (1983) 9-13. Matula, D.W., Marble, G. and Isaacson, J.D., Graph colouring algirithms, from Graph Theory and Computing, (Ed. Read, R.C., Academic Press, 1972). I15 1 Nijenhuis, A. and Wilf, E., Combinatorial Algorithms, (I1 Ed., Academic Press, 1978). Sedgewick, R., Algorithms, (Addison-Wesley, 1984). Speranza, F., Colorazioni di specie superiore d'un grafo, Boll. Un. Mat. Ital., (4) 12, supp. fasc. 3 (1975) 53-62. Speranza F., Sur les colorations des graphes orientgs, Boll. Un. Mat. Ital., (5) 1 6 - ~(1979) 517-522. 1191 gang, J., ACM 21, (1974), 385.
I
I
39 1
Annals of Discrete Mathematics 37 (1988) 39 1-394 0 Elsevier Science Publishers B.V. (North-Holland)
A BLOCKING SET
IN PG(3,q),
q
2
5
Sandro Rajola Dipartimento di Matematica Ist ituto “G. Cas telnuovo” Universitl degli Studi di Roma “La Sapienza” 1-00185 Rome, Italy
z
A blocking set i s constructed in PG(3,q), q 5, which turns out to be the first example of a blocking set when q = 5.
1. INTRODUCTION A blocking set S in PG(3,q), GF(q),
the 3-dimensional projective space over the field
is a set of points meeting all lines and containing none.
In [2] it was proved that, for any prime power q , there exists an integer b(q)
< b(q), such that blocking sets exist in PG(r,q) provided that r -
whereas r>b(q)
rules out their existence. Furthermore, in [61 it was shown that b(2) =1,b(3)= 2, and b(q)
2
3 f o r any q > 7 . The last result was achieved producing examples of blocking sets in PG(3,q), q 2 7.
=
Therefore, the existence of blocking sets in P G ( 3 , 5 ) was an open problem. Here
> 5. Consequently, b(q) 2 3 , a blocking set is constructed in PG(3,q) for any q q
2
5.
In sect. 3 it is shown that a blocking set exhibited in [ 6 ] is reducible and an irreducible one is constructed starting from it.
2. THE CONSTRUCTION Let 01,02,03, and V be four independent points in PG(3,q), vertices of
a
q
2
5, i.e. the four
thetrahedron T . Take a point O4 on the edge OIV of T , other than
O1 and V, which is possible as q
5. The plane through 0203and 04 meets the
faces 0 0 V and 0 0 V in the lines 0204 and 0304,respectively. Next, take any 1 2 1 3 point A1 on 0102\ {01,02] and this is possible by the previous remark. Similarly, take any point B1 on 0204\ {02,04}. Let Z = AIBl n 0 V and C be a point on 2 1 0 2 V \ {Z,02,V]. Next, take A2 on 0103\ {O ,O } and B2 on 0304\ {O ,O 1. Let 1 3 3 4 T = A2B2 n 0 V and C2 be a point on 0 V \ {T,O ,V}. Finally, choose a point D on 3 3 3 the line 0203 other than O2 and O3 and non-collinear with any of the pairs A1A2, B1B2, and C1C2. Such a choice is possible by the following argument.
S. Rajolu
392
Denote by A ' , B', and C ' the points on 0203 collinear with the pairs A A 1 2' B1B23 and ClC2, respectively. Even if the five points 02, 0 3 , A ' , B', C' are all distinct, at least one point remains on 0203to be taken as D. Denote by E
1' the points on 0 V \ {O ,O ,V}, and n(L,M,N) the plane spanned by the 1 1 4 three independent points L, M, and N; whenever it is the case, the plane is
E2,
...,Eq-2
considered only as the set of its points. Define the set S by
\(O 10 2 u 0203u 0301u U {A , A
1
2
, B , B ,C
1
2
1
olv u
02v
u 03v u
0204
" 0304))
,..., Eq-2 1 .
, C ,D,E1,E2
2
Notice some symmetries in S: the faces 0 0 V and 0 0 V are similarly constructed, 1 2 1 3 as well as 0203V, 020304, and 0 0 0 1 2 3' We claim that S is a blocking set. Firstly, we prove that S contains no line. Since the points of S on the faces V and 0 0 V are distributed in a similar way, it suffices to show that one 1 2 1 3 of them contains no line. A line Q on n ( 0 ,O ,V), (1 # 0102,OIV, 02V, 0204, is 1 2 contained in S iff it passes through A 1, B1, C1, a contradiction as thosepoints
0 0
L # 0203,02V, 03V, belongs to S iff it is on C1, C2 and D which is impossible since such points are not collinear. The same argument proves that neither the face 020304nor the are not collinear. Similarly, a line R on r(02,03,V),
face 0 0 0 contains lines. 2 3 1 Next, any line R in PG(3,q), not on a face of T, meets S at five points at most since R\
S
consists of points on five planes; thus, there is at least one point on
s.
Finally, we show that any line R meets S. This is obviously true for the sides of the triangle 0 0 V and for the lines on E j 1 2 j'
=
1,2,. . . , q - 2, as well as for
0 0 on n ( 0 ,O ,V). On the other hand, by the construction o f S , on any other 2 4 1 2 line of n ( 0 ,O ,V) at most three points lie off S. The same argument applies to
1 2 n(01,03 ,V). Next, we turn to n(02,03,V).
Of course, the lines 0203, 02V and 03V meet S and
the same occurs for the lines on Cl, C2 and D, respectively. Keeping in mind the construction of S , on any other line of n(0 0 ,V) three points at most lie 2' 3 off S. Similarly, all lines on n(02,03,04)and n(02,03,01) meet S. Since S consists of points on five suitable planes, any line Q meeting those planes off the lines O.O., 0 V meets S , (i,j) 1 J
S
=
(1,2),(1,3),(2,3),(2,4),(3,4),
s = 1,2,3. On the other hand, it is easy to check that all lines incident with
both VOi and ojoh meet S , (i,j,h) a permutation of (1,2,3) or (i,j,h)=(3,2,4),
A Blocking Set in PG(3,q), 4 2 5
393
(2,4,3). F i n a l l y , i f R i s a l i n e i n c i d e n t w i t h 0102 and 0304, t h e n e i t h e r R i s on one of and n(01,03,V),
t h e p l a n e s T I ( O ~ , O ~ , OT~ I)(, O ~ , O ~ , O n(Ol,02,V), ~), meets S , o r R meets t h e p l a n e n(02,03,V)
so t h a t R
a t a p o i n t on S. (Take i n t o account
t h a t S i s c o n s t r u c t e d by d e l e t i n g some p o i n t s from t h e edges of T . ) The same argument shows t h a t any l i n e meets S which meets b o t h 0103 and 0204. Obviously, any l i n e on a v e r t e x of T meets S. F i n a l l y , a l i n e R n o t on any v e r t e x of T and i n c i d e n t w i t h one edge of T meets one f a c e of T a t a p o i n t on S . Thus, S i s a b l o c k i n g set i n PG(3,q) f o r ’ a n y q > 5.
3. AN IRREDUCIBLE BLOCKING SET F i r s t of a l l , o b s e r v e t h a t t h e b l o c k i n g s e t c o n s t r u c t e d i n s e c t . 2 may be red u c i b l e . I n [ 6 ] a b l o c k i n g s e t was e x h i b i t e d i n PG(3,q)
f o r any even q
2
8.
Here we show t h a t such a b l o c k i n g s e t i s r e d u c i b l e and produce a n i r r e d u c i b l e
one s i t t i n g i n i t . The c o n s t r u c t i o n i n [ 6 ] i s a s f o l l o w s . Take a p l a n e TI and a h y p e r o v a l on n, s a y
C. Denote by 0
j’
j = 1,2,3,
any t h r e e p o i n t s on
c
and d e l e t e from ?r t h e p o i n t s
on O . O . \ {Oi,O.}, i , j = 1,2,3; t h i s y i e l d s a b l o c k i n g s e t S on n. L e t O4 be any 1 J J p o i n t o f f TI and r t h e cone p r o j e c t i n g c from 0 4 . D e l e t e from r the p o i n t s on
r*.
0.0 \ { O . } ; t h i s gives a s e t C a l l nT t h e p l a n e OiOh04, ( i , j , h ) a permutaJ 4 J I t i o n of ( 1 , 2 , 3 ) , from which t h e above mentioned p o i n t s have been removed. The
T
*U ( n * .
j = 1,2,3) i s a b l o c k i n g s e t i n PG(3,q). j’ T h i s b l o c k i n g s e t i s r e d u c i b l e ; namely, w e can d e l e t e from K t h e p o i n t s on set K = S U
* and s t i l l have a b l o c k i n g s e t , s a y K*. * * Obviously, S \ C = S i s a b l o c k i n g s e t on n. S i n c e K w a s a b l o c k i n g s e t , i t * * s u f f i c e s t o show t h a t any l i n e R o f f n and T and m e e t i n g IT on C a c t u a l l y * * * meets K . S i n c e R meets a t most one of 040j, i t m e e t s n. f o r some i. Hence, K C\
( 0 ,O
I
Z
,O
3
} = C
i s a blocking set. F i n a l l y , we show t h a t K
*
is irreducible, i.e. that there is at least onetangent
* , PO4
a t each p o i n t . For any p o i n t P on S
n;;
t h e p l a n e w = P040j meets
r*
i s a t a n g e n t a t P. Next, t a k e P on
a t a g e n e r a t o r g . L e t G = g n C*.
t o check t h a t GP i s a t a n g e n t a t P. Now suppose P i s on P040j, any j; w . l . o . g . , A =
R n 0203 #
02,03;
assume j = 1. This p l a n e meets
AP i s a t a n g e n t a t P.
It i s e a s y
T*; t a k e t h e p l a n e i n a l i n e R on O1 and
IT
394
S. Rajola
REFERENCES Bruen, A.A., Blocking sets in finite projective planes, S I A M J . Appl. Math. 21 (19711, 380-392. Mazzocca, F., and Tallini, G., On the non-existence of blocking sets in PG(n,q) and AG(n,q) f o r all large enough n, Simon Stevin 59 (1985), 43-50. Rajola, S . , Un esempio di blocking set in PG(3,q) per ogni q > 5, Quad. Sem. Geometrie Comb. n. 57, Nov. 1985, Dipart. Mat. Univ. Romz, "La Sap ienza" . Rajola, S., Blocking sets, k-insiemi e fibrazioni i n PG(r,q), Tesi, Univ. Roma "La Sapienza", a.a. 1985-86. Tallini, G., Problemi e risultati sulle geometrie d i Galois, Kelaz. n . 30, 1st. Mat.,Univ. Napoli, 1973. Tallini, G., k-insiemi e blocking sets in PG(r,q) ed in AG(r,q), 1st. Mat., Univ. L'Aquila, 1982.
Annals of Discrete Mathematics 37 (1988) 395-398 0 Elsevier Science Publishers B.V. (North-Holland)
395
A CHARACTERIZATION OF ALL ABELIAN GROUPS WHOSE LATTICE OF PRECOMPACl GROUP TOPOLOGIES REPRESENTS A PROJECTIVE GEOMETRY Dieter Remus c/o Institut fur Mathematik, Lehrgebiet D Universitat Hannover, Welfengarten 1, 0-3000 Hannover, F.R.G. First, two structure theorems in the sense of the title are proved. Then all projective spaces are characterized for which the correspondent lattice of subspaces can be embedded in the lattice of precompact group topologies of some abelian group. AMS Subject Classification. Primary 52 A05; Secondary 06 CIO, 22 A99. 1.
NOTATION AND CONVENTIONS
For lattices notation and definitions are used as in L4] . The order relation on a lattice L is denoted by I If equality does not hold, one writes < .vX means the supremum for every Xc L. For X = {x,, x2) one writes x, v x2 instead o f VX. Let a, beL. Then a
.
2.
RESULTS
DEFINITION 1. A topological group is said to be precompact if it is precompact with respect to its left (right) uniformity. REMARK. A Hausdorff precompact topological group is isomorphic to a dense subgroup of a compact group.
D.Remus
396
It is easy to show that the set of (not necessarily Hausdorff) precompact group topologies on a group G forms a complete lattice PK(G) with respect to In this paper some results of geometrical nature on the strucinclusion [5] ture of the lattice PK(G) and an embedding theorem for the lattice o f subspaces of Desarguesian projective spaces are stated. From lattice theory one needs - see 14) -
.
DEFINITION 2. Let L be a lattice. (a) arL is called cmpact iff a i V X for some X c L implies a< VX, for some f i nite X1 c x. (b) L is called ( 1 ) semimodular iff it satisfies the upper covering condition, that is, a
.
To get a relation with projective spaces one needs from [4] , P. 204 3 - A geometry is called projective if the associated geometric lattice is modular.
Precompact Group Topologies
391
Observe that the notion "projective space" in [4] , Def. 3, p. 202 includes the degenerated case "a line has exactly two points". Then one has a one-to-one correspondence between projective spaces, defined by points and lines, and projective geometries, defined by modular geometric lattices ( 143 , Theorem 6, p. 204). Since the lattice PK(G) is modular, one obtains from Theorem 1 COROLLARY. Let G be an abelian group. The lattice PK(G) represents a projective geometry iff G is an elementary group of bounded order. Now it suggests itself to examine the existence of a class of groups G with the property that the lattice PK(G) represents a nondegenerated projective space for which a coordinatization by some vector space is possible. Theorem 2 gives an answer. For the proof one uses LEMMA 2 ( [5] , (2.9)(b), p.40). For every abelian group G the lattice PK(G) is isomorphic to the lattice of subgroups of the (algebraic) dual group G* o f G. THEOREM 2. Let G be an abelian group of cardinality m. The lattice PK(G) represents a Oesarguesian nondegenerated projective space of dimension at least two iff G is an elementary p-group with m2p3. Then the coordinatizing vector space equals Z(P)~if m is infinite, and Z(pIn if m is finite, where pn=m with some natural number n13.
PROOF. (a) If PK(G) represents a projective space, then PK(G) is geometric. By Theorem 1 G is an elementary group of bounded order. Let L be a lattice which is isomorphic to the lattice of subspaces of a vector space. Then it holds ( [2] , p.108): For distinct atoms p,, p2cL there is an atom pfL being distinct from p,, p2 such that p, vp2+. If G is not a p-group, then it is easy to see by the aid of Lemma 2 that PK(G) does not fulfil the quoted property. Therefore, G is an elementary p-group if PK(G) possesses a representation described in the assertion. (b) Let G be an elementary p-group. Then G ?' @Z(p) for m2%, and for m<%, , where pn=m with some some & ( [3] , Theorem 8.5, p.43). G = @Z(p) G for ~ m>%, and G* the dual group G* of G one infers G* 2 Z ( P ) for For m< h, . [3] , Theorem 8.5, p.43 implies that every subgroup of G* is also a subspace of the vector space G* over the field Z(p). Combining this with Lemma 2 one concludes that PK(G) i s isomorphic to the lattice of subspaces of the vector space G*. Since only nondegenerated projective spaces o f dimension at least 2 are considered, one gets the condition m>p3. 0 Theorem 2 shows that the lattice of subspaces of a Desarguesian nondegenerated space can be identified with the lattice of precompact group topologies
D.Remus
398
of an abelian group only in a special case. In general, one has THEOREM 3. Let P be a (not necessarily nondegenerated) projective space and L the lattice of subspaces of P. Then L can be embedded in the lattice of P P precompact group topologies of some abelian group iff P is Desarguesian. PROOF. From [4] one needs the following results: (1) Let L be a complemented modular lattice. Then L can be embedded in the lattice of subgroups of an abelian group iff L is arguesian ( Theorem 20, p.213). (2) Let L be a modular geometric lattice. Then L satisfies the arguesian identity iff Desargues' Theorem holds in the associated projective geometry ( Theorem 8, p.205).
(a) If P is Desarguesian, then ( l ) , (2) imply that L can be embedded in P the lattice L(G) of subgroups of some abelian group G. By [5] , (2.52), p.59 the lattice L(G) is isomorphic to a sublattice of PK(H) of some abelian group H. (b) Let L be isomorphic to a sublattice o f PK(H) of some abelian group H. P Using Lemma 2, L can be embedded in the lattice of subgroups of the dual group P H*. NOW apply (l), (2) to complete the proof. 0 REFERENCES [I]
N. Bourbaki: General Topology, Part 1. Addi son-Wesley Pub1 i shing Company, Reading , 1966
[23
P. Crawley, R.P. Dilworth: Algebraic Theory of Lattices. Prentice-Hall, Englewood Cliffs, N.Y., 1973
]3[
L. Fuchs: Infinite Abelian Groups, Vol. I Academic Press, New York, 1970
[4]
G. Gratzer: General Lattice Theory. Birkhauser-Verlag, Basel, 1978
]5[
D. Remus: Zur Struktur des Verbandes der Gruppentopologien. Dissertation, Universitat Hannover, 1983. English summary: Resultate Math. 6 ( 1983), 151-152
.
Annals of Discrete Mathematics 37 (1988) 399-404 0 Elsevier Science Publishers B.V. (North-Holland)
3 99
Hans-Peter SEIDEL Mathematisches Institut der Universitat Tiibingen Auf der Morgenstelle 10
D-7400 Tubingen. W-Germany
It is shown that in a 4-dimensional stable plane (M.9) every group of r with common center c E M and common axis A E Se is CC.Al
homologies
isomorphic to a closed subgroup of the multiplicative group Cx of
complex numbers. 1. DEFINITIWS
AND STAlElIEHF OF RESULTS
1.1.Definition
A stable plane is a pair (M,Y) consisting of two locally compact Hausdorff topological spaces M and 9 of finite positive topological dimension, the point space M and the line space Y , respectively, such that the following axioms hold: (u)
Any two distinct points x,y E M are on a unique line x
u
y E 9, and the
maPP ing u:
M
x
M
\ {(x.x):xEM}
+ 9: (x,y)
--f
x
u
y
is continuous. (n)
Any two distinct lines K.L E 9 contain at most one common point K
the set 9):= {(K,L)€S?x%:KflL#0}
0
L E M.
of pairs of intersecting lines is nonempty and
open in 9 x 9. and the mapping J W: (K,L) * K n L 0:
is continuous.0 Classical examples of locally compact stable planes are the projective planes over IR. C , M. 0 and their open subsets like the classical affine or hyperbolic planes. But there also exists a variety of nonclassical stable planes and even planes that cannot be embedded in any topological projective plane at all. For further information see e.g. [14].[4]
and references given there.
It is known in general that the point space of a stable plane can only have the classical dimensions 2. 4.
8. 16 and that
the group of all continuous
collineations of (M.9). if endowed with the compact open topology, is a locally
H-P. Seidel
400
compact topological group, see [6],[4].
If dim M
<4
then M and all its lines
L E Y are topological manifolds. In this paper we shall study groups of homologies
r CC*Al consisting
of all
collineations of (M.9) having a comon center c E M and a common axis A E Y with c
e
A . It is easy to see and well known that in a locally compact stable plane
with dim M = 2 every such homology group
r cc.A1
is isomorphic to a closed
subgroup of the multiplicative group Rx of the reals. In this paper we shall study the case dim M = 4 and prove the following theorem: 1.2.Theorem Let (M.9) be a stable plane with dim M = 4 and consider c E M, A E Y with
e A.
r[c.Al
is isomorphic to a closed subgroup of the is multiplicative group C" of complex numbers. More precisely, Cc.al isomorphic to one of the groups Z. %. Z x %, R . S0,R. R x SOJR x Z, C x . where k is any integer.0 c
Then the homology group
4.
The following example shows that there are in fact nonprojective 4-dimensional stable planes (M.9) admitting a nontrivial group of homologies c E M and A E 9:
r [..A1
with
1.3.kample
Let P,C be the classical projective plane over the complex numbers and consider a point
c and a line A with c k? A. Let I: Au= t(P,C)
denote the full
automorphism group of P,C and let Z be any nontrivial compact subgroup of the homology group I
[..A1
= C". Consider the orbit K of a point x E P2C\(A U {c})
under I:. Define M:= PzCW and let 9 denote the lineset that is induced on M by PzC.
Then (M,Y) is a 4-dimensional nonprojective stable plane with c E M and
A E 3, and the group of homologies
given group Z . 0
r
Cc.Al
< r:=Aut(M.2)
is isomorphic to the
It is an open question to what extent the analogue to Theorem 1.2 remains true for 8- and 16-dimensional planes. There are however some partial results which indicate that this m y be the case at least for topological translation planes
PI. 2. DISCXEIE SWXIUPS
OF
r
[CAI
2.1 .w
discrete subgroup of
r [c,A]'
<
r[c * A1 be a closed Furthermore let a be a point on A. and define
Let (M.9) be a positive dimensional stable plane and let C
Groups of Homologies in 4-Dimensional Stable Planes
L:= c
u
40 1
a. Then the following holds:
(1) Z acts properly discontinuously on X:= L\{c,a}. (2) The orbit space Y:= x/c is a locally compact Hausdorff space. (3) The canonical projection p: X + Y is a regular covering with group C. Proof: We only have to prove (1)
and (2). Since C operates freely on L,
according to a criterion given in [lO:p.166,Ex.8.5] it suffices to check that for every compact subset S of X the set R:= {u€C:u(S)fE#0} is finite. Otherwise R contains a sequence tun} such that {un(x)}
converges to c for every
X, see [5:3.12]. Now consider points xn E S with un(xn) E un(S) n S # 0 . Since S is compact we may assume xn + y , un(xn) + z with y,z E S. But since C operates equicontinuously on X [5:3.13] this would imply that u (y) cannot n x
E
converge to c, a contradiction.0 2.2.Proposition
Let (M.9) be a 4-dimensional stable plane and let Z
< r cc.'41
be a closed
subgroup with dim C = 0. Then C is either cyclic or isomorphic to the direct product Z x
% of
the group of integers Z with some finite cyclic group
% of
order k. Proof: (1) Consider a and L as in Lemma 2.1 above. According to [4]. L is homeomorphic to an open submanifold of the 2-sphere S,, and since effectively on L, both
r
CC.Al
r [c.A]
and C are Lie groups, i.e. C is discrete. If C is
finite, the assertion follows by a theorem of P. A. Smith [15] (Consider the is homeomorphic to action of Z on the line pencil Y ). Otherwise X:= L\{c.a} a Rz\{O}, see [5:3.18], and Lemma 2.1 above implies that the fundamental group r(Y)
of the surface Y:= x/c contains an infinite cyclic normal subgroup
(7)=
p,(r(X))
<
r(Y) such that 2 is isomorphic to the quotient group ~(Y)/(T).
(2) I f Y is noncompact. its fundamental group r(Y) is free [l].
hand r(Y) operates on the cyclic normal subgroup relation a - 2 v - a 2 =
7
(7)
On the other
of r(Y) and we get the
for every a E r ( Y ) . Together this implies that r ( Y ) must
be infinite cyclic itself, and we obtain that Z = r(Y)/(7)
is cyclic of finite
order. (3) We are left with the case that the surface Y is compact. Since according to
[5:3.0]
the action of Z on L is orientation preserving Y is an orientable
surface, of genus g. say. We claim that Y is homeomorphic to the torus T: The
....ag'PJ3 :r) where r is the fundamental group r(Y) is given by r(Y) = (a,,P,, [ai.Pi]. If Y has genus g 2 2. then the Freiheitssatz [9:p.252]
relation r =
implies that the subgroup 3:= (ai.Pi)of H(Y)
is freely generated by ai and
and the Same argument as in (2) above shows that the generator infinite
cyclic
normal
subgroup
(7)
<
r(Y)
satisfies
the
7
pi
of the relation
H. -P.Seidel
402
a,-2*7*a12= -r = pl-2-~*p,2.Thus -r cannot be contained in 3 . But then nE according to a theorem of G. H. Bagherzadeh [8:p.203] the group 8:= T-Z-T-' is cyclic and contains both a,' and Pl2.Hence E cannot be generated freely by a, and p , , a contradiction. Therefore Y is in fact homeomorphic to the torus T and our group Z = r ( Y ) / ( - r )
= (Z x Z)/(T)
i s isomorphic to a subgroup
of Z x %.O
3. PROOF OF THE MAIN THEORM We now complete the proof of Theorem 1.2:
Proof: (1)
r [c.A].
Throughout this section C will denote the full homology group
E A and define L:= c u a. We consider the action of C on X:= L\{c,a}. According to section 2 and since C acts freely on X we are left with the case that C is a Lie group of dimension 1 or 2. (2) Case 1: dim 2 = 2. It is easily seen that in this case 2 is noncompact. Thus according to [5:3.18]
Take a point a
the set X:=L\(c,a} must be homeomorphic to IR2\{O}.
Therefore Z is isomorphic to
Ex and the assertion follows.
(3) Case 2: dim Z = 1. Let C1 denote the identity component of Z. Then both C and Ci act freely and equicontinuously on X and have closed orbits [5:3.8].
A criterion given in
[3:p.140.1.6] thus implies that the quotient space Y:= X/C'
is a locally
mi also
operates freely
compact Hausdorff space and since the quotient group
on Y we obtain that V C 1 is homeomorphic to each of its orbits C(x)fl'.
C' is isomorphic to the circle group S O . $ . Since the isotropy groups :C are all trivial it follows from a result of Mostert [ll] that 2' has no singular orbit on X. Hence Y must be homeomorphic to the open unit interval (0.1). But then, according to a theorem of Holder [12:p.8].[13]. the group is isomorphic t o some subgroup of the additive group IR of the real numbers and, since V C i has closed orbits, a result of (4) We first consider the case where
mi = ( j i ) is infinite cyclic. It is now easy is isomorphic to the semidirect product of its
R. Lowen [7:Th.2.6] implies that to check that C = Z'
x
connected component Z' Proposition 2.2
p
(ji)
with an infinite cyclic group (p). According
to
commutes with every element 6 E Z1 = SOJR of finite order,
hence this product must be even direct, and C = S02R x Z is isomorphic to a closed subgroup of C x . (5) We are left with the case that 1% is isomorphic to the additive group IR of the reals. Since Z'
X:= L\{c.a}
is noncompact. i t follows again from [5:3.12.3.18] that
is homeomorphic to R2\{O}
and that the connected orbits Z'(x)
(x E X) have nontrivial intersection with every neighbourhood of c or a. The
Groups of Homologies in 4-Dimensional Stable PIanes
403
restriction of the canonical projection p: X +EL to the standard unit is therefore surjective and Y:= E' is compact. Hence the
sphere St of IR2\{O}
discrete quotient group Z/Z' finite, and
it
being homeomorphic to each of ist orbits is
follows e.g. from a theorem of Gaschiitz [16:p.231]
that
F is a semidirect product of Z' with some finite group F. According to section 2, F is cyclic and centralizes every 6 E 2 ' . Therefore Z is isomorphic
X = Z'
x
to the direct product IR x Q of the additive group of reals with some finite cyclic subgroup
% and Theorem 2.1
is proved.0
c11 Ahlfors, L.V. and Sario. L . , Riemann Surfaces (Princeton University Press, Princeton N.J.. 1960)
PI
Buchanan, T. and HZhl. H., On the nuclei of 8-dimensional locally compact quasifields. Arch. Math. 24 (1977) 472-480
c31 Dugundji, J., Topology (Allyn and Bacon, Boston, 1970) c41 Lowen. R.. Vierdimensionale stabile Ebenen. Geom. Ded. 5 (1976) 239-294
c51 Lb'wen, R., Central collineations and the parallel axiom in stable planes, Geom. Ded. 10 (1981) 283-315 [GI
Lb'wen,R., Topology and dimension of stable planes: On a conjecture of H. Freudenthal, Journ. reine u. angew. Math. 343 (1983) 108-122
c71 Lowen, R., Lacunary free actions on the real line, Topol. Appl. 20 (1985) 135-141
PI
Lyndon, R.C. and Schupp, P.E., Combinatorial Group Theory
(Springer,
Berlin, Heidelberg, New York, 1977)
191 Magnus, W.,
Karrass. A.
and
Solitar, D., Combinatorial
Group Theory
(Interscience Publishers, New York. London, Sydney, 1966)
[lo]
Massey. W.S.,
Algebraic Topology: A n
Introduction (Springer, Berlin,
Heidelberg. New York. 1977) [ll] Mostert. P.S.,On a compact Lie group acting on a mifold. Ann. Math. 65 (1957) 447-455;Errata. Ann. Math. 66 (1957). 589
[12] Priess-Crampe. S . . Angeordnete Strukturen (Springer. Berlin, Heidelberg, New York, 1983) [13] Salzmann. H.. Kompakte zweidimensionale projektive Ebenen. Arch. Math. 9 (1958) 447-454 [14] Salzmann, H., Topological planes, Adv. Math. 2 (1967) 1-60
[15] Smith, P.A., New results and old problems in finite transformation groups, Bull. Am. Math. SOC.66 (1960) 401-415 [l6] Suzuki, 8 . . Group Theory I (Springer, Berlin. Heidelberg. New York, 1982)
This Page Intentionally Left Blank
Annab of Discrete Mathematics37 (1988) 405-412 0 Elsevier Science Publishers B.V. (North-Holland)
405
POLYNOMIAL SPECIES AND CONNECTIONS AMONG BASES OF THE SYMMETRIC POLYNOMIALS
Domenico SENATO A n t o n i e t t a M. V E N E Z I A
1.
U n i v e r s i t h di N a p o l i Univ. d i R o m a "La S a p i e n z a "
INTRODUCTION
I n t w o p a p e r s p u b l i s h e d in 1968 ( s e e [S] ) G . C . R o t a , c a r r y i n g t o the limit the a l g e b r a i c p r o c e s s e s w h i c h B a x t e r a n d o t h e r s introd u c e d for t h e r e s o l u t i o n of p r o b l e m s g e n e r a t e d by t h e o r y of p r o b a b i l i t y , s h o w s that e v e r y i d e n t i t y in a B a x t e r a l g e b r a is e q u i v a l e n t to a n i d e n t i t y b e t w e e n s y m m e t r i c f u n c t i o n s . I n h i s s e c o n d p a p e r t h e Author proves,through combinatorial methods, classical identities b e t w e e n s y m m e t r i c f u n c t i o n s w h i c h t r a n s l a t e i d e n t i t i e s in B a x t e r a l g e b r a s of p r o b a b i l i t y i n t e r e s t . T h e c o n c e p t of g e n e r a t i n g f u n c t i o n o f a f u n c t i o n s e t , c o n v e y e d in these p a p e r s , is d e v e l o p e d later by D o u b i l e t , R o t a a n d S t a n l e y (see [ 4 ] , [9]).These A u t h o r s i n t r o d u c e a p r o c e s s f o r t h e c o n s t r u c t i o n o f a l g e b r a s of g e n e r a t i n g f u n c t i o n s , b o t h c l a s s i c a l a n d innov a t i v e , for the r e s o l u t i o n of e n u m e r a t i v e problems. U s i n g the c o n c e p t of g e n e r a t i n g f u n c t i o n of a f u n c t i o n s e t a n d t e c h n i q u e s i n v o l v i n g the l a t t i c e o f p a r t i t i o n of a s e t , D o u b i l e t (see [ 3 ] ) d e r i v e s m a n y of t h e k n o w n r e s u l t s a n d n e w o n e s a b o u t s y m m e t r i c f u n c t i o n s . I n m a n y c a s e s D o u b i l e t u t i l i z e s the M o b i u s i n v e r s i o n f o r m u l a , but h e a l s o s u c c e d s in g i v i n g b i j e c t i v e p r o o f s of i d e n t i t i e s b e t w e e n s y m m e t r i c functions. T h e s e p r o o f s c o n s i s t e s s e n t i a l l y i n a n i n t e r p r e t a t i o n o f t h e f u n c t i o n s t h a t o c c u r in t h e i,dentities i n t e r m s of s e t s a n d i n f i n d i n g a b i j e c t i o n s b e t w e e n them.
I n t h i s p a p e r w e u s e t h e t h e o r y of p o l y n o m i a l s p e c i e s (see [ l ] ) w h i c h g i v e s a s y s t e m a t i c a p p r o a c h to t h i s k i n d o f proof. We p r o v e with bijective arguments, some identities which occur among the c l a s s i c a l b a s e s of s y m m e t r i c p o l y n o m i a l s of d e g r e e n. T h e l a n g u a g e is t h a t o f c a t e g o r i e s t h e o r y a n d t h i s e m p h a s i z e s t h e g e n e r a l i t y d e g r e e of the c o n c e p t of species.
2. Let
POLYNOMIAL SPECIES
9
ApecieA
be the c a t e g o r y of f i n i t e s e t s a n d b i j e c t i o n s . A , L i n i f e ( s e e [ 5 ] ) is a f u n c t o r M f r o m 9 to
L e t E , F E Ob( @ ) , w e s h a l l d e n o t e by t u r e s o n E a n d by M[u], uEHom(E,F),
.
t h e s e t of M-structhe bijection between
M[E]
D. Senato and A.M. Venezia
406 N[E] on
, ~ n d M[F]
obtained "transforming i n t o a n M-structur? on F.
E
via
u
"
every M-structure
The c o n c e p t of p o l y n o m i a l s p e c i e s i s a g e n e r a l i z a t i o n of f i n i t e species i n the sense t h a t the polynomial species f in addition t o carrying the structures, defines a subset t i o n s from a f i n i t e set t o a set of v a r i a b l e s as s p e c i f i e following.
t h a t of unctor,
ot d
tunciri
the'
X = { x . : i E y } b e a f a m i l y o f v a r i a b l e s w i t h i n d i c e s i i i il non e m p t y aAd t o t a l l y o r d e r e d s e t 3 Let I b e t h e f u n c t o r From t c the category Ens o t sets and f u n c t i o n s d e f i n e d by: Let
.
and,
for
A 5 E
each
dnd
I [ u ] ( t ) Let M from
be
9 to
a
U
ttom(~,x) A L E f : A + X ,
I[EJ=
=
fou
-1
: u ( A ) +
f i n i t e s p e c i e s . We s h a l l Ens defined as follows:
denote
M[E]
x I[E]
Pol(M)[E]
~
X.
by
Pol(M)
the
functor
pu4ynurnia~! 4 p e c i e 4 i s a n y s u b f u n c t o r P of Pol(M), i.e. for each E O b ( @ ) , P[E] i s a s u b s e t of Pol(M)[E] s u c h t h a t i f u€llom(E,t') (s,f)EP[F:] t h e n (M[u](s), I [ u ] ( f ) ) E P[F]. I t P i s a s u b t u n c t o r and of Pol(M) we s h a l l write P c Pol(M).
A
E
The
c a t e g o r y of
polynomial
species
is
defined as
follows.
Let
M
and N be f i n i t e species, and let y be a natural transformation of M t o N . We d e f i n e a n a t u r a l t r a n s f o r m a t i o n of t h e polynom i a l species Pol(M) t o the polynomial species Pol(N) as follows. Pol(M)[E] is the set (YE(s),f) , a5 (s,f) The image of t h e s e t ranges over Pol(M)[E]. It P and Q a r e polynomial species, we define Horn( P , Q ) t o b e t h e s e t of n a t u r a l t r a n s f o r m a t i o n s of P t o Q which are r e s t r i c t i o n s t o P and 0. o f s o m e n a t u r a l t r a n s t o r Y : Pol(M) Pol(N), where P E Pol(M) and Q c _ Pol(N). In mation particular, we w r i t e P=a , when P a n d a r e naturally equivalent i n t h i s category.
-
a s s o c i a t e t o each polynomial species d generating function which c o n s i d e r s b o t h t h e s t r ~ c t u r ea n d t h e s u b s e t o f f u n c t i o n s d e t e r m i n e d by t h e s p e c i e s .
We
.
Let C b e a c o f i n i t e s u b s e t of X We w r i t e , i f t o denote t h e polynomial obtained from p by s e t t x E C. I f p E Z[X] we let N(p,C) b e t h e s e t of such t h a t L qIc=". T h i s d e f i n e s a topology Let Z[(X)] the completion ot Z[X] and al(X)
P/~,"
p 6 Z[X], PIC=" ing t o 0 a l l a l l q E Z[X] on the ring the algebra
Z[X]. over
Polynomial Species 2
P:
g e n e r a t e d by X. A f u n c t i o n E +al(X) d e f i n e d by: ice)
f: =
407
E-+ X determines a function for each e E E .
f ( e )
We shall call genenafuzy monomial of the f u n c t i o n of Z[(X)] d e f i n e d by:
Let
P
C_
be a polynomial
Pol(M)
gen(PIE1)
=
f
the element
s p e c i e s . W e set:
c
p[E]se"(s>f)
( S , f )
g e n ( s , f ) = genif). W e note that gen( P[E]) depends o n the where cardinality of E a l o n e a n d so w e set: gen(P[E]) = gen( P , n ) for E such that / E l = n . We shall call gen( P , n ) n t h - c o e f each s e t
ficienf
poiynomial uf
P.
.the 4 p e c i e 4
T h e y e n e n a t i n g f u n c f i o n Gen( P , z ) o f t h e p o l y n o m i a l s p e c i e s the formal power series Gen( P , z ) with
coefficients in
=
gen( P , n )
n 2 0
p is
X:
2 [(X)].
T H E O R E M 2.1. P o i y n o m i a l 4 p e c i e 4 i 4 o m u n p h i c h a v e t h e 4ume y e n e n u f A n g Luncfion.
3.
THE S P E C I E S OF THE A S S E M B L I E S
W e i n t r o d u c e n o w t h e n o t i o n of a s s e m b l y of f i n i t e s p e c i e s . L e t N "01 = 0. W e s h a l l b e a f i n i t e s p e c i e s w i t h o u t c o n s t a n t t e r m , i.e. c a l l a44em64y o f 4 f / r u c L u n e 4 o f 4 p e c i e 4 N o n the finite set E a E o n every block of w h i c h a s t r u c t u r e of species N partition of is d e f i n e d . F r m a l l y a n a s s e m b l y is a p a i r (",SJ where n=(B: B S E } is a p a r t i t i o n o f E and S,= ( s : s E N[B]}. B
{assemblies
Expk(N)[E]
B
Exp (N) og f h e a 4 4 e m b 4 i e 4 is i e f i n e d a s follows:
The 4pecie.l of onden k :
T h e 4pecie.1
Exp(N)
(.Z
on
,SJ
01 4 f n u c f u n e 4 og
E
1x1
{assemblies on
Let
be the set of the partitions of
E }
4pecie4 N
.
=
with
E
k
block and
a p o l y n o m i a l s p e c i e s w i t h o u t c o n s t a n t t e r m (i.e.P[@]= P C POl(F1) 0 ) . A n a44embly o n E o f onden k o{ o p e c i e d p is every pair (s,f) s i) Sa
where: =
( a , S a )
={
sg:
is a n a s s e m b l y of species
B € a
,
sB E
M[B]
),
M
with
N
k).
=
or f h e a 4 4 e m b l i e . 1 o,f 4 f n u c t u n e 4 o f
Exp(N)[E] P/k/
with
4pecie4
Z E $(k)
and
is
D.Senato and A.M. Venezia
408
there exist, for every B f n, a function f B (sB,fB) E p[B], f is defined o n a set A n B , then f i i i ) if A = u ( A n B) and f/A A = fB. B E Jc ii)
The 4pecie4 uc a44emhiie4 uc polynomial species Exp ( P ) k Expk( P)[E] = { assemblies o f T H E O R E M 3.1.
{-on a n y
Gen(Exp(
P K O P O S I T I O N 3.1.
is defined o n
k is the defined a s follows: k P - s t r u c t u r e s of o r d e r k). k Cen( P , z ) k E N id: Gen(Exp ( P ) , z ) = k k !
P - d n u c t u a e 4 0; Pol(Exp (M))
P),z)
uaden
C_
T h e ~ i i p e c i e i u f -the u 4 4 e r n 6 l i e d o c 4 p e c i e 4 Exp( €')[El s p e c i e s Exp( P ) d e f i n e d b y T H E O R E M 3.2.
that
e
=
is the polynomial Expk( P)[E]. = k
P
Gen( P , z )
yf P and Q a n e i 4 o , n o / c p h i c , i h e n ~ x p( P ) = ~ x p ~ ( t
a).
an isomorphism between P and a . An a s s e m b l y of structures of species p is a partition n such that o n each block B f n a structure ( s , f ) EP[B] is d e f i n e d . The bijection associates to PE: E x p ( P)[E] E x p ( aB[Ef the assembly of species a t r e l a t e d to partition n s u c h ( ( n ,Sn),f) that o n each B the structure eg(s fB) is defined. The bijection Q E determines the requested natural flomorphism. Proof. (
Let
@
( n, S n , f )
-
4.
Let
C O N N E C T I O N S A M O N G B A S E S OF S Y M M E T R I C P O L Y N O M I A L S n
be a n i n t e g e r . A p a n f i t i o n o f
ri
is any sequence ( h )
( A , , . . . I , ) o f n o n n e g a t i v e i n t e g e r s i n d e c r e a s i n g o r d e r A,
T
> ...> . L q
s u c h t h a t t h e i r s u m i s n. T h e n o n - z e r o h i a r e c a l l e d t h e p a n 2 4 of ( h ) and the number of the parts is the l e n g t h o f ( A ) . S o m e t i m e it is convenient to u s e a n o t a t i o n which indicates the n u m b e r s of time ) means that exactly e a c h i n t e g e r o c c u r a s p a r t : ( A ) = (I" 2r2 ri o f t h e p a r t of ( A ) a r e e q u a l t o i.
...
ile n o t e t h a t e v e r y p a r t i t i o n n o f a s e t B w i t h / E l = n d e t e r m i n e s ) o f n , w h e r e ri is the number of t h e p a r t i t i o n ( n ) = (Ir]2r'2 blocks of w i t h i e l e m e n t s . W e s h a l l c a l l ( n ) c l a s s of n .
...
The symmetric polynomials of degree n in the variables x 1 w i t h r a t i o n a l c o e f f i c i e n t s h a v e f o u r c l a s s i c a l bases.
T h e e i e m e n t a a g o y r n n e f n LC { u n c t c o n 4 :
...
Let p E N and a = 2 x i sequences i, i p p s u c h t'hat
,...
b e set:
ah
=
aA,
aA2...aAs
x
i
, . . .x t
aL
w h e r e the sum is over a l l the < il, I t.
i,<
...
Polynomial Species
The m o n o m i a l 4 y m m e f n i c L u n c L i o n 4 :
409
kA
w h e r e the s u m is o v e r a l l d i s t i n c t m o n o m i a l s w i t h d i s t i n c t indices.
The homogeneun
elementafiy f u n c t i o n 4 :
Ex il...xit 1
L e t P E N and h p = il, i such that
...
i
hA
w h e r e the s u m is o v e r all the s e q u e n c e s = p. W e set:
+.t.+it 1 hA
h4".hAq
=
The p o w e n 4 u m , l u t z c t i 0 ~ 4 : s1 t
L e t P E N and
s
p
=
2
.
xp
i=l
I
s
1.
We set: =
s
1,
... A, 5
When(A)ranges o v e r all p a r t i t i o n s of t h e i n t e g e r n , the s e t s {aA}, {kA}, {hA}, { s A } a r e the c l a s s i c a l b a s e s of the s y m m e t r i c p o l y n o m i a l s of d e g r e e n. Let be the f i n i t e s p e c i e s d e f i n e d by I[E]={E}. We d e n o t e w i t h S G P o l ( T ) the p o w e n 4 u m d p e c i e d defined by f:E-X constant]. T h e n t h - c o e f f i c i e n t of the g e n e r a t i n g S[E]= {(E,f): f u n c t i o n of S is:
L e t t E N and P t h e f i n i t e s p e c i e s d e f i n e d by t P t [ E ] = { p a r t i t i o n s of E w i t h t blocks}. W e d e n o t e w i t h K t t h e polynom i a l s p e c i e s K t [E]={(n,f): In( = t , f:E--X, k e r f 2 IT} C Pol(Pt)(E]. T h e n t h - c o e f f i c i e n t of the g e n e r a t i n g f u n c t i o n s of K t is: gen(Kt,n) T h e o t h e r hand:
c
=
E P t [El
kerfBII
w h e r e ( a ) = (1" 2 " . alone, hence setting
sl!
=
s2!
O P n
.. ) . bA=
gen(Kt,n) =
k e r f ? II
gen(n,f)
...k
(0)
T h e s u m o n the r i g t h d e p e n d s o n s2!. .kg), w e h a v e
,EF[E]
sl!
b.,=
.
j,,+'.+At
=n
(Ai...Lt)
(A)=
(I)
!at !
from which
THEOREM 4 . 1 .
Expt(S)
=
Kt
P r o o f . T h e b i j e c t i o n t h a t t o any p a i r ((n,Sn),f) EExpt(S)[E] assoc i a t e s the p a i r (n,f)EKt[E] d e t e r m i n e s the r e q u e s t e d i s o m o r p h i s m .
D. Senato and A.M. Venezia
410
COROLLAKY
4 . 1.
yf!
4e.t ( n ) = ( I ) , w e h u v c :
CJ@
Proof.
3.1,
F r o m t.he t h e o r e m s
5 be
We s h a l l ned b y :
sl!
s*!
...k 2.1
and
4.1
sum F u n c t i o n s
‘The ( i ) g i v e s t h e p o w e r metric functions. Let
E
sA-
( i )
in
(0 )
we
have:
terms
t h e kiriite s p e c i e s defined by: S[E]={permutations u n E}. denote w i t h H C _ P o l ( E x p ( S ) ) t h e d i 4 p u 4 i i i u n 4 p ~ ~ i e d- ei t i -
H[E]= { ( s , f ) : s = ( x , S n ) E E x p ( S ) [ E ] arid f : E 4 X s u c h t h a t The n t h - c o e f f i c i e n t o f t h e g e n e r a t i n g f u n c t i o n of H i s :
hence
gen(H,n)
=
Gen(H,z)
.=
11
n!
23
n>,O
(see
n n!
h ri
kerf
zn 11 !
1.
gen(c,ri)
We
Geri(C,z)
shall
any
L
c
that
now for
n
partition
xn
i e 3
n b a (n-1):
calculate
Expt(Exp(C)) 3 . 1 we h a v e :
For
(n-l)!
=
the
i
E
(n-l)!
he
sIl,
11
-L
s
n
n!
11th-coefficient
b r e v i t y we
of
:
=a
[l]
‘The cyciic 4 p e c i e 4 C C_ Pol(S) i s d e f i n e d b y C[L] = { ( / L , f ) : 11 c y c l i c p e r m u t a t i o n (it E , ~ : E + x c o n s t a n t 11th-coefficient of t h e g e n e r a t i n g f u n c t i o n of s p e c i e s C i s :
hence:
sym-
t h e monomial
nf
of
t h e
s h a l l d e n o t e by
( \ E / = n )s u c h L h a t
( n ) :=
C
species
.
From
theorem
().,,...A[)
we
1-1 ii v e :
wliere
,... ) .
( o ) = ( V , , v2
The
sum
-7
THEOKEM
4.3.
yr
(X) =
(
A,,
...A, )
011
the
right
depends
on(n)alone,
1.
Polynomial Species
L,!
( i i )
P r o o f . From t h e t h u s , using t h e
1
..hi
A,!.
t h e o r e m s 4.2 theorem 2.1,
ocn
41 1 -1)!
( Y
1
a n d 3 . 3 , we we o b t a i n :
( Y , - l ) ! . .
*
have:
s(o)
Expt(Exp(C))=Exp
The ( i i ) g i v e s t h e homogeneus e l e m e n t a r y f u n c t i o n s power-sum f u n c t i o n s .
i n
terms
of
t
(H)
the
We d e n o t e w i t l i A c _ P o l ( ~ ) t h e e4er;2en.:ufly ~ g m m e t n i4 ~ pecieo d e f i n e d A[E] = { ( i < , f ) : f : 1.: + X m o n o m o r p h i s ~ i i The c o e f f i c i e n t of t h e by: g e n e r a t i n g [ u n c t i o n o t t h e s p e c i e s A i s : g e n ( A , n ) = n! a herice:
}.
n
Let
t EN.
he
{
n,t)
A,[k] =o f At i s :
denote
=
c
gen(n,f)
s u m on t h e s, ! s2! n=O
The OA
t ,
f :
C P o l ( P ) t h e s p e c i e s defined by: E +X}tand k e r f A IT=; The c o e f f i c i e n t
.
-
gen(n,f).
k e r f An=O
nEPt[E]
=
At
by
:
uAn-O
=
s, !
d e p e n d s on we h a v e :
right
...ktu)
gen(A1,n)
s*!.
. .k(o)
(n)a l o n e .
The
other hand:
where
( 0 )
:
Hence s e t t i n g
(1s12s2 di
... ) .
=
=
from which
1HEOREM 4.4.
Exp
(A) = A ,
P r o o f . The b i j e c t i o n t h a t t o any p a i r ( ( u , S U ) , f ) E Expt(A)[E] t h e r e q u e s t e d isomora s s o c i a t e s the p a i r ( 0 , f ) E At[E]determindtes phism. C O R O L L A R Y 4.2.
( i i i )
ill!
Poof.
From t h e
The
( i i i )
gives
.?f
(n)=(i),w e h a v e :
,I2!. . . ai
=
aAn=O
theorem 3.1,
the
4.4
s1 ! and
s2'!.
2.1
elementary symmetric
..k(u)
follows:
functions
in
terms
of
the
D. Senato and A.M. Venezia
412
monomial symmetric functions.
REFERENCES
B o n e t t i , Y., R o t a , G.C., S e n a t o , D., V e n e z i a , A.M., O n t h e f o u n d a t i o n of C o m b i n a t o r i a l T h e o r y , X: a c a t e g o r i c a l s e t t i n g f o r S y m m e t r i c F u n c t i o n s . To a p p e a r B o n e t t i , F . , K o t a , G.C., S e n a t o , D., V e n e z i a , A.M., S y m m e t r i c Functions and Symmetric S p e c i e s , A n n a l s of Discrete Math. 30 (1986)
~4
1
107-114.
D o u b i l e t , P., O n t h e f o u n d a t i o n o f L o m b i n a t o r i a l T h e o r y , VII: S y m m e t r i c F u n c t i o n s t h r o u g h t h e T h e o r y of D i s t r i b u t i o n a n d O c c u p a n c y , S t u d i e s i n App. M a t h . v. 51 ( 1 9 7 2 ) D o u b i l e t , P., R o t a , G.C., S t a n l e y , R., O n t h e f o u n d a t i o n of C o m b i n a t o r i a l T h e o r y , VI: 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 , Sixth Berkeley Symposium on Math. Statistic and Probability, V. 2 , B e r k e l e y U n i v . P r e s s ( 1 9 7 2 ) . J o y a l , A., U n e T h e o r i e C o m b i n a t o i r e d e s s b r i e s f o r m e l l e s , A d v . i n M a t h . V. 4 2 n.1 ( 1 9 8 1 ) M a c d o n a l d , I.G., S y m m e t r i c F u n c t i o n s a n d H a l l p o l y n o m i a l s , ( C l a r e n d o n p r e s s , O x f o r d , 1979) Mac L a n e , S . , Categories for the Working Mathematician, 1971) (Springer, New York, Heidelber, Berlin, Rota, G . C . , Baxter Algebras and combinatorial identities I and 1 1 , B u l l . A m e r . M a t h . Soc. 7 5 ( 1 9 6 9 ) S t a n l e y , R., T h e o r y a n d A p p l i c a t i o n o f P l a n e P a r t i t i o n , p a r t 1 a n d 2 , S t u d i e s i n App. M a t h . V. L , n. 2 a n d 3 ( 1 9 7 1 )
Annals of Discrete Mathematics 37 (1988) 413-420 0 Elsevier Science Publishers B.V. (North-Holland)
SET AND
SEQUENCE CLOSURE FOR
413
PERMUTATION GROUPS
FINITE
Johannes Siemons"
INTRODUCTION
1. A
k
permutation group
<
In1
G
has a natural faithful action on
k-element subsets of n. G(k)
of
G
n
acting on a finite set
for each integer
n(k), the collection of all
We now define (see p.399 in [81) the (k)-closure
to be the largest subgroup in Sym(n) which has the same
as G. Thus G is a subgroup of G(k) and the map orbits on n(k) G G{k) satisfies the usual properties of a closure operator as we
-
shall see in Section 2.
Set closure is relevant for automorphism groups of
incidence structures:
if
(G,n)
denotes the point action of some
automorphism group G of an incidence structure S
-
(n,B) with block set B contained in n{k) (i.e. all blocks are incident with precisely k points) then G { k ) in particular preserves the blocks of S and hence consists of automorphisns. For this reason the full automorphism group of S coincides with its {k)-closure. This can be interpreted as a A group for which completeness property of full automorphism groups. G G(k) will becalled (k)-closed. Such groups are the subject of Section 3 .
-
By analogy we may consider the action of a group of k-element sequences of distinct points of (k)-closure
G(k)
G
n.
on the collection n ( k ) Here we define the
of G to be the largest permutation group on n
-
that has
Then G G(k) is also a closure the same orbits on n ( k ) as G . operator in the usual sense. Sequence closure was first investigated by The definition of k-closure Uielandt in the Ohio Lecture Notes (111. there, given In terms of the action on the Cartesian product of k copies of n , does agree uith the definition given above.
*
Financial support from SERC through grant GR/D/30181 is acknowledged.
414
J. Siemons
In [ll] the completeness properties of G(2) formed an important step in the classification of uni-primitive groups of degree p2. In Section 2 we review the basic properties of closure groups such as inclusions in general, primitivity and permutation rank, both for set and for sequence closure. In Section 3 we shall deal with conditions for a group to be {k)-closed for some k. A useful criterion (Result 4 ) depends on the existence of regular sets for closed over-groups. This problem has been investigated in collaboration with Jennifer Key, a survey [ 7 1 will be given at this conference. 2. GENERAL PROPERTIES OF CLOSURE
For the remainder of this note we shall use the following notation. G is a permutation group on a finite set n of n elements. The orbits The orbitals of G, denoted by r l , . . ..rr , are of G are n,, . . . ,nt. Uhen G is transitive, then there is j u s t the orbits of G acting on n x n. one diagonal orbital, say T i = I ( u , u ) ~ u E ~ ) so that r 2 ,... ,rr are the In this case r(G) = r is the rank of G. When orbits of G on O('). r 1. = ( ( a g , B g ) l g E GI is an orbital then (Bg,ag)lg E G) is the orbital paired to r j , denoted by r j f . If r . = rjt then r j is said to be 1 self-paired. In the transitive case the number of self-paired orbitals is given by the Frobenius-Schur indicator V G = IGI-'EgE~ n ( g 2 ) where n denotes the permutation character of G. As a reference to these facts see for instance Chapter 1 in Cameron's article [ 3 1 . A non-diagonal orbital r . may J be viewed as a directed graph with vertex set n and (a.13) an edge if and only if ( a , O ) belongs to r j . Its full automorphism group is denoted by Aut(r.1 ). The undirected graph underlying r j will be denoted by ( r j t r j t with automorphism group Aut(T. tr.1 1. Thus r j and ( r j t r j t ) represent 1 1 the same graph if and only if r j is self-paired. Let F be a field and H some subgroup of Sym(n). Then the centralizer algebra Cn(H) over F consists of all nxn matrices A (with entries in F) for which AM,, = MhA for all h in H where M is the permutation matrix representing h. As a vector space the centralizer algebra is spanned by the adjacency matrices associated to the orbitals of H , including diagonal orbitals. In For a particular, if H is transitive, then the dimension of Cn(H) is r(H).
Set and Sequence Closure far Finite Permutation Groups
415
reference see for instance Chapter V in Uielandt's book [lo 1. All the remaining notation is standard. Comparing the actions of G on n(k) and nk, the Cartesian product of k copies of 0 , it is clear that G-orbits on n ( k ) determine G-orbits on nk and vice versa. This shows that our definition of (k)-closure agrees with the definition of k-closure given by Wielandt in [111. The first result to be stated now shows that both set- and sequence-closure satisfy the usual properties of a closure operator, RESULT 1:
PROOF: (1) follows directly from the definitions. The second statement is theorem 5.10, page 16 in [ll], while ( 3 ) follows from theorem 5.1 in [81.U The next theorem is concerned with the lattice of closure groups in general. RESULT 2: Let G f Sym(n), let n* be the integer with n - 1 E 2n* < n and suppose that 2 E k 4 1 < n - k. Then
(4)
PROOFt The inclusions in (1) and (2) are theorem 5.8, page 15 in [ll] and The equations in ( 3 ) and ( 4 ) can be established theorem 5.1 in 181. directly from the definitions.0 REMARKS: A. It is worthwhile to comment on the inclusions in Result 2.U) and 2.(2). While 2.(1) is straightforward to establish due to the inductive nature of sequence closure the inclusions in 2.(2) are due to rather different arguments: no obvious inductive arguments apply. The proof given
J. Siemons
416
in [ 8 ] relies upon the fact that the incidence matrix of k-element subsets versus I-element subsets of s l has maximal linear rank. Note also that 2.(2) remains true for infinite permutation groups. This result, relying on a weak form of Ramsey's theorem, is due to Bercov and Hobby [l]. No information in general seems to be known about the index of G ( k ) in G(k). Its range can be seen from the following two examples. The cyclic group G = Cn , generated by an n-cycle, has G(' ) G 4 G( = D zn' the dihedral group of order 2n , while G = PSL(2,q) , acting on the q t l points of the projective line, leads to G G ( ' ) c G(') = Sym(qt1) if q E 3 mod 4 .
B.
-
-
As a consequence of Result 2.(2) the group G(n*) has the same orbits as G when acting on the collection of subsets of any size and it is the largest group with this property. In [ a ] , page 401, I have conjectured that, under suitable general hypotheses, G should coincide with G("*). In this direction theorem A in [ 9 1 shows that a primitive group G for which some prime divides the order of G("*) but not that of G must be one of seven groups of degree at 'most nine. Result 4 in Section 3 will give more information about this question.
C.
We now turn to the questions of primitivity and permutation rank of the closure groups.
RESULT 3:
Let G be a transitive group on n and suppose that 2 6 k < n - 2. and G(k) all share the same block systems of imprimitivity. Thus G is primitive if and only if any of its closure groups (1 < k < n - 1 ) is primitive. (2) For the centralizer algebras we have Cn(G) Cn(G(k)) and in particular r(G) r(G(k)). ( 3 ) When OG denotes the Frobenius-Schur indicator of G then r(G) 3 r(G(k)) 3 2-' (r(G) t OG). (1) The groups G , G(k)
-
-
PROOF: (1). As G c G(k) c G(k) c G(') by Result l . ( 1 ) and 2.(2), this follows from theorem 5.2 in [ a ] where it is shown that G and G(') have the same block systems of imprimitivity. See also theorem 4.10, page 10 in [ll]. ( 2 ) . As G H implies Cn(H) c C,(G) in general, by Result 2(1) we only have to show that C,(G) C,(G(')). By definition G and G(')
-
Set and Sequence Closurefor Finite Permutation Groups
417
have the same orbitals, and hence, by the remark in the introduction of this section, their centralizer algebras coincide. (3). As G c G(k) c G('), the rank of these groups satisfies r(C) > r(G( k, ) 3 r(G( ) ) so that only the lower bound for r(G( * needs to be established. Hence compare orbitals of G with orbitals of G(*). An orbital of G t Z J can be the union of at most two G-orbitals and this happens is at least the only if they are paired orbitals for G. Thus r(G(')) number of self paired G-orbitals plus the number of orbital pairs of G. This number is 2-' (r(G)tnG) by the remarks zt the beginning of this Section.0
3.
CLOSED
GROUPS
-
-
A group G on fl is (k)-closed or (k)-closed if G G(k) or G G(k) As a consequence of Results 1 . ( 1 ) , 2 . ( 1 ) and 2 . ( 2 ) a respectively. (k)-closed group is also (k)-, and (&)-closed for k d P < n - k is always (k)-closed. In this section we are concerned and of course G(k) with conditions for a group to be (k)-closed for some k. As already mentioned in the introduction, examples of (k)-closed groups are provided by full automorphism groups of structures on n in which blocks have constant size k. Our main criterion in general is Let G c Sym(n) be given and suppose that some permutation group H on n containing G satisfies I : H is (k)-closed for some k , 2 d k d n - 2 , and 11: for some subset A C n of size 1 , k 6 1 S n - k, the set stabilizer of A in H is the identity on n . Then G is (n)-closed and a group on n having the same orbits on n(') as G coincides with C.
RESULT 4:
-
Then G c K c G(k) E H(k) H by Results I. Therefore K ( A ) 1 = G(A) by hypothesis I1 and as AG AL we have G = K . Suppose now that some group L has the same orbits on n(') as G. Thus, by definition, G(') L(') so that But then, as above, L L(') so that L G.0 L is a subgroup of H. PROOF:
Let
K = GI1).
l . ( l ) , 2.(1) and hypothesis
-
-
-
-
-
Hypothesls I in Result 4 in itself can be satisfied quite easily by choosing H to be the (k)-closure of any group Containing G. The relevant
418
J . Siemons
requirement now is the existence of a subset A as in hypothesis 11. Such a set is said to be a regular set for H. In the survey [ 7 1 by J.Key the existence of regular sets for several infinite families of groups will be discussed, including ArL(d,q), PrL(d,q), the Ree groups R(q) and Pru(3,q2 ) in their natural representations. These groups generally are (3)or {4)-closed and are shown to have regular sets of sufficiently small size. These results are independent of the For details see also [ 6 1 and [ 5 ] . classification theorem of finite simple groups. A theorem in [41 which however does depend on the classification shows that all primitive groups of sufficiently large degree not containing the alternating group of the same degree have regular sets. Note finally that a group G as in Result 4 is uniquely identified through its orbits on the power set over n. A strengthening of the notion of {k)-closure is that of a k-geometric action, Here a permutation group G on f2 is said to be due to Betten [ 2 ] . k-geometric if there is a system B of k-element subsets of n such that G is the full automorphism group of the structure (n,B). Clearly a k-geometric group is (kl-closed but the converse does not hold in general.
RESULT 5: Let G be a subgroup of ArL(d,q) or PTL(d+l,q) where d > 2 if q > 3 and d 3 6 if q = 2. Then G acts k-geometrically for some k on the points of the corresponding affine or projective space. The values for k are given in [ 6 1 and [ 5 1 but are too cumbersome to reproduce here. Ue give an indication of the PROOF : There is a regular set A for ArL(d,q) or PrL(d+l,q) Now let 9, AG and let B, be the respectively, say of size k. collection of all k-element subsets of points in the corresponding affine (projective) space that are contained in a do-dimensional coset (subspace), where do < d is taken to be minimal. Setting B = B, u B, and A = Aut(n,B) it can be shown that BZA = B, so that A is a group of collineations. As A{A) = 1 = G{A) and AA = AG it follows that G = A. For the details of this proof see theorems 3.1 and 3.2 in [ 5 ] and theorem 5.1 in [61.0
-
Set and Sequence Closure for Finite Permutation Groups
419
BIBLIOGRAPHY [l] R.D.Bercov and C.R.Hobby, 'Permutation groups on unordered sets', Math. 2 . 1 1 5 , 165-168,
(1970).
[ 2 ] D. Betten, 'Geometrische Permutationsgruppen', Mitt. Math. Gesel lschaft
Hamburg, 1 0 , 317-324, ( 1 9 7 7 ) . 'Suborbits in transitive permutation groups', "Combinatorics", Part 3 , Mathematical Centre Tracts, Amsterdam, 1975. P.J. Cameron, P.M. Neumann and J.Saxl, 'On groups with no regular orbits on the set of subsets', Arch. Math. 43, 295-296, ( 1 9 8 4 ) . F. Dalla Volta, 'Regular sets for the affine and projective groups over the field of two elements', to appear in Journal of Gnometry. J.D. Key and I.J. Siemons, 'Regular sets and geometric groups', Resultate der Hathematik, 11, 97-116 ( 1 9 8 7 ) . J.D.Key, 'Regular sets in Geometries', Proceedings of this conference. I.J.Siemons,'On partitions and permutation groups on unordered sets', Arch. Math. 38 391-403, ( 1 9 8 2 ) . 1.J.Siemons and A.Wagner, 'On finite permutation groups with the same orbits on unordered sets', Arch. Math. 45, 492-500, ( 1 9 8 5 ) . H.Wielandt, "Finite permutation groups", New York 1984. H.Wielandt, "Permutation groups through invariant relations and invariant functions", Lecture Notes, Ohio State University, Ohio 1969.
[ 3 1 P.J.Cameron,
[41 [5] 161
[71 [81 [91
[lo
111
Department of Mathematics, University of Birmingham, Birmingham 815 2TT United K ingdom
School of Mathematics and Physics, University of East Anglia, Norwich NR4 7TJ United Kingdom
This Page Intentionally Left Blank
Annals of Discrete Mathematics 37 (1988) 421-426 0 Elsevier Science Publishers B.V. (North-Holland)
42 1
P-CYCLIC HYPERGROUPS WITH THREE CHARACTERISTIC ELEMENTS Thomas N.VOUGIOUKLIS and Stefanos H.SPARTALIS Democritus University of Thrace, 67100 Xanthi, Greece. Let (G;) be a group and P a non-empty subset of G. The set G with hyperoperation P* given by the relation xP*y=xPy becomes a hypergroup. These hypergroups have been introduced by the first author and have been called P-hypergroups. In this paper we study the simplest case of 3-dimensional P-hypergroups.
5 1. Let (G;) be a group and can define hyperoperation P*:GxG
P
a non-empty subset of
B(G):(x,y) >-
G.
In
G
we
xP*y=xPy.
becomes a hypergroup in the sense of Marty (1934), i.e. Then G associative and the reproduction axiom xP*G=GP*x=G, [6y :Ad The above hypergroup denoted
P* is is valid. generalized P* are
In this paper we focus our attention on P-hypergroups when G is a finite cyclic group, say (H,,.). The hypergroup
Hn
such that
Hn=h[llU...Uh[vl.
The minimal such number
v
is
called the period of h and if all generators have the same period then we call the
if we can write
H = h [vl;
see
[6 and 21.
is a singleton then P* becomes an operation. The simplest case If P of P-hypergroups with two elements is P = {e,x], e#x and this case In this paper we study the for cyclic hypergroups was studied in [61. simplest case of 3-dimensional P-cyclic hypergroups
5 2. Henceforth we denote a generator of elements of
~€{ ...,[: l,
n
Hn
1)
are ;
as,
s E{0,1,.
(Hn,-)
. .,n-l).
so we study the hypergroups
where
is an even, we take
The powers of the element
by
notation
a; therefore
the
We observe that
is the integer part of
>
.
with Obviously
if
...,[$] -1).
~€{1,
as
with respect to the hyperoperation
a:
are
T.N. Vougiouklis and S.H. Spartalis
472
The s e t of u n i t elements (two-sided) of i.e.
if
P = te,aw,a-'I,
is
<~,,a:>
we have
xEP
as E Hn with r e s p e c t The s e t of i n v e r s e elements (two-sided) of -s -1 unit xEP is a x P and t h e s e t of i n v e r s e elements of -sp2 r e s p e c t t o some u n i t i s a . F i n a l l y , t h e only t o t a l i n v e r s e -S
of as is a , see [7]. We note t h a t every and r i g h t i n v e r s i b l e i n i t s e l f ; s e e [ 3 and 71.
A subset X
{e,a ,a
--W
H
of
is a
i s called a
Hn
}CHCHn
Hn
then
S
of
Hn
is a
element left
< H n , a:>
if
i s a subgroup of
P-subhypergroup of a hypergroup i s
PROPOSITION 1 The union of t h e powers, with r e s p e c t t o t h e a
H
with
I t i s obvious t h a t
i s a hypergroup.
P-subhypergroup of
F i n a l l y w e n o t e t h a t every see [7 and 51.
the
as
P-hypergroup i s
P-subhypergroup o f
and
to
if
(H,,.).
ultraclosed;
P-hyperoperation, of an element
P-subhypergroup of
<~,,ai>.
Pro0f
The union of t h e powers of S=aS[11"a"[21"...={a
For
t = n
Now l e t
and
atos+vo-W E S;
r =0
(ii)i f
I
0
for
(to+t)S+(Vo+V)-W ~
tErn0,
V E X } . W
--W
we o b t a i n
{e,a ,a
such t h a t
to romodn,
}CS.
then
at o S + V o M
obviously
r i 0 ,
is
v~{-l,O,l}
Moreover when we t a k e ( i )i f
as
t S+VW
t E{n-r,+l,
aSf(Y~+Y)-W
,
a:S=S, n-ro+2,
and
...I
.
a 2 ~ + ( v ~ +,v , ). ~ ;
Therefore from t h e commutativity of axiom i s v a l i d .
a:
then
we have .toS+VOX
s o again
we conclude t h a t
the
a H* S = S .
reproduction
PROPOSITION 2 In
i f t h e r e e x i s t s an element
as
and a number
t>2
such
P-Cyclic Hypergroups with Three Characteristic Elements
that
It],
as
[llu...uaS[t-ll
‘
I;
s t+l
Let y€a We have two c a s e s . ts+s+vw ( i )y = a
then
with
ats+(v-l)wS a s [ t 1 ts+(v-I)-H
y=a
O
-r < u < r .
with
for
l < r gt-1
where
r
si t-11
the
element
-t < v < t .
where
From t h e hypothesis
t h e r e e x i s t s an
rs+uw = a
...
a s [ t + l l ~ a S ~ l lu ya
then
( t + lS+VH )
423
such
Consequently
... U a sit-11-
y = ats+(v-l)w+s+w= a(r+l)s+(u+l)wE , s [ l ]
V
ts+s+vw ( i i )y = a
Then i n a s i m i l a r way we have
with
a t s + ( v + l ) w€ a s [ t l ; =
Therefore
so
ats+ ( u + l )H-
-t
at s + ( v + l ) w
rs+uw
= a
H+
that
s = a ( r+1) s+ ( u-I ) w
,
lbr4t-1,
as [ 11
-r
...u a s [ t - 1 ]
5 3. THEOREM 1 The
P-cyclic
hypergroup
i s single-power c y c l i c
and i n t h i s c a s e every element of
Hn
iff
(k,n) = 1
[9]+1.
i s a g e n e r a t o r with period
Proof t o be a g e n e r a t o r of t h e single-power i t must s a t i s f y t h e r e l a t i o n
We observe t h a t f o r an element as c y c l i c hypergroup Hn of period
Ias[tli
=n.
For
= {atS+vw
But, s i n c e l a s [ t l 1G2t-1
we have
t = [+]+1,
t,
I
-t
t 2[$1+1.
so
we have
21:)~
Now, i f
2’:’’
=a
n=2w (n-l)w
Also, f o r every
;
we have
therefore
rETiNo,
T h i s means t h a t t h e element hypergroup Hn
= e,
a
and i f
=a
a
n = 2w+l
te,aH,.
we
have
. .,a (n-1) n 1.
we have
as
i s a g e n e r a t o r o f t h e single-power
cyclic
T.N. Yougiouklis and S.H. Spartaris
4 24
We conclude t h a t every element
as
rq1 +l.
5 4. THEOREM 2 In t h e P-cyclic
(s,-H,n) =I.
iff
hypergroup
of
i s a generator
Hn
t h e element
with
period
i s a generator
as
Pro0 f We observe t h a t if a v € a[*] a
Therefore
I
as.
power of
as
iff
THEOREM 3
(x,n) = 1
Pro0f a “t’
We have
a-H[tl
=aH
= tar’
a
[z]
for
P-cyclic
n
But
.I:[
n=2w,
w2 - w + l
H =a
of t h e
{e,a
aw ( w - 1 )
a(2t-1)-H}
so and
...,I:[
+I}.
t E ~ 2 ,
“[;I
w e deduce t h a t a
--H
+
13 = Hm.
a
has p e r i o d
hypergroup
[z]
w>2, 2
‘w,
i.e.
aw
)..., E1w[w-l]
;
so
[Wj
n>5,
I:[
+1.
t h e generator
But
aw ( w - 1 ) - 1
, a
w(w-1)
!21
a
has
then
u{aW -w-1+2w-1 Hn=aw [ w l u
1.
...u aw i l l .
..
aw[wlyI$w-l)-l
tEINo,
P-cyclic hypergroup
( i i )I f n=2w+l, i.e. [k]=w, w>2, then ,an-ll = ,w[w] uIaw2-w+1+2w-~ w2 -w+1+2w W2-W 1 H =a te,a,. , a I =
we can f i n d
h
p e r i o d a t most
( i )If
aH,a-’
r = 1,
I n a s i m i l a r way we deduce t h a t T’HEORFM In t h e
belongs t o
a iff
t s + vw :Imodn,
iff
...,2t-11;
1
(-H,n) = 1,
Moreover, s i n c e
we have
+l.
[t-l]u{a(2t-2)H,
aH’t-l’gax‘tl
iff
Hn
i s a generator
as
then t h e g e n e r a t o r s
have p e r i o d
v€Wo
t h e n f o r every
i s a generator of
That means t h a t
VEX such t h a t ( s , H , n ) = 1.
tEBo,
If
t€Bo,
a€aS[tl,
=
1.
, aw ( w - l ) E a W [ w - l ]
, ,
so
H n = a w [ W l u . . . uw a[ 1 1
Therefore t h i s theorem i s v a l i d for n>5. N o t i c e t h a t t h i s theorem i s also v a l i d f o r n-4. i t i s not v a l i d a r e n~{3,5}.
.
So t h e o n l y cases f o r w h i c h
P-Cyclic Hypergroups with Three Characteristic Elements
425
THEOREM 5 n >5,
If
a’
then t h e p e r i o d of t h e g e n e r a t o r
>1,
H
of t h e
P-cyclic
[I:.
< ~ ~ , a z >is a t most
hypergroup
Proof
W e have t h e following four cases. ( i )n > 9
and
.
1
a[11ua[21u.. u a
..
r E {I[:
for
+ 1,. , n ) ,
if
r-[”]=wd, 2
if
I--[”]2 ]:[
and
r€{l
Also, f o r
1.
-t
Moreover, with t h e above assumptions
a =a
then
151 +Hd
arE a
implies
and
then
- w+u+(d+l)n
x=[:]
,..,[$I
n
then
arEa”];
even, we have
,
( i i i )n > 9
-1.
1,
n
n-1 a €a
if
and
=a
( i i )n > 9
even
.I[:,
O
]:[
+xd+u
t = l,. .,[:I
we observe t h a t
d€Mo
= Hd+u,
a =a
If
r €{l,. ,
for
a r E a[.]
So
We o b t a i n
1<~<[:] -1.
and
an€a
“$1
a
n
odd,
we
have
-11 We examine t h i s c a s e o n l y f o r odd
H=[:].
we have a degenerate case f o r
r€{[g]+1,
and f o r
2 [ $ ] +l),
then
n,
since f o r
P.
ar€ a
“Yl
I n t h e above t h r e e cases we can s e e t h a t t h e p e r i o d of
a’
(iv) We can s e e by i n s p e c t i o n t h a t t h e theorem i s v a l i d f o r
i s a t most
[$I.
nE{6,7,8,91.
5 5 . THEOREM 6 If (H,n) = 1 then t h e c y c l i c with p e r i o d (c.f.
P-cyclic hypergroup
<~,,a:>,
n>5,
is
not
[4]).
Proof From theorem 3 we d e r i v e t h a t , s i n c e
( x , n ) = 1,
t h e generator
ax
has
T.N. Vougiouklis and S.H. Spartalis
426
period
+l.
generator
a
I;[
the generator Therefore
In theorem
4 it was shown that, for
has period at most
n’
]:[
also has period at most
a
;
.I:[
for
the
n = l , H>I,
from theorem 5
is not cyclic with period.
Remark
If (n,n)= 1, then the only hypergroups are for n E {3,51.
cyclic with
period
REFERENCES Corsini,P., Sur les homomorphismes d’hypergroupes, Rend. Sem. Mat. Univ. Padova, V o l . p.p. 117-140 (1974). De Salvo, M.-Freni,D., Semi-ipergruppi ciclici, Atti Sem. Mat. Fis. Univ. Modena, XXX,- P- . P- . 44-59 . . (1981). . Dresher,M.-Ore,O., Theory of Multigroups, h e r . J. Math. V . p.p. 705733 (1938). Konguetsof,L.-Vougiouklis,T.-Kessoglides,M.-Spartalis,S., On cyclic hypergroups with period, to appear in Acta Un. Carolinae - Math. et Physica. Sureau,Y., Sous-hypergroupe engendr’e par d e w sous-hypergroupes et soushypergroupe ultra-clos d’un hypergroupe, C.R.Acad. Sc. Paris, t . a (2Mai 1977) S’erie A. p.p. 983-984. Un. Vougiouklis,T., Cyclicity in a Special Class of Hypergroups, Acta NO I, p.p. 3-6 (1981). Carolinae-Math. et Physica, Vol. g, Vougiouklis,T., Generalization of P-hypergroups, to appear in Rendiconti del Circolo Matematico di Palermo. Wall,H.S., Hypergroups, American Journal of Mathematics, V o l . p.p.7798 (1937).
z,
a,
z,
Annals of Discrete Mathematics 37 (1988) 427-432
0 Elsevier Science Publishers B.V. (North-Holland)
427
ORDER AND UNIFORM STRUCTURE IN PROJECTIVE GEOMETRY Horst SZAMBIEN Kaiser Wilhelms Gymnasium See lhorststraRe 52 D-3000 Hannover 1 Fed. Rep. of Germany Riassunto. In questa nota studieremo uno spazio proiettivo ordinato: Esiste una struttura uniforme compatibile. I1 complementamento di questo spazio 6 strutturabile a sopraspazio proiettivo ordinato. AMS-Subject-Classification: Primary 51H05, secondary 54E15, 06B30.
1.
INTRODUCTION
Uniform structures have been studied in planar geometry by Guggenheimer [Gu 591, Breitsprecher [Br 671 and Szambien [ S z 811, [ S z 861 mainly in the context of projective planes. In this paper the notion of uniform projective geometry is defined for the case of arbitrary dimension. It shall be a projective geometry (i. e. subspace lattice of some vector space) provided with a uniformity, such that the following condition and its dual hold: Joining of skew subspaces is uniformly continuous outside an arbitrary uniform neighbourhood of the closed set of non-skew pairs of subspaces. Any uniform projective geometry becomes a topological projective geometry when it is provided with the uniform topology, [ S z 861. Proposition 1 shows that every finite-dimensional projective geometry over a topological field is a topological projective geometry. So, in this class the converse problem arises: For which topological projective geometries does there exist a compatible uniformity, which turns it into a uniform projective geometry? The problem will be solved for ordered geometries: In a projective space (i. e. the set of I-dimensional subspaces of some vector space) one can consider an order structure for instance in terms of a 4-ary separation relation or some equivalent concept; cf. [Le 651 . The so-called special separation relations are then equivalent to orderings of the coordinatizing division ring. In this paper finite-dimensional pappian projective geometries, whose correspondent spaces carry an order structure, are considered. They are of course topological projective geometries w. r. to their order topology, [Mi 68;S.5IlSatz11]. It is shown that they can be provided with a uniformity derived from the order structure to become uniform projective geometries. The construction of this uniformity is carried out using the well known Grassmann-mapping. This mapping embeds the set of subspaces of a given vector space into the point set of some projective space of higher dimension. It then suffices to define a suitable uniformity on this projective space. This is done in proposition 2 . In proposition 3 the existence of the Weil-completion is proved for any ordered pappian projective geometry; as a corollary to this one recognizes in proposition 4 the Dedekind-completion and the Weil-completion as isomorphic objects.
H. Szambien
428
2.
RESULTS
[Bo 661, [ L e 6 5 1 , and [ B i 7 9 1 a r e t a k e n as g e n e r a l r e f e r e n c e s f o r n o t i o n s o f g e n e r a l t o p o l o g y , a l g e b r a , geometry, and l a t t i c e t h e o r y , Throughout t h i s p a p e r " I " w i l l d e n o t e t h e r e s t r i c t i o n of resp. mappings, t o p o l o g i e s , and u n i f o r m i t i e s , Mn t h e n - f o l d C a r t e s i a n p r o d u c t of t h e s e t M; f u r t h e r m o r e L i s a l a t t i c e , where s+t and s a t denote t h e j o i n , r e s p . m e e t o f l a t t i c e e l e m e n t s s and t . If L i s an i r r e d u c i b l e , meet-continuous, a t o m i s t i c , modular l a t t i c e , t h e n it w i l l be termed p r o j e c t i v e geometry. The l e n g t h of t h i s l a t t i c e minus 1 w i l l a s u s u a l be c a l l e d t h e dimension of t h e geometry. I f L h a s bottom 0 and t o p then S:={(s,t)EL2: s * t = O } is the set of s k e w p a i r s and d u a l l y S : = { ( s , t ) E L 2 : s + t = l } . L e t CJ be a t o p o l o g y on L. ( L , o ) i s c a l l e d a t o p o l o g i c a l p r o j e c t i v e geometry (TPG) i f f L i s a p r o j e c t i v e geometry, and + IS and * I S * are c o n t i n u o u s mapp i n g s w i t h open domains. Examples of T P G s are c o n s t r u c t e d as f o l lows : L e t ( K , R ) be a c o m m u t a t i v e , t o p o l o q i c a l f i e l d w i t h t o p o l o g y R , r t h e e x t e r i o r a l g e b r a on K n l t h e n-dimensional v e c t o r s p a c e o v e r K , n a n a t u r a l number 2 3 , rk t h e (:)-dimensional K-vector s p a c e of k v e c t o r s : t h u s r i s t h e d i r e c t sum of i r k : OSkSn) L e t O k c r be t h e s e t of decomposable k - v e c t o r s f o r l s k l n , O0:= K\ 1 0 ) , O:=U{0 k :Olksn} Now t a k e t h e l a t t i c e L r = L ( K ) of s u b s p a c e s of K , L := { s E L : dim s = k ] , P t h e s e t of I-dimensional s u b s p a c e s of t k e v e c t o r s p a c e r;rx:= r\{o}, K X : = K \ { O } ,r X / K X : = iKXg : g E 1-1 , D := O;/KX, D : = O X / K X . P w i l l be i d e n t i f i e d w i t h T*/KKX L e t r carry t k e p r o d u c t t o p o l o g y of R . P r o v i d e P w i t h t h e t o p o l o g y 7 1 , t h e f i n e s t t o p o l o g y which r e n d e r s t h e c a n o n i c a l p r o j e c t i o n r + T X / K X cont i n u o u s . S i n c e t h e r e e x i s t s a b i j e c t i o n between L and D, one g e t s an embedding a : L + P ; p r o v i d e L w i t h 0 , t h e i n i t i a l t o p o l o g y w. r . t o @. u i s c a l l e d t h e coordinate topology.
.
A,
.
.
( L , o ) i s a TPG.
PROPOSITION 1 .
d e n o t e t h e e x t e r i o r m u l t i p l i c a t i o n on , and 0 ) ; then A maps C i n t o 0 ; furthermore it i s c o n t i n u o u s w i t h C open, because t h e c o o r d i n a t e s of s A t are c a l c u l a t e d by r a t i o n a l o p e r a t i o n s from t h e c o o r d i n a t e s of s and t . Hence i t i n d u c e s a c o n t i n u o u s mapping S + L , which c o i n c i d e s w i t h t h e l a t t i c e - j o i n +, making t h e f o l l o w i n g diagram commutative: Proof. C:=
Let
A
{ ( s , t ) E 02 : s A t f.
c s
2 0
+
Z L
T h e r e i n a i s t h e map, which a s s o c i a t e s w i t h e a c h a E 0 t h e subspace o f K d e t e r m i n e d by it. + becomes c o n t i n u o u s , b e c a u s e a , and a l s o a x a I C are q u o t i e n t mappings. The c o n t i n u i t y of remains t o be shown. T o t h i s end c o n s i d e r I ? ' , t h e e x t e r i o r a l g e b r a on t h e d u a l v e c t o r s p a c e of K : l i k e w i s e i n r ' , mapping t r o d u c e A ' , L ' , s ' , E', 0 ' a s b e f o r e . Now 6 : ra m u l t i v e c t o r t o i t s d u a l , i s a homeomorphism and i n d u c e s a n o t h e r homeomorphism L __.z L ' , mapping a subspace t o i t s a n n u l l a t o r . D e f i n e C*:= (6X6)-'[E'] Now t h e f o l l o w i n g diagram commutes - making t h e lower row mapping S* d L c o n t i n u o u s . T h i s i n t u r n c o i n cides with IS*
-
.
.
Order and Uniform Structure in Projective Geometry
c*
XI--------,
0'
S-------,S'-----+-L'-------+
------+>
1
A'
429
L
+'
q . e . d.
L e t 1-1 be a u n i f o r m i t y on L , N E 1-1 an e n t o u r a g e ; S N : = L Z \ N o ( L Z \ S ) O N , SN*:= L2\No(L2\S*)oN ( L , p ) i s c a l l e d a uniform p r o j e c t i v e geometry (UPG) i f f L i s a p r o j e c t i v e geometry such t h a t f o r e v e r y N E )-1 t h e mappings + I S and a r e uniformly c o n t i n u o u s , and furthermore S and S* are open'wSN*r. t o t h e uniform topology T ( p ) . A c l a s s of examples i s f u r n i s h e d by
.
-
L e t L be an o r d e r e d pappian p r o j e c t i v e geometry of f i n i t e dimension a t l e a s t two. Then t h e r e can be c o n s t r u c t e d a uniform s t r u c t u r e 1-1 on L i n d u c i n g t h e c o o r d i n a t e topology, such t h a t ( L , u ) i s a UPG. 1-1 i s c a l l e d t h e c o o r d i n a t e u n i f o r m i t y on L.
PROPOSITION 2.
Proof. With t h e n o t a t i o n s of p r o p o s i t i o n 1 one c o n s i d e r s t h e t o t a l l y o r d e r e d c o o r d i n a t e f i e l d ( K , < ) of L which of c o u r s e is a t o p o l o g i c a l f i e l d w. r . t o i t s o r d e r topology Take a n a t u r a l numb e r n 2 3 , m:= 2n, g : = ( g , , . . . , g ,) E Km; Ri:= {g : -1 Sg < 1 f o r 1 5 j S i - I , g i = l , -1 < g < 1 f o r i+l<j<m] and There i s a b i j e 2 t i o n r : R --+ P R:=R(K):=U{Ri :'T=l,...,m] whose i n v e r s e a s s i g n s t o each KXg E P t h e v e c t o r
.
.
(*I
sgn(g) max(lgli , . . . , l g m l ) - ' g
where s g n ( g ) d e n o t e s t h e
E R,
s i g n of t h e component g . of h i g h e s t index j such t h a t t h e e q u a t i o n lg I = max( I g I , . ,1g '1) h o l d s . r I R n 0 t h e r e f o r e maps o n t o L. O n j t h e t o p o l a g i c a l gr8up Km t h e r e i s g i v e n t h e l e f t ( o r r i g h t ) u n i f o r m i t y w. r. t o a d d i t i o n ; r e s t r i c t t h i s u n i f o r m i t y t o R n O and t a k e on L t h e f i n a l u n i f o r m i t y w. r. t o r I R fl 0 ; c a l l t h i s p, t h e c o o r d i n a t e u n i f o r m i t y on L. Since t h e c o o r d i n a t e s a r e taken o n l y from a bounded s u b s e t of K , t h e e x t e r i o r product s A t of s and t depends on ( s , t ) E ( R II E l l 2 i n a uniformly continuous manner. Take an entourage N i n ( R n C~:={(s,t)€(Rno)~:N(s)AN(t) # 01, SN:= ( r x r ) [ C ~ ] : then c e r t a i n l y t h e r e e x i s t s a neighbourhood V of z e r o i n I' , such t h a t EN i s mapped by A i n t o B \ V; any image m u l t i v e c t o r g can be normed as i n d i c a t e d by ( * ) above; t h e combined mappingEN __t R n 0 i s uniformly continuous and induces a uniformly continuous mapping SN L , which c o i n c i d e s w i t h + :
..
-
1 EN
rxr
+
sN
*
r
R
~
O
L
One now proceeds a l o n g t h e l i n e s of t h e proof i n p r o p o s i t i o n 1 t o show t h e uniform c o n t i n u i t y of t h e m e e t : Define E N ' , R ' , S N ' a s above, and CN := (6k6)-'[cN'] S i n c e 6 [ R ] c R ' , w e have t h e f o l l o w i n g commutative diagram:
.
430
H. Szam bien
S"
'N
+
>-
L'
___)
L
f'
By the same arguments as above it contains a uniformly continuous upper row mapping. Now r is uniformly continuous and rxr is a quotient mapping; hence the lower row mapping which coincides with is continuous, too. Finally, p induces the coordinate topology, because scalar multiplication is continuous. 4 - e. d.
-
Any uniform space can be densely embedded into a unique complete uniform space, called its Weil-completion; continuation of the geometric structure is possible for UPGs, as is shown by PROPOSITION 3. The Weil-completion (w. r. to its coordinate uniformity) of an ordered pappian projective geometry of finite dimension at least two is again a UPG. Proof. Let (K,<) be the coordinatizing, field, K ^ its Weil-completion, which again carries the structure of an ordered field by fNi 65;p.14,Thm.2]. L" will denote the lattice of subspaces of (K^)" , n t 3, m:= 2 " ; Lc, R" : = R ( K " ) , and PA are used analogously to the notations in the proof of proposition 2. (KA)m is certainly complete, R" as a closed subset as well. Thus P"carries a complete uniform structure. Since the Grassmann-manifolds Lc are Zariskiclosed projective varieties in P , they are closed in the projective coordinate topology as well, [Le 65;~.337,Satz 1.21; thus they are complete. Hence L A is complete, and it is a UPG by proposition 2. Since it contains L densely, L A is uniformly isomorphic to the q . e. d. Weil-completion of L. It is shown by Massaza that the Dedekind-completion by cuts of an ordered field is an extension field K- of K. Furthermore K- is isomorphic to the Weil-completion K A of K; [Ma 69;p.331,3331 or [Pri 83;p.72,Satz 91. Considering ordered projective geometries L and L- of finite dimension over K resp. K-, one terms L the Dedekind-completion of L. Then propositiom 3 yields an analogue of Massaza's theorem: The Dedekind-completion of an ordered pappian PROPOSITION 4. projective geometry of finite dimension not less than two is isomorphic to its Weil-completion. Proof.
L^
=
L(K^) = L(K") = L".
q . e. d.
From Sz 81;(3.11) one derives that the coordinate field of any pappian UPG is of type V (called locally retrobounded in [Bo 66; TG, Ch.III.6, Exerc.1). Following [Pre 78;(5.4)1 a field of type V is orderable iff the set of all sums of squares is disjoint with some neighbourhood of -1. This condition singles out the orderable UPGs among the pappian UPGs, and it can of course be translated into geometric language.
Order and Uniform Structure in Projective Geometry
43 1
REFERENCES [ B i 791 B i r k h o f f , G.: L a t t i c e t h e o r y , h e r . Math. S o c i e t y C o l l o q . P u b l i c a t i o n s V o l . 25, Providence R I 1960
[Bo 661
Bourbaki, N.:
A l g s b r e , T o p o l o g i e GBnBrale; P a r i s 1966
[ B r 671 B r e i t s p r e c h e r , S . : Uniforme p r o j e k t i v e Ebenen; Mathematis c h e Z e i t s c h r i f t 95 ( 1 9 6 7 ) 139-168
[ G u 591 Guggenheimer, H . : The t o p o l o g y o f e l e m e n t a r y g e o m e t r y ; M a t h e m a t i c a J a p o n i c a e 5 ( 1 9 5 9 ) 1-26
[ L e 651 Lenz, H.: L e i p z i g 1965
V o r l e s u n g e n iiber p r o j e k t i v e G e o m e t r i e ;
[ M a 691 Massaza, C . : S u l completamento d e i campi o r d i n a t i ; R e n d i c o n t i d e l S e m i n a r i o Matematico, U n i v e r s i t 5 e P o l i t e c n i c o d i T o r i n o 2 9 ( 1 9 6 9 / 1 9 7 0 ) 329-348
[Mi 681 Misfeld, J.: Topologische p r o j e k t i v e Raume, D i s s e r t a t i o n Hamburg 1968 [ N i 651 Nieminen, T . : The c o m p l e t i o n s o f o r d e r a b l e t o p o l o g i c a l g r o u p s a n d f i e l d s ; A n n a l e s Academiae S c i e n t i a r w n F e n n i c a e S e r i e s A 365 ( 1 9 6 5 ) 1-15
[ P r e 781 P r e s t e l , A . , Z i e g l e r , M . : Model t h e o r e t i c m e t h o d s i n t h e t h e o r y of t o p o l o g i c a l f i e l d s ; J o u r n a l f i i r R e i n e und Angewandte M a t h e m a t i k 2 9 9 / 3 0 0 ( 1 9 7 8 ) 318-341 [ P r i 831
PrieB-Crampe,
S.:
Angeordnete S t r u k t u r e n ; B e r l i n 1983
Szambien H . : D e s a r g u e s s c h e u n i f o r m e E b e n e n ; Disserta[ S z 811 t i o n , Hannover 1 9 8 ; summary (German) i n R e s u l t a t e der M a t h e m a t i k 5 ( 1 9 8 2 ) 96-98 [ S z 861 Szambien H . : Uniform s t r u c t u r e s i n p r o j e c t i v e g e o m e t r y ; P r o c e e d i n g s of t h e S i x t h P r a g u e T o p o l o g i c a l Symposium 1 9 8 6 , B e r l i n ca. 1987
This Page Intentionally Left Blank
Annals of Discrete Mathematics 37 (1988) 433-450 0 Elsevier Science Publishers B.V. (North-Holland)
433
ON BLOCKING SETS I N FINITE PROJECTIVE AND AFFINE SPACES G i u s e p p e TALLINI 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 a ' d i Roma,"La
Sapienza" ( I t a l i a )
T h i s p a p e r i s a new v e r s i o n o f t h e f i r s t "Quaderno" [ 9 1 ( w r i t t e n i n i t a l i a n ) of t h e " S e m i n a r i o d i G e o m e t r i a C o m b i n a t o r i a " p u b l i s h e d by t h e U n i v e r s i t y o f L ' A q u i l a i n 1982. I n t h i s Quaderno w e proved s e v e r a l new r e s u l t s f o r b l o c k i n g s e t s ( w i t h r e s p e c t t o l i n e s ) i n P G ( r , q ) and AG(r,q). S i n c e t h i s p a p e r h a s found g r e a t i n t e r e s t (see r e f e r e n c e s [ll], C421). i t seems r e a s o n a b l e t o make i t a v a i l a b l e t o a w i d e r a u d i e n c e . ( N o t e t h a t i n [24] a r e s u l t o f C91 was proved i n a d i f f e r e n t way).
...,
1. k-SETS I N PROJECTIVE GALOIS SPACES L e t S, = P G ( r , q ) b e a p r o j e c t i v e G a l o i s s p a c e o f d i m e n s i o n r _ > 2 and q=ph ( p p r i m e ) . I f Sd d e n o t e s a d - d i m e n s i o n a l s u b s p a c e o f S, , w e have:
order
rl
.
I t i s well-known
.
t h a t t h e number o f d - d i m e n s i o n a l s u b s p a c e s Sd i n Sr i s
n
given
d
by (1.2)
Yr,d
=
;=" ~.
Denote by K a k - s e t o f P G ( r , q ) w i t h K + @ a n d K + P G ( r , q ) . D e n o t e by m and n r e s p e c t i v e l y t h e s m a l l e s t and t h e g r e a t e s t p o s i t i v e number o f p o i n t s i n which K is i n t e r s e c t e d by a d-dimensional s u b s p a c e Sd ( l < d S r ) o f S, In o t h e r words w e s e t :
.
Knsd
# gj}
(1.3)
m=min{lKnSdI
:
(1.4)
n=max{lKnSdI
: Knsd # @
We h a v e , o b v i o u s l y (1.5)
Isrn
}
=Od.
..
I f s=O, I , , ,% , by t,=t! w e d e n o t e t h e number o f d - d i m e n s i o n a l s u b s p a c e s o f S,, e a c h o f which c o n t a i n s e x a c t l y s p o i n t s of a f i x e d k - s e t K . The numbers t, a r e c a l l e d t h e c h a r a c t e r s o f i n d e x s o f s e t K , w i t h r e s p e c t t o d i m e n s i o n d. Fix h+lintegers mo,m, mh w i t h O < m , < m , <...<mh, A k-set K i s s a i d t o be of c l a s s [ m a , ml m,, w i t h r e s p e c t t o t h e d i m e n s i o n d , i f t , =O f o r a n y s d i f f e r e n t from mB,m,, mh.Moreover,a k - s e t K of c l a s s [mg,m,,...,mh]d, w i t h t h e a d d i t i o n a l c o n d i t i o n t,#-0 f o r s=m, ,ml , .mh, i s s a i d t o b e o f t y p e (mo,mI m h ) d , w i t h r e s p e c t t o d i m e n s i o n d , cf. [ 7 ] . Note t h a t , i f m=n,each Sd o f S r i n t e r s e c t s K i n z e r o o r m p o i n t s . S o i f r = 2 , e i t h e r we h a v e K = @ o r K=S2 or K i s a set o f t y p e (O,m), w i t h m=p' ( 0 5 t < h ) (cf. 171). If r 2 3 , w e h a v e two p o s s i b i l i t i e s : ( i ) d - d i m e n s i o n a l s u b s p a c e s e x t e r n a l t o K d o n o t e x i s t . ( I n t h i s case w e know t h a t e i t h e r K = @ o r K=S,). (ii) Suppose t h a t t h e r e i s a n s d w i t h o u t p o i n t s i n common w i t h K . (Then e i t h e r K i s o n e p o i n t o r t h e complement o f a h y p e r p l a n e ( c f . [ 7 ] , Prop. I , XIV)). So. w e suppose:
,...,
,...,
,..., Id,
...,
..
G. Tallini
434 (1.6)
.
l<m
.
Let K be a k-set of S r Denote by m and n , r e s p e c t i v e l y t h e s m a l l e s t and t h e g r e a t e s t c a r d i n a l i t y of t h e set Sd n K , i f Sd n K k $.Then K i s c a l l e d a ( k ; O,m,n)d-set. The c h a r a c t e r s of a k - s e t K o f S, s a t i s f y Lhe f o l l o w i n g c o n d i t i o n s ( c i . [7]) :
(1.7)
(1.7),,
and (1. 7)1,1
imply:
e
(1.8)
S-m
s2 ts =
[(lC-l
r-
2
+ Yr-
,d .
I ,
d-
I
1
F i x two i n t e g e r s M,N. By ( 1 . 7 ) and (1.8) we have: s-m &(N-s)(s-Ff)t,=-F
(1.10)
,
F=F(k,M,N,to,r +
Yr-z.d-
Suppose 14Tn
,
2
d , q ) = Yr-s,d-ok2-
] +MI{( r , d
-to )
k [(M+N-~)Y~-~,~-~+
-
M l m , then F S 0
(1.11)
Moreover, w e claim t h a t
<=>
F=O
(1.12)
N=n, M=m,
t,+,=
...=t
=O.
"-1
I n f a c t N > n i m p l i e s F # 0 ( s i n c e ' t n + 0) and M < m i m p l i e s F#O t o o . So, when F=O i t f o l l o w s N=n and M=in. C o n s e q u e n t l y , w e h a v e t,+, = . . . = tn - i =O. The c o n v e r s e is o b v i o u s . By (1.12) w e h a v e (1.13)
F=O
+>
N=n, M=m,
K i s of c l a s s [O,m,nId
.
S i n c e F L U , i f N a n and M g m i t f o l l o w s
A=
[( M f N - l ) Y r - l , d - l
Then, i f N 2 n and M < m ,
(1.15)
to 2
b r , d -to)
tYr-z,d-~]Z-4MNYr-z,d-Z
w e have:
yr,d - [(M+N-l)yr-t,d-t
+Yr-z,d-z
3'
/4MNyr-z,d-2
I n (1.15) t h e e q u a l i t y h o l d s i f , and o n l y i f , A =O. S i n c e F S O , i n (1.15) h o l d s i f , and o n l y i f , F=O and k=k,:
=
[(m+n-1) yr-
I
,d-,
'Yr-n,d-z]
the
'
equality
I2 Y r - z , d - z
t h a t i s i f , a n d o n l y i f , (1.13) h o l d s w i t h k=ko. C o n d i t i o n (1.15) d o e s n o t depend on k. When
its
right-hand
side
is
On Blocking Sets in Finite Projective and Affine Spaces
.
p o s i t i v e , (1.15) g i v e s u s a l o w e r bound f o r t o s i g n i f i c a n t , w e c a n e x c l u d e t h e e x i s t e n c e o f some I f M=m and N=n, i t f o l l o w s F S O . So w e h a v e a h o l d s and c o n s e q u e n t l y ( 1 . 1 5 ) . Then w e c a n d e d u c e
<
k, I k
(1.16)
43 5
When t h i s l o w e r bound i s k-sets. s p e c i a l case i n which ( 1 . 1 4 ) t h e bound f o r k:
k,
where k, and k, a r e t h e s o l u t i o n s o f t h e e q u a t i o n F=O ( i n which w e p u t M=m, IJ=n). I n ( 1 . 1 6 ) k = k , ( o r k=k,) h o l d s i f , and o n l y i f , F=O. T h a t i s i f , and o n l y i f , (1.13) h o l d s . Moreover, w e n o t e t h a t i f l = n , M=m, ( 1 . 1 5 ) becomes: (1.17)
t o 2 Y r - 2 , d - z {[4mn -
Yr,d/Yr-z.d-2
1
-
/ Y ~ . , , ~ - ~ I ~/4mn. }
[l+(m+n-l)yr-,,d-,
W e set:
(1.18)
D
2
:= 4m-(m+n-l)
By (1. 2 ) , we h a v e : ,
Yr-,,d-, /yr-z,d-z=br-i/6d-i
Yr,d / Y r - z , d - z =
3r6r-i'Gdad-?
So (1.17) c a n b e w r i t t e n i n t h e form: (1.19)
t o 2 Yr-z,d-2
[Dq
2r+d-2
+P(q)
1 /4mn
adai-1
where (1.20)
P(q):=D
6d
+ q d @ r - z (fir-, +2q'-')
-l
- 4mnqd6r-1 6 r -d
-1
-6d 6d
- 1
1
-
[2(m+n-1) f i r - , + 6 d - ,
i s a p o l y n o m i a l f u n c t i o n whose d e g r e e i s less t h a n 2r+d -2. (1.21)
A(q):=D q
zr
+ d--2
I
Now w e set
+ Kq).
So ( 1 . 1 9 ) becomes: (1.22)
t o >Yr-z,d-z
A(q)/4mn +d
*:-1
-
S u p p o s e D=4mn - (m+n-l)'>O. S i n c e A(0)=-(m-njz< 0, t h e r e e x i s t s i n t h e s e t of r e a l p o s i t i v e numbers R f , a t l e a s t o n e s o l u t i o n o f t h e e q u a t i o n A(q)=O. D e n o t e by q, t h e g r e a t e s t s o l u t i o n of t h e e q u a t i o n A(q)=O i n R f We h a v e A(q)>O f o r e a c h q > q o , and
.
D=4mn-(m+n-l)' S i n c e n-( h - 1 ) ' > (1.23)
=- [.-(
.
0 , w e c a n write:
n<(hh+l)'+>D>O
C o n d i t i o n (1.23)
[
h-1j2]. n-( &+1)')
=>
to>O,
W
q>qo.
i m p l i e s t h e f o l l o w i n g theorem:
I. If q > q , i n P G ( r , q ) t h e r e i s no (k;O,m,n)d-set w i t h t,=O and n < ( h h + l ) ' . (We s h o u l d remember t h a t q o is t h e g r e a t e s t real p o s i t i v e s o l u t i o n of t h e e q u a t i o n A(q)=O, w i t h A ( q ) d e f i n e d i n ( 1 . 2 1 ) ) . S u p p o s e r = 2 , d = l . By ( 1 . 2 2 ) ,
we h a v e
G. Tallini
436
The i m p o r t a n c e o f ( 1 . 2 2 ) , (1.23) a n d ( 1 . 2 4 ) c a n b e shown i n s e v e r a l s p e c i a l cases. The f o l l o w i n g are e x a m p l e s of s u c h cases. L e t K b e a k - s e t i n pG(2,q), w i t h m = l and n=3. Then c o n d i t i o n ( 1 . 2 4 ) h o l d s and q:4,309 So t h e f o l l o w i n g i s a C o r o l l a r y o f Theorem I.
... .
11. I n PG(2,q) i f q > 4 , a k - s e t w i t h m = l , n=3, a d m i t s a t l e a s t to e x t e r n a l l i n e s , w i t h to>[3q2-12q-41/12. (Note t h a t a n i r r e d u c i b l e c u b i c i s s u c h ‘.-set. When q 4 , t h e e x i s t e n c e of i r r e d u c i b l e c u b i c s w i t h o u t e x t e r n a l l i n e s i s w e l l known. One o f t h e s e i s t h e u n i t a l of e q u a t i o n x3+y3+z3=~). It f o l l o w s t h a t i n P G ( r , q ) , q > 4 , e a c h k - s e t o f c l a s s [ O , 1 , 2 , 3 I 1 , w i t h r e s p e c t t o l i n e s , always admits e x t e r n a l l i n e s .
<
L e t K b e a k - s e t i n PG(2,q) w i t h m=2 and n < 5 . C o n d i t i o n (1.23) we have qo=11,20 Then
....
holds
and
111. A k - s e t o f PG(2,q) o f class [ 0 , 2 , 3 , 4 , 5 ] , admits a t least t o e x t e r n a l of l i n e s i f q > 11,where t,>[4q2-44q-9]/40,Then i n P G ( r , q ) , q > 11, a k - s e t c l a s s [O,2,3,4,51,a l w a y s a d m i t s e x t e r n a l l i n e s .
Let u s now c o n s i d e r a p a r t i c u l a r case o f Prop.111. Suppose q>:1 and q e v e n i n P G ( 2 , q ) ; l e t K , and K, d e n o t e t w o d i s t i n c t (q+Z)-arcs i n s u c h a p l a n e . S u p p o s e t h a t a n a l g e b r i c c u r v e %‘ o f d e g r e e 5 i n PG(Z,q),q >11, s a t i s f i e s t h e following conditions: ( i ) $9 c o n t a i n s n o l i n e s , ( i i ) $9 h a s n o e x t e r n a l l i n e s . Then % a d m i t s t a n g e n t l i n e s and K , !J K, a d m i t s e x t e r n a l l i n e s . 2 . THE k-SETS OF AG(r,q) L e t A,=AG(r,q) b e a n a f f i n e Galois s p a c e o f d i m e n s i o n r > 2 and o r d e r q. L e t K be a k - s e t of A , w i t h K?$6 and K # A , . Denote by m and n r e s p e c t i v e l y t h e s m a l l e s t , a n d t h e g r e a t e s t p o s i t i v e number of p o i n t s common t o K and a d - s u b s p a c e Ad o f A , , w i t h A d n K= $6 Let u s put t h e r e f o r e
.
{ 41 n=max I K n {
(2.1)
m=min 11: f l
(2.2)
:
Ad(
O b v i o u s l y , we h a v e
l<m
Ad
n K *$6}
Ad
nK
f
$6).
l A d l =qd.
Denote by t,=t: ( s = O , l , . . . , q d ) t h e number of d - d i m e n s i o n a l s u b s p a c e s of A , , e a c h of which c o n t a i n s e x a c t l y s p o i n t s of set K.The numbers t , are c a l l e d t h e c h a r a c t e r s . As w i t h t h e p r o j e c t i v e case, t h e k - s e t K i s s a i d t o b e o f c l a s s Em,, m l mkId, w i t h r e s p e c t t o t h e d i m e n s i o n d , i f t , = O f o r e v e r y s # m , , , m , , ..,mp Suppose K i s o f class[mo,ml , If i n a d d i t i o n tSZ0 when s=mo , m 1 , v , t h e n K i s s a i d to b e of t y p e ( m o , m l , . . . , mejd. If m=n, w e h a v e e i t h e r t O = O o r t o > O . I n t h e case t O = O , K i s o f t y p e (m)d C o n s e q u e n t l y , n e i t h e r K= @ n o r K= AG(r,q) ,a c o n t r a d i c t i o n . Suppose now t o> 0. I n t h i s case K i s o f t y p e (0, m)d.So,if w e h a v e t h e a d d i t i o n a l c o n d i t i o n 1 >3, set K i s o n e p o i n t (cf.[5]). I n t h e sequel l e t u s suppose
,...,
.
.. .,meld.
...,
(2.3)
.
l < m <
n
The c h a r a c t e r s o f a k-set K s a t i s f y t h e f o l l o w i n g e q u a t i o n s , ( t h e p r o o f
is
On Blocking Sets in Finite Projective and Affine Spaces
similar t o t h e p r o j e c t i v e case).
(2.4)
C o n d i t i o n s (2.4)11 and (2.4),11 i m p l y :
(2.5) Now we f i x two i n t e g e r s P4 and N. We have:
I n t h e s e q u e l , w e suppose N z n and M s m . These c o n d i t i o n s imply 0
<
n
2
(N-s)(s-M)ts
=-Yr-23d-2
F
I
s=m
So, i t f o l l o w s F <_ 0.
(2.8)
We h a v e ( a s w i t h (1.12)):
(2.9)
F=O
N=n,
M=m,
tm+l=...=tn-l=O
t h a t is,
(2.10) As F
F=O
5 0,
N=n, M a , and K is e i t h e r of t y p e (O,m,n)d or of t y p e (m,n)d a c c o r d i n g t o t o >0, o r t o = 0.
<=3
we have
C o n d i t i o n s (2.11) and (2.12) h o l d i n t h e case M=mand N=n, t o o . I n (2.12) e q u a l i t y h o l d s i f , and o n l y i f , A=O. A s F G O , w e have: In
(2.12)
e q u a l i t y h o l d s i f , a n d o n l y i f , F=O w i t h k=k,
:=
t h a t i s i f , and o n l y i f ,
[ l+(m+n-l)$r,_l / G - , ] /2; (2.10) h o l d s w i t h k=k, .
437
438
G. TaElirzi
I f t h e right:-hand s i d e o f ( 2 . 1 2 ) i s p o s i t i v e , we g e t a l o w e r bound f o r t o . I n t h i s c a s e we can exclude t h e e x i s t e n c e of p a r t i c u l a r k-sets. I f M=m a n d N=n, i t f o l l o w s F < 0 a n d e i t h e r c o n d i t i o n ( 2 . 1 1 ) o r (2.12) h o l d s . So, w e c a n d e d u c e t h e b o u n d s f o r k
kl 5 k I k ,
(2.13)
w h e r e k, a n d kZ a r e t h e s o l u t i o n s o f ( t h e e q u a t i o n ) N=n)
F=O
M=m
and
I n ( 2 . 1 3 ) w e h a v e k=k, or k=k, i f , and o n l y i f , F=O; t h a t i s i f , a n d o n l y (2.10) holds. W e o b s e r v e a g a i n t h a t , i f N=n a n d M=m, ( 2 . 1 2 ) becomes
if,
.
(in
which
By ( 1 . 1 8 ) a n d ( 2 . 1 4 ) w e h a v e :
where P ' ( q ) is t h e f o l l o w i n g polynomial o f q w i t h
d e g r e e l e s s t h a n 2r-2:
8
Put
,
B ( q ) := D q 2 ' - * + P ' ( q )
(2.17)
D :=4mn-(m+n-1) 2
.
So ( 2 . 1 5 ) becomes t o >/
(2.18)
.
2
Y ~ d -- 2 ~B ( q ) / 4 m n a d - ,
I f r = 2 a n d d = l , ( 2 . 1 8 ) becomes t o >B(q)l4mn=
(2.19)
{
Dqa, -
9,
[ (mtn)
2
-11 -1
j
/4mn.
S i n c e B(O)=-(n-m)'< 0 , i f D > C ) , t h e e q u a t i o n B(q)=O h 2 s r e a l p o s i t i v e s o l u t i o n s . D e n o t e by qb t h e g r e a t e s t p o s i t i v e s o l u t i o n o f B(q)=O. Then B ( q ) > O f o r e a c h q>qA We h a v e :
.
D=4mn-( m+n-1 S i n c e n-(Cn-l)
(2.20)
2
)2= -
[n-( d'T1-1)~].[n-(
> 0, by p r e v i o u s e q u a l i t y
e> D
n<(h+1)'
> O
=>
fit1 ) 2
].
it follows:
to>3,
v
q>qb
.
By ( 2 . 2 0 ) w e o b t a i n
I V . I f q>qA
,
t h e n i n AG(r,q)
t h e r e i s no k - s e t w i t h t , = O
and n < ( h + l )
2
.
3. SPREADS WITH LINES I N P G ( r , q ) L e t S , = P G ( r , q ) b e a p r o j e c t i v e G a l o i s s p a c e o f d i m e n s i o n r a n d o r d e r q=ph ( p p r i m e ) . A f a m i l y y ( # @) of l i n e s o f P G ( r , q ) m u t u a l l y skew i s c a l l e d a s p r e a d i n P G ( r , q ) . If t h e u n i o n S o f t h e l i n e s o f 9 c o i n c i d e s w i t h P G ( r , q ) ,
On Blocking Sets in Finite Projective arid Affine Spaces
439
t.he s p r e a d 9 i s s a i d 1.0 b e 1.ot.al or a s p r e a d o f P G ( r , q ) ; i f i t is not^, Y i s s a i d 1.0 b e p a r t i a l . A s p r e a d .v i n P G ( r , q ) i s s a i d 1.0 b e m a x i m a l i f i n P G ( r , q ) t . h e r e e x i s t ~ sn o s p r e a d .’f‘ s u c h t.hai. ,Y’ C .Y’: L?t. b e a s p r e a d i n P G ( r , q ) , r r 4 . C o n s i d e r t.he s u b s p a c e s S , o f P G ( r , q ) t h a t c o n t . a i n at. least. o n e l i n e o f .Y.Den0t.e by m a n d n r e s p e c t i v e l y t h e T h a t is, s m a l l e s t . a n d t.he g r e a l . e s t number o f l i n e s o f .Y c o n t a i n e d i n a S , i f Y‘nS, dcnot.es t h e set of l i n e s of Y c o n t a i n e d i n S, , t.hen:
.
ii.yns,i
(3.1)
m=min
(3.2)
n=max{i
~ , n g~} z
:
ms, I
S,
n ,Y i’. @
}.
Obviously, it h o l d s
(3.3)
.
1rin1nsl+q 2
...,
We d e f i n e c h a r a c t e r t , (s=O,l, I+q2) of a spread Y i n PG(r,q) with r e s p e c t . 1.0 d i m e n s i o n 3 , t.he number of s u b s p a c e s S , of P G ( r , q ) c o n t a i n i n g e x a c t l y s l i n e s of c Y ( c f . [ 8 ] ) . A s p r e a d .Y is s a i d t o b e o f c l a s s [m,,m, , .,m,] w h e r e O S ! J , < ~ ~ < < m e-; mo,m,,. ,mp.1fgy 5 q%l , w i t h r e s p e c t . t o t h e d i m e n s i o n 3 , i f t,=O f o r e a c h s . , m p , t.he:l :/is s a i d i s of c l a s s [m,,m, meland m o r e o v e r t , f O V s = m o , m l t o b e of t y p e (mo,m,,.-.,mg) w i t h r e s p e c t t o dimension 3. I f m=n i n ( 3 . 3 ) , n e c e s s a r i l y .Y i s of c l a s s [0,m] w i t h r e s p e c t t o d i m e n s i o n 3. I n P G ( r , q ) t h e r e a r e n o s p r e a d s wit-h a s i n g l e c h a r a c t . e r d i f f e r e n t f r o m z e r o . M o r e o v e r , s p r e a d s of t y p e ( 0 , m ) e x i s t o n l y i.f r=4 ( c f . [ 8 ] ) . Consequent . ] ~ , i f r s 5 , then rncn i n (3.3). I< The c h a r a c t e r s o f a s p r e a d Y s a t i s f y t h e f o l l o w i n g e q u a t i o n s , w h e r e (cf. [Sl):
..
..
,..
,...,
...
lyl=
(3.4)
1 stm
s(s-l)t,=
Conditions (3.4)
(3.5)
and (3.4),,
k(k-1)
imply
s-m
However we f i x t w o i n t e g e r s 1l. a n d N , i t h o l d s
z n
(3.6)
5-m
t, =
(h-S)(S-!l)
-9
where
(3.7)
! P = 91 ( k , M , N , q , r , t ,
) = k 2 - k [l+(M+N-l)Yr-z,~
]
I n t h e s e q u e l , we s u p p o s e N T n , b k m . T h e c o n d i t i o n s i m p l y So, i t f o l l o w s
z (N-s)
s-m
(s-Pl)
spco.
(3.8) Suppose m
(3.9) that is
t,lO.
A s w i t h (1.12) and ( 2 . 9 ) ,
9, =O
e
N=n,
M=m,
we have
tm+,=. ..=t,-,=0
+ MlV(
Yr.3 -to
)
.
440
C. Tallini
If N=n and M=m, (3.11)
to
N=n, M=m, Y o f t y p e (O,m,n) i f t o > O , o r Y o f t y p e (m,n) i f t o = O , b e i n g r = 4 , 5 n e c e s s a r i l y , [81.
<=>
y=0
(3.10)
( 3 . 6 ) and (3.8) imply:
> {
k2-k Ll+(m+n-l)yr-2s, A
+my,,,}
/mn,
where t h e e q u a l i t y h o l d s i f , and o n l y i f , ( 3 . 1 0 ) h o l d s . Since
I p l O , w e have 2
(3.12)
A
=
- 4 m ( Y , , ~ - t o )2 0 ,
[(MtN-1) 7, - ~ , , +1]
which i m p l i e s
(3.13)
t o2
{ 4Mi y r , 3 -
[lt(M+N-l) y r - 2 , 1
]'} /4MN.
I n p a r t i c u l a r ( 3 . 1 2 ) and ( 3 . 1 3 ) h o l d i f M=m and N=n. I n (3.13) t h e e q u a l i t y h o l d s i f , and o n l y i f , A=O. S i n c e that : e q u a l i t y h o l d s i n (3.13) i f , and o n l y i f , Tp=O w i t h (3.14)
k=
l+(mtn-1)
)'r-2,
11 2
k,
I
we
and
(3.12)
,
t h a t i s i f , and o n l y i f , ( 3 . 1 0 ) and ( 3 . 1 4 ) h o l d . I f , i n p a r t i c u l a r , w e p u t N=n, M=m i n ( 3 . 7 ) , since (3.13) are s a t i s f i e d , w e o b t a i n t h e bounds (3.15)
k <
Ips0
or
k , ,
where k, and k, a r e t h e s o l u t i o n s o f Sp =O. I t r e s u l t s k=k, or k=k2 i f , and o n l y i f , (3.10) h o l d s . Moreover, w e n o t e t h a t , i f N=n, M=m, m < n , r 2 4 , (3.13) becomes (3.16)
have
y C 0,
t o 2{4mnyr,, - [l+(m+n-l)yr-2,,
in
(3.15)
I
I }
/4m.
Condition (3.16) is s i g n i f i c a n t , i f its right-hand s i d e i s p o s i t i v e . I t can b e w r i t t e n ( c f . (1.18)):
ED
q 4 r - 6 + F ( ~ ) I /4nn
6,q26,
where F ( q ) i s a p o l y n o m i a l i n q w i t h a degre'e l e s s t h a n determined e a s i l y ) . I f w e p u t
C( q )=Dq'
(3.17)
r-
4r-6
(which
can
be
+P( q ) ,
we obtain to 2
(3.18)
c(q)/4m6$6263
,
Since C(q)= 6, 6,6,
{
4mnyr,, - [ I + ~ m + n - i J . y , . 2 , , J2}
2
2
,
t h e n C(O)=-(n-m)< 0. So, i f D=4mn-(m+n-l) > 0, t h e e q u a t i o n C ( q ) = 0 h a s r e a l p o s i t i v e s o l u t i o n s . Denote by T o t h e g r e a t e s t r e a l p o s i t i v e s o l u t i o n o f 2 C(q)=O. Then C ( q ) > O f o r e a c h q > ? O . We h a v e D=4mn-(m+n-l) = [n-(6-1)2f* .[n-(h+l)*]
. By
t h i s , s i n c e n-(&l)'
>
0, w e o b t a i n
On Blocking Sets in Finite Projective and Affine Spaces (3.19)
n<
(&n+l)
2
G> D > 0
=>
to
4,
, there
v
q>;io
44 1
.
By (3.19) i t f o l l o w s : q >
V. I n PG(r q ) w i t h r > 4 , f o r e a c h and n < ( S m + l ) j
.
are no s p r e a d s w i t h t,=O
4. BLOCKING SETS I N P G ( r , q ) A s u b s e t S o f P G ( r , q ) is c a l l e d a b l o c k i n g s e t w i t h r e s p e c t t o l i n e s , i f i t s a t i s f i e s the following conditions. ( i ) Each l i n e o f P G ( r , q ) i n t e r s e c t s S. ( i i ) S c o n t a i n s n o l i n e s of P G ( r , q ) . It f o l l o w s i m m e d i a t e l y t h a t :
V I . The complement o f a b l o c k i n g set i s a b l o c k i n g set. The s e t H n S i s a b l o c k i n g set of
V I I . Let H be any subspace of PG(r,q). w i t h r e s p e c t t o t h e l i n e s o f H.
H
A b l o c k i n g s e t S i s s a i d t o b e i r r e d u c i b l e i f f o r a n y p o i n t P E S t h e set S - ( P l i s n o t a b l o c k i n g set, i . e . i f a t l e a s t o n e t a n g e n t p a s s e s t h r o u g h each p o i n t P of S. C l e a r l y , e a c h b l o c k i n g set S c o n t a i n s some i r r e d u c i b l e b l o c k i n g set.
VIII. I n PG(2,q) t h e r e d u c i b l e b l o c k i n g set.
complement
of
an
irreducible
blocking
set
is
a
PROOF. Let S be a n i r r e d u c i b l e b l o c k i n g set of PG(2,q), s u c h t h a t i t s complement S' is i r r e d u c i b l e t o o . I f P E S , t h e r e e x i s t s a t l e a s t o n e t a n g e n t t of S t h r o u g h P. Denote by Q , , Q z two p o i n t s w i t h Q,$ Q2 and Q , , Q z E t - { P I A t a n g e n t ri t o S ' n e c e s s a r i l y p a s s e s t h r o u g h Q , ( i = l , Z ) , i . e . a q - s e c a n t of s. Moreover, r , # r 2 and ri + t. S i n c e S i s i r r e d u c i b l e , n e c e s s a r i l y S = ( r , - ( Q , l ) U U(r2 - ( G 2 ) ) U ( P ) ( I f w e add a p o i n t , w e h a v e a c o n t r a d i c t i o n ) . Then q > 2 and S'=PG(2,q)-S. Hence, S' i s r e d u c i b l e , a c o n t r a d i c t i o n . Another p r o o f o f VIII i s t h e f o l l o w i n g , L e t S and S'=PG(Z,q)-S b e i r r e d u c i b l e b l o c k i n g sets of PG(2,q). Through e a c h p o i n t P o f S t h e r e i s a t l e a s t o n e t a n g e n t t o S and t h r o u g h e a c h p o i n t P' of S' t h e r e i s a t l e a s t a q - s e c a n t S Moreover, t h e l i n e s p a s s i n g t h r o u g h d i s t i n c t p o i n t s o f t h e s e l i n e s are d i s t i n c t . Denote by t S t h e c h a r a c t e r of i n d e x s of S (Cf.R.1). Then t1+tq>az2. S i n c e t , t ...+t q = Q 2 , w e h a v e t , + t q = Q 2 , t 2 = = t q - l = O , t h a t i s S i s of t y p e ( 1 , q ) . T h i s i s a c o n t r a d i c t i o n , s i n c e s e t s of t y p e ( 1 , q ) d o n o t e x i s t ( c f . [ 7 ] ).
.
.
.
...
Problem. I n P G ( r , q ) , 9 2 3 , are t h e r e i r r e d u c i b l e b l o c k i n g sets whose comp l e m e n t s are i r r e d u c i b l e ? A n example o f i r r e d u c i b l e b l o c k i n g s e t i n P G ( 2 , q ) , w i t h q s q u a r e , is t h e Baer s u b p l a n e . I n f a c t , P G ( 2 , G ) i s a ( q + q t l ) - s e t o f t y p e (l,l+ Q). O t h e r e x a m p l e s c a n b e found i n t h e p r o o f o f X I I . The f o l l o w i n g t h e o r e m o f Bruen ( [ 2 ] , [ 3 ] ) c o n c e r n s t h e b l o c k i n g sets of PG( 2 ,q 1. I X . Let S be a b l o c k i n g s e t of PG(2,q).
(4.1)
q+dj+l
<
I SI <
Then q2-
sq
.
The e q u a l i t y h o l d s o n t h e l e f t - h a n d s i d e i f , and o n l y i f , S i s a B a e r s u b p l a n e ( t h a t on t h e r i g h t - h a n d s i d e h o l d s i f , and o n l y i f , S i s a complement o f a
G. Tallbzi
442 Baer s u b p l a n e ) .
No b l o c k i n g set. e x i s t s i n P G ( 2 , 2 ) . From V I I i t l o l l o w s
I t . can b e p r o v e d d i r e c t . l y o r by ( 4 . 1 ) .
X. No b l o c k i n g set. e x i s t s i n P C ( r , Z ) , r 2 2 . Now w e p r o v e t h a t
X I . I n PG(2,3) t h e r e a r e e x a c t l y two b l o c k i n g t h e s e h a s 6 p o i n t s , t.he o t h e r h a s 7 p 0 i n t . s .
set.s,
u p t.o i s o m o r p h i s m s . O n e o f
PROOF. Each b l o c k i n g set. o f P C ( 2 , 3 ) h a s 6 or 7 p o i n t s by ( 4 . 1 ) . S i n c e P C ( 2 , 3 ) h a s 13 p o i n t s , i f S i s a b l o c k i n g set. w i t h I S I = 6 , t h e comSlemen1 S ' o f S i s a b l o c k i n g s e t . wit.h 7 p o i n t s , a n d v i c e v e r s a . T h e f o l l o w i n g e x a m p l e p r o v e s t.he C o n s i d e r tthe p o i n t s e x i s t e n c e o f a b l o c k i n g s e t . S w i t h ISI=6 (see a l s o [ l ] ) . U, ( O , l , l ) , ( u s i n g homogeneous c o o r d i n a t . e s ) : O , ( l , O , O ) , 0, (0,1,0),0,(0,0,1), U,(l,O,l), U , ( l , l , O ) , V ( O , l , - 1 ) , W ( l , O , - 1 ) . I t . i s v e r y e a s y t o v e r i f y t h a t t h e s e t { O,, U , , U , , U,, V,W i s a n i r r e d u c i b l e b l o c k i n g set. w i t h 6 p o i n t - s . N o w w e p r o v e t h a t a n y ot.hcr b l o c k i n g set. wit.h 6 p 0 i n t . s is p r o j e c t i v e l y e q u i v a l e n t t o t h e p r e v i o u s b l o c k i n g s e t . I t i s s u f f i c i e n t t.0 p r o v e that. e a c h b l o c k i n g s e t S w i t h 6 p o i n t s n e c e s s a r i l y h a s t.wo 3 - s e c a n t l i n e s , w h i c h mret. at. a p o i n t o f S. S i n c e S h a s t h e s m a l l e s t . c a r d i n a l i t y , S i s i r r e d u c i b l e . Then t - h r o u g h e a c h p o i n t O3 t h e r e i s a t l e a s t . o n e Langent t.o S. Two t . a n g e n t s t o S c a n n o t p a s s t.hrough 0, , b e c a u s e t.he ot.her 5 p o i n t s o f S s h o u l d l i e on t.he o t h e r two l i n e s t-hrough 0,. T h i s i s a c o n t r a d i c t i o n , b e c a u s e a l i n e s h o u l d b e c o n t a i n e d i n S. H e n c e , e v e r y l i n e t h r o u g h 0, , d i f f e r e n t . f r o m t.he ttangenl:, c o n t . a i n s a t least a f u r t . h e r p o i n t o f S a n d at. l e a s t o n e p o i n t . o f t h e complement of S. Then t h e r e m a i n i n g t w o p o i n t s of S l i e n e c e s s a r i l y on two d i s t i n c t . l i n e s t h r o u g h 0, , w h i c h a r e 3 - s e c a n t o f S. Let. 0, b e ( 0 , 0 , 1 ) . Assume t.he t.wo Moreover, p o i n t s o f t h e 3 - s e c a n t a n d o u t s i d e S a s O t = ( l , O > O ) and 0 , = ( O , l , O ) . l e t U , = ( l , 1 , O ) b e t h e point: common t o S arid tto a l i n e t h r o u g h 0, , w h i c h i s n o t a 3 - s e c a n t o f S. So w e o b t a i n e x a c t l y t.he b l o c k i n g s e t o f t h e p r e v i o u s example. A d i i f e r e n t . p r o o f of t h i s t h e o r e m c a n b e f o u n d i n H i r s c h f e l d [5). We r e c a l l t.he f o l l o w i n g :
1
THEORZM ( M a z z o c c a - T a l l i n i [6]). For a n y p r i m e power q t h e r e e x i s t s a n i n t - e g e r b = b p ( q ) , which d e p e n d s o n l y on q s u c h t h a t i n P G ( r , q ) no b l o c k i n g s e t e x i s t s i f , and o n l y i f , r i s great-er t h a n b .
By X a n d XI1 i t . is:
No\" w e p r o v e t h e f o l l o w i n g X I I . In a n y P G ( 2 , q ) w i t h q 2 3 , t h e r e a r e b l o c k i n g s e t s . PROOF. W e g i v e s e v e r a l e x a m p l e s of b l o c k i n g s e t s f o r e v e r y 4 2 3 . a ) D e n o t e by 0 , , 0 , , 0, t h r e e i n d e p e n d e n t p o i n t s o f P G ( 2 , q ) . Define t h e set
I t . i s e a s y t o v e r i f y t h a t S i s a b l o c k i n g set. w i t h lSl=qz -2q+4. The complement of S is a n i r r e d u c i b l e b l o c k i n g set. b ) Let C b e a c o n i c , T a p o i n t o f C a n d t t h e t a n g e n t on T t.o C. The set S=Cu L - { T } , i f q i s o d d , i s a n i r r e d u c i b l e b l o c k i n g s e t w i t h ISI=2q. S e t S is of t y p e (1,2,3,q) with r e s p e c t t o t h e l i n e s .
On Blocking Sets in Finite Projective and Affine Spaces
443
c ) Suppose q odd. L e t C be a c o n i c o f PG(2,q). d e n o t e by S t h e set o f e x t e r n a l p o i n t s o f C. C i s o f t y p e ( ( q - l ) / Z , ( q + I ) / Z , q ) . Then S i s a b l o c k i n g set w i t h IS1 =q(q+1)/2. d ) Suppose q s q u a r e . I n PG(2,q) a h e r m i t i a n form H and a B a e r s u b p l a n e S a r e b l o c k i n g sets w i t h IHI=qJ-+l and / S I = q + @ + l . They a r e i r r e d u c i b l e . e ) L e t T be t h e p o i n t - s e t of t h e s i d e s o f a t r i a n g l e i n PG(2,q). Let u s c o n s i d e r T-IVl, V 2 l , where VI , V2are two o f t h e t h r e e v e r t i c e s o f T, t h i s i s a b l o c k i n g set w i t h 3q-2 p o i n t s . From Prop. X I 1 i t f o l l o w s t h a t (4.3)
q23
bp(q)>2.
Now w e p r o v e t h a t
XIII. I f 1-23, t h e r e i s no b l o c k i n g s e t i n P G ( r , 3 ) . Hence, w e h a v e bp(3) = 2
(4.4)
.
PROOF. By VII, i t i s s u f f i c i e n t t o p r o v e t h a t no b l o c k i n g s e t e x i s t s i n PG(3,3). Suppose t h a t S is a b l o c k i n g set i n PG(3.3). By VII, i f ?G i s a p l a n e Then S i s of t y p e ( 6 , 7 ) w i t h o f P G ( 3 , 3 ) , rC ll S i s a b l o c k i n g set o f rC r e s p e c t t o t h e p l a n e s , a s a c o n s e q u e n c e of X I . Denote by t s , t7 t h e c h a r a c t e r s of S w i t h r e s p e c t t o t h e p l a n e s . W e have
.
I
(4.5)
By (4.5),
1
tY3
t g + t7=
=
40
6 t , + 7 t 7 = IS1tY2= 13151 3 0 t 6 + 4 2 t , = 4 IS1 ( I S 1 -1)
we obtain
and ( 4 . 5 ) , 1
,
t,= 280 - 13 IS(
(4.6)
I f w e s u b s t i t u t e t h e s e i n (4.5)lll I S 1 2 - 4 0 IS1
(4.7)
, we
+
t, = 13 ( S I - 240
.
have
420 = 0
.
So w e o b t a i n a c o n t r a d i c t i o n , s i n c e ( 4 . 7 ) h a s n o r e a l
solution
Consequently
t h e a s s e r t i o n i s proved.
We n o t e t h a t e a c h b l o c k i n g set S o f PG(2,4) i s s u c h t h a t 7 < I S ( I 1 4 . Moreover t h e p r o o f of X I 1 i m p l i e s t h e e x i s t e n c e o f a b l o c k i n g s e t f o r e a c h possible cardinality
.
X I V . I n P G ( 3 , q ) , w i t h q e v e n and q > 4 , t h e r e a r e b l o c k i n g sets. Hence, (4.8)
q>4,
q even
=>
b p ( 9 ) 2 3.
PROSF. The f o l l o w i n g i s a n example of a b l o c k i n g set i n PG(3,q) w i t h q e v e n and q > 4 . L e t W b e a ( q + 2 ) - a r c i n a p l a n e n a n d O , , 0, , 0, t h r e e of i t s t h e c o n e p r o j e c t i n g %? froin p o i n t s . L e t 0, b e a p 0 i r . t o u t s i d e $ Z . Denote by U 0,3,) and 0, C o n s i d e r t h e s u b s e t K g i v e n by -(O,O, u a ) t h e p o i n t s of n o u t s i d e l i n e s Ol 0,, O,O,, U301 and i n a d d i t i o n t h e p o i n t s 0, I 02, 0, ; b ) t h e p o i n t s o f p l a n e s 0, 020,, 02030,, O,O, 0, d e p r i v e d of t h e p o i n t s o f t h e e d g e s o f t h e t e t r a h e d r o n 0, O,O,O,, e x c l u d e d a b o v e . I f n, i s t h e face of t e t r a h e d r o n 0, 02030, o p p o s i t e t o 0, , K c a n b e w r i t t e n a s
.
r
r
444
G. Tallini
K:=[
rL'mlun2V",
u m4 - ~ o , o , u o 2 0 3 u 0 3 0 1 u o , o l u o 4 0 ~ u o , 0 3 ) l u {o,, 0 1 ,0 3 } ,
Now w e p r o v e t h a t K i s a b l o c k i n g set i f q.4. We b e g i n by o b s e r v i n g t h a t no l i n e i s c o n t a i n e d i n K . T h i s ' i s t r i v i a l f o r t h e l i n e s of t h e f a c e s of t h e t e t r a h e d r o n 0,0,030, and f o r t h e l i n e s o f t h e c o n e . The o t h e r l i n e s h a v e a t most s i x p o i n t , i n common w i t h K . So no l i n e i s c o n t a i n e d i n K , s i n c e q 2 8. Moreover, e a c h l i n e i n t e r s e c t s K . T h i s i s o b v i o u s f o r t h e l i n e s o f t h e f a c e s of t h e t e t r a h e d r o n , f o r t h o s e o f t h e c o n e and a l s o f o r t h e l i n e s which i n t e r sect t h e f a c e s o f t h e t e t r a h e d r o n o u t s i d e t h e e d g e s . Now s u p p o s e t h a t f i s a l i n e t h r o u g h a p o i n t o f t h e e d g e 0102 and a p o i n t o f t h e e d g e 0, (diffe. r e n t from O4 ). The p l a n e s p a n n e d by O , , and f c o n t a i n s a l i n e f o f Then f ' - { 0 1 ) i s c o n t a i n e d i n K , and so f n f ' € K . The same a r g u m e n t s h o l d f o r O,O, , O,O,. So, t h e a s s e r t i o n f o l l o w s . t h e p a i r of e d g e s O,O,, 0, 0, and
p,
XV.
r
I n P G ( 3 , q ) , w i t h q odd and q 3 7 , t h e r e a r e b l o c k i n g sets. Hence,
(4.9)
,
q 2 7
q
odd
=>
bp(q)
2 3.
PROOF. We g i v e a n example o f a b l o c k i n g set i n PG(3,q) w i t h q odd and q 2 7 . Let b e a p l a n e o f P G ( 3 , q ) , '$ a c o n i c o f n , 0, , O,, 03 t h r e e p o i n t s on v Denote by t,.,t 2 , t, t h e t a n g e n t s t o V on 0 , . O , , 03 r e s p e c t i v e l y . Put TI = t2n , t 3 , T, = t, n t, , T , = t , n t , . L e t 0, b e a p o i n t o u t s i d e m C o n s i d e r t h e set K g i v e n by: a ) t h e p o i n t s o f m o u t s i d e t h e l i n e s O,O, , 02g,O, 0, b u t w i t h t h e p o i n t s 0 1 . 02, 0 3 ; b ) t h e p o i n t s d i f f e r e n t from O4 o f t h e c o n e f! p r o j e c t i n g from 0, t h e s e t ( W U { T , t T,, T3 1 ) - { O l , 0 2 , 031. c ) The p o i n t s of t h e p l a n e s Ol O,O, , 0 1 0 ~ 0 ,, O,O,O, which a r e n o t o n t h e e d g e s b u t w i t h t h e p o i n t s 3,. O , , 0,. o f t h e t e t r a h e d r o n O,O,O,O,,
.
.
The p r o o f t h a t I! i s a b l o c k i n g set i s similar t o t h e p r o o f o f X I V .
5. BLOCKING SETS I N A G ( r , q ) Let AG(r,q) b e a n a f f i n e Galois s p a c e o f d i m e n s i o n r and o r d e r q. A
subset if
S of AG(r,q) i s c a l l e d a b l o c k i n g s e t w i t h r e s p e c t t o t h e l i n e s of AG(r,q) it s a t i s f i e s t h e following conditions: ( i ) Each l i n e o f AG(r,q) i n t e r s e c t s S.
( i i ) No l i n e of AG(r,q) i s c o n t a i n e d i n S. T h e M a z z o c c a - T a l l i n i Theorem a l s o h o l d s i n t h e
a f f i n e case: f o r any q , t h e r e e x i s t s a p o s i t i v e i n t e g e r b,(q) s u c h t h a t i n AG(r,q) no b l o c k i n g s e t e x i s t s i f , and o n l y i f , r i s g r e a t e r t h a n b,(q). ( c f . [ 6 ] ) . I t i s easy t o p r o v e t h a t
X V I . Let S r - , b e a AG(r,q)=PG(r,q)-S,-l set o f P G ( r , q ) .
h y p e r p l a n e of P G ( r , q ) . I f K i s a blocking set o f and K , i s a b l o c k i n g set of S , - , , t h e n K U K O i s a b l o c k i n g
Now w e p r o v e t h a t
X V I I . L e t bp(q) b e t h e g r e a t e s t v a l u e of r f o r which t h e r e i s a i n P G ( r , q ) ( c f . n . 4 ) . Then (5.1)
b,(q)
I bp(q)
blocking
set
.
PROOF. Suppose b,(q) > b p ( q ) . C o n s i d e r a h y p e r p l a n e S ( b p ( q ) ) = P G ( b p ( q ) , q ) of PG(bp(q)+l,q). L e t K ' be a b l o c k i n g s e t o f S ( b p ( q ) ) . I n AG(bp ( q ) + l , q ) = PC(b,(q)+l,q)-S(b,(q)) t h e r e i s a b l o c k i n g set K , s i n c e b p ( q ) + l l b , ( q ) . So, by X V I , K U K ' i s a blocking n e t of PG(bp(q)+l,q). T h i s i s a c o n t r a d i c t i o n , s i n c e
On Blocking Sets in Finite Projective and Affine Spaces b,(q)+l
445
'> b p ( q ) . Then t h e a s s e r t i o n i s proved.
It is easy t o prove t h a t
X V I I I . No b l o ck i n g set e x i s t s i n AG(2,q) w it h q=2,3. Hence
(5.2) Now w e prove t h a t
X I X . I n AG(2,q), w i t h q 2 5 , t h e r e are blocking sets. PROOF. W e prove t h e e x i s t e n c e w i t h t h e f o ll ow i n g example. Denote by ( x , y ) t h e a f f i n e c o o r d i n a t e s i n AG(2,q). Consider t h e set K whose p o i n t s a r e on t h e l i n e s of eq u at i o n s : x=O, y=O, x+y=l r e s p e c t i v e l y , e x c e p t t h e p o i n t s (0.0) ,(O, I), ( 1 , O ) and i n a d d i t i o n t h e p o i n t s ( 1 , l ) . (-1,l). ( 1 , - I ) . I f q i s odd, K i s a (3q-3)-set, w h i l e i f q i s even, K i s a (3q-5)-set. I t is easy t o prove t h a t i f q > 4 , K is a b l o c k in g set. I f q=4, K i s a 7 - set , which i s n o t a blocking set s i n c e , f o r example, i t c o n t a i n s t h e l i n e y = i x t i + l (where i i s such t h a t i2 +i+l=O).
XX. I n AG(Z,q), w i t h q t 4 , t h e r e are blocking sets. PROOF. We prove t h e e x i s t e n c e w i t h t h e f o ll ow i n g example. Denote p a r a l l e l l i n e s . L e t A,B be two p o i n t s w it h AEa, B E b . S e t
by
a,b
two
K :=(a- ( A 1 ) u ( b - ( B t ) u ( A B - ( A , B ) ). K i s a blocking set w i t h 3q-4 p o i n t s . P r o p o s i t i o n s X I X and XX imply t h a t ba(q) 2 2
(5.3)
if q > 4
.
X X I . Each blocking set of A G ( 2 . 4 ) h a s e i g h t p o i n t s .
.
PROOF. L e t K be a b l o c k i n g set of AG(2,4), t h e n K i s of c l a s s C1 ,2 ,3 ] Necessarily t 3 f 0, o t h e r w i s e K would b e a n arc, which a l w a y s h a s e x t e r n a l l i n e s . Moreover, t l P 0 s i n c e t h e complement of K h a s a l s o 3-secant l i n e s (it i s a blocking set t o o ) . P u t I K I =k. Then by (2.8) and (2.10) w e have kz-16k+60 I 0
=>
6 g k S 10
and k = 6
Moreover,we have ( K ' k = 6 k = 7 k = 8
K i s o f t y p e (1.3).
o r k = l O
i s t h e complement o f K ) :
I K'I 1 K'f I K'I
= 10
(K of t y p e ( 1 , 3 ) )
=
9
( K of t y p e ( 1 , 2 , 3 ) )
=
8
(K of t y p e ( 1 , 2 , 3 ) )
Hence, w e can suppose k = 6,7,8; i n f a c t , w e o b t a i n c a r d i n a l i t i e s b y c o n s i d e r i n g t h e complements. We have
.
the
other
possible
G. TaNini
446
\ t, + (5.4)
I
tl
+
t2
+
t3
=
20
2 t , t 3t3 = 5k t 2 + 3t3 = k ( L 1 ) / 2 .
I f k=6, f r o m ( 5 . 4 ) w e h a v e t , = 5 , t , = O , t l = 1 5 . L e t a , b b e two l i n e s w h i c h are 3 - s e c a n t o f K . N e c e s s a r i l y t h e y meet a t a p o i n t o f K , o t h e r w i s e t h e l i n e s t h r o u g h t h e i r common p o i n t would b e e x t e r n a l t o K. Then t h e p o i n t s of K a r e t h o s e o n two 3 - s e c a n t a , b a n d a n o t h e r p o i n t P , w h i c h n e c e s s a r i l y l i e s o n l i n e AB, where { A ) = a \ K , {B) = b \ K . S e t C : = a f l b , t h e n CP i s a 2 - s e c a n t o f K , a c o n t r a d i c t i o n (K is o f t y p e ( 1 , 3 ) ) . So K c a n n o t h a v e 6 p o i n t s . I f k=7, f r o m (5.4) w e h a v e t 3 =6, t 2 = 3 , t , = l l . L e t a , b b e two l i n e s whi.ch are 3 - s e c a n t o f K . They meet n e c e s s a r i l y a t a p o i n t 0 of K . S e t I A ) := a \ K , l a ) := b \ K , d e n o t e by a ' t h e l i n e t h r o u g h B a n d p a r a l l e l t o a , a n d by b ' t.he l i n e t h r o u g h A p a r a l l e l t o b. F i n a l l y p u t C : = a ' n b ' . F i v e p o i n t s o f K l i e on a U b. N e c e s s a r i l y o n e o f t h e r e m a i n i n g two p o i n t s o f K , s a y D , l i e s o n AB, a n d t h e o t h e r l i e s on a ' a n d b ' n a m e l y it i s p o i n t C. CD i s d i f f e r e n t f r o m t h e l i n e s a ' , b ' a n d f r o m OC t o o . ( I t i s p o s s i b l e t o c h o o s e O=(O,O), A=(l,O),B=(O,l) and C = ( l , l ) , w h e r e ( x , y ) a r e a f f i n e c o o r d i n a t e s i n AG(2,4). I t f o l l o w s t h a t A B a n d CC h a v e e q u a t i o n s x+y=l a n d x-y=x+y=O, r e s p e c t i v e l y ) . T h e n CD i n t e r s e c t s a a n d b a t two d i s t i n c t . p o i n t s , w h i c h are d i f f e r e n t from A a n d B, namely a t two p o i n t s of K . So CD i s c o n t a i n e d i n K , a c o n t r a d i c t i o n . So, K c a n n o t h a v e 7 p o i n t s . F i n a l l y , i f k = 8 , t h e e x i s t e n c e o f b l o c k i n g s e t s i s p r o v e d by t h e e x a m p l e g i v e n i n t h e proof of XX. From p r o p o s i t i o n X V I , i t f o l l o w s t h a t
X X I I . No b l o c k i n g s e t e x i s t s i n A G ( r , 4 ) , w i t h r 2 3 . H e n c e ,
(5.5)
ba ( 4 ) = 2
PROOF. S u p p o s e t h a t K i s a b l o c k i n g s e t o f A G ( r , 4 ) w i t h r 2 3 . Each p l a n e a i n t e r s e c t s K i n a b l o c k i n g s e t . Hence Ian K l =8, by X X I . Then K would h a v e a u n i q u e c h a r a c t e r d i f f e r e n t from 0 w i t h r e s p e c t t o t h e p l a n e s . Consequently,we h a v e K = @ or K=AG(r,4) ( c f . [ 7 ] ), a c o n t r a d i c t i o n .
Now we p r o v e :
XXIII. S u p p o s e t h a t i n AG(2,q) t h e r e e x i s t s a b l o c k i n g s e t K s u c h Then l i n e h a s a t l e a s t t w o p o i n t s i n K a n d two p o i n t s o u t s i d e K . s e t e x i s t s i n A G ( r + l , q ) . C o n s e q u e n t l y , e a c h bloc!cing s e t o f s u c h t h a t e i t . h e r t , # 0, o r t q - , $0. PROOF. L e t n = A G ( r , q ) b e a
hyperplane
o f AG(r+l,q),
t h a t every a blocking AG(b,(q),q) is
and K a
b l o c k i n g set i n
.
a , h a v i n g t , = O a n d t,-,=O. Fix a direction 6 , non-parallel to n For each p o i n t P E 7G -K c o n s i d e r t h e l i n e rp t h r o u g h P , p a r a l l e l t o 6 Define the set
.
K'
KU(
(J
(rp -(P))). P€n-K We p r o v e t h a t K ' i s a b l o c k i n g s e t . Each l i n e p a r a l l e l t o 6 i n t e r s e c t s e i t h e r K ' o r i t s complement i n a t l e a s t one p o i n t . L e t S b e any l i n e of AG(r+l,q) n o n - p a r a l l e l t o 6 D e n o t e by a the plane through S p a r a l l e l t o 6 Obviously a i s n o t p a r a l l e l t o n and Ft t h e p o i n t s common t o S' a n d L n t e r s e c t s rC i n a l i n e S ' . D e n o t e by Pl ,P2 K is t h e complement of K ) , d e n o t e by Q t + l , Q, +2 , ,Q, t h e p o i n t s comiilon t o S' a n d K , w i t h 2 It < q-2. =
.
(z
Then w e h a v e
.
,...,
.. .
On Blocking Sets in Finite Projective and Affine Spaces
447
S i n c e 2 5 t l q-2, K ' f l a i s a b l o c k i n g s e t o f a , and s o S ( c a ) i n t e r s e c t s e i t h e r K ' o r i t s complement; namely K ' i s a b l o c k i n g set o f AG(r+l,q). So, t h e a s s e r t i o n follows. Now w e c o n s t r u c t a n example o f b l o c k i n g s e t i n AG(2,q), where q 2 7 , w i t h t,=O, tq-l=O. F i x i n AG(2,q), f o u r d i s t i n c t l i n e s a , b , c , d w i t h a // c , b /I d and a n o n - p a r a l l e l l i n e t o b. S e t A : = a n d , B : = a n b , C : = c n b , D : = c n d , E:=DBnAC. F i x a l i n e b' // b w i t h E # b ' , b'S- b, b'S: d. F i n a l l y s e t F : = B D n b ' , G:=AC n b ' . It is easy t o v e r i f y t h a t
K
=
a u b u c u d u{E.F,G}-{A,B,C,D]
i s a blocking set with t h e required property.
By t h e p r e v i o u s p r o p o s i t i o n i t f o l l o w s t h a t
(5.6)
b,(q)
2
C o n d i t i o n s (5.6) and ( 5 . 1 )
3
if
q r 7
a g a i n imply ( 4 . 8 ) , ( 4 . 9 ) .
6. A CHARACTERIZATION OF HERMITIAN ARCS I n t h i s s e c t i o n w e c h a r a c t e r i z e h e r m i t i a n arcs. So, w e g e n e r a l i z e a r e s u l t o f Bruen and T h a s which i s a b o u t i r r e d u c i b l e b l o c k i n g sets ( c f . C4I). L e t nq be a p r o j e c t i v e p l a n e o f o r d e r q , d e s a r g u e s i a n o r n o t . Denote by S a s-set o f Sq w i t h t,=O and tl 2 IS I (Each i r r e d u c i b l e b l o c k i n g set. o f nq i s a n example o f s u c h a s e t ) . C o n s i d e r t h e c h a r a c t e r s t , o f s e t S. Denote by n Then t h e c h a r a c t e r s s a t i s f y t h e maximum o f Ir n S I , where r i s a l i n e o f rCq ( 1 . 7 ) . I f w e m u l t i p l y ( 1 . 7 ) j by n a n d s u b t r a c t (1.7)llfrom i t , w e o b t a i n
.
.
n- 1
(6.1)
E2 (n-i)t,=n(q2+q+l)-s(q+1)-(n-l)tl
1-
,
where IS I =s. The r i g h t - h a n d s i d e o f ( 6 . 1 ) is n e v e r l e s s t h a n 0; b u t i t i s 0 i f , a n d o n l y i f , t2=...=tn-2=0, namely i f , and o n l y i f , S i s o f t y p e ( 1 , n ) . It f o l l o w s t h a t
(6.2)
t, < ; n ( q Z + q + l ) - s ( q + l ) 1
/ (n-l),
where e q u a l i t y h o l d s i f , and o n l y i f , S i s o f t y p e ( 1 . n ) . i s t , 2 s a n d by ( 6 . 2 ) w e h a v e (6.3)
s 5
By o u r a s s u m p t i o n i t
n(q*+q+l)/(q+n),
where e q u a l i t y h o l d s i f , and o n l y i f , s = t , and e q u a l i t y h o l d s i n ( 6 . 2 ) , i s o f t y p e ( 1 , n ) ; i n t h i s case, f r o m ( 1 . 7 ) w e h a v e t h a t t,= q 2 + q + I - s
=
then S
sq/n = s(s-I)/n(n-l),
by which w e o b t a i n s=qJ-+1, n=Fq+l, namely S i s a ( q J - + l ) - s e t S i s a h e r m i t i a n arc. Then w e proved
of t y p e (l,l+Q),
i.e.
X X I V . L e t S a s-set of nq w i t h o u t e x t e r n a l l i n e s and s u c h t h a t t h e number t l of 1 - s e c a n t l i n e s o f S i s g r e a t e r o r e q u a l t o t h e c a r d i n a l i t y s of S. Denote by n t h e g r e a t e s t number o f p o i n t s common t o S and a l i n e . Then i t f o l l o w s t h a t
(6.4)
s
I n(q2+q+l)/(q+n).
E q u a l i t y h o l d s i f , a n d only i f , S i s a h e r m i t i a n arc.
448
G. Tallini
I t i s easy t o prove t h a t
n(q*+q+l)/(q+n) 5 qG+1
(6.5)
<=>
n <Sqtl
From X X I V w e o b t a i n
XXV. L e t S be a s-set o f PG(2,q) w i t h o u t e x t e r n a l l i n e s and such t h a t s 5 tl and n 5 q+l. Then i t i s :
s
(6.6)
5
qh+l
,
where e q u a l i t y h o l d s i f , and o n l y i f , S i s a h e r m i t i a n arc. W e c o n c l u d e t h i s paper w i t h t h e f o l l o w i n g t a b l e s a b o u t blocking sets (y=yes).
the
existence
of
W r , 4)
Y
Y
Y
Y
?
?
?
?
~
3
4 -
8
9,.
. ., I 3
16
_e
NO
Y
Y
Y
NO
NO
Y
Y
NO
NO
?
Y
NO
NO
?
7
REFERENCES [l]
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[2]
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[33
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643
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[5l
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449
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G. T a l l i n i , F i b r a z i o n i i n r e t t e d i PG(r , q ) , "Le
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37 ( 1 9 8 2 ) , 8 -27. [93
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Quaderno L'Aquila
d i un p i a n o g r a f i c o f i n i t o , con 1103 M . T a l l i n i S c a f a t i , S u i { k ; n j - a r c h i p a r t i c o l a r e r i g u a r d o a q u e l i i con due c a r a t t e r i , n o t a I e I1 Rend. Acc. Naz. L i n c e i , 8 , 4 0 (1966) 812-818 and 1020-1025.
REFERENCE I1 [l;j L.M. B a t t e n , Embedding t h e complement of a minimal p r o j e c t i v e p l a n e , t o appear i n Discrete Math.
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[12]
L.Berardi, I r r e d u c i b l e b l o c k i n g s e t s i n a Mat. I t a l . 1-A (1987), 39-47.
1131
L . B e r a r d i and A . B e u t e l s p a c h e r , B l o c k i n g s e t s of t h e known appear.
symmetric
in
set
design,
Boll.
biplanes,
a
Un. to
114) L. B e r a r d i , A.Beutelspacher and F. Eugeni, On ( s , t ; h ) - b l o c k i n g sets i n F i n i t e P r o j e c t i v e and A f f i n e S p a c e s , A t t i Sem. Mat. F i s . Univ. Modena, 32 (1983). 130-157.
[151 L.Berardi and F.Eugeni, On t h e c a r d i n a l i t y of b l o c k i n g sets J. Geom. 22 (1984), 5-14.
in
[16] L. B e r a r d i and F. Eugeni, On Blocking sets i n a f f i n e p l a n e s , (1984), 167-177.
PG(2,q),
J.Geom.
C17] L. B e r a r d i , F.Eugeni and 0. F e r r i , S u i b l o c k i n g sets n e i sistemi S t e i n e r , B o l l . Un. Mat. I t a l . Algebra e Geometria 1 (1984). 141-164.
22 di
[18] L . B e r a r d i and F. Eugeni, Blocking sets i n p r o j e c t i v e p l a n e of o r d e r f o u r , I n these Proceedings. 1193 L. B e r c r d i , Blocking s e t s i n t h e S(3,6,22), I n t h e s e Proceedings. E203 L. B e r a r d i , Blocking s e t s i n t h e S ( 4 , 7 , 2 3 ) , P r e p r i n t 1986.
large large
Mathieu Mathieu
design, design,
I:
the
case
11:
the
case
1211 L. B e r a r d i and F. Eugeni, Blocking sets i n t h e l a r g e Mathieu d e s i g n , 111: t h e c a s e S ( 5 , 8 , 2 4 ) , P r e p r i n t 1986. 8221 A.Beutelspacher and F.Eugeni, On t h e t y p e of p a r t i a l t - s p r e a d s i n p r o j e c t i v e s p a c e s , Discrete Math. 54 (1985), 241-257.
finite
t231 A. B e u t e l s p a c h e r and F. Eugeni, S u i b l o c k i n g s e t s d i d a t o i n d i c e con p a r t i c o l a r e r i g u a r d o a l l ' i n d i c e t r e , B o l l . Un. Mat. I t a l . 4-A (1985), 441-450.
G. Tallini
450 [24]
A . B e u t e l s p a c h e r and F. Eugeni, A l o w e r bound f o r p a r t i a l t - s p r e a d s o f l e v e l t-1 i n P G ( 2 t + l , q ) , A t t i Sem. Mat. F i s . Univ. Modena, 33 (1984),1-8.
[25] A . B e u t e l s p a c h e r and F. E u g e n i , On n - f o l d blocking sets, Proceedings " C o m b i n a t o r i c s 84" Bari, i n A n n a l s of Discrete Math. 30 (1986), 31-38. [26]
A. B e u t e l s p a c h e r and F. E u g e n i , On b l o c k i n g sets i n p r o j e c t i v e and a f f i n e s p a c e s o f l a r g e order,Communicated i n O b e r w o l f a c h , O c t o b e r 1985, Rend. Mat. (Roma), t o a p p e a r .
r271 A. B e u t e l s p a c h e r and F. Mazzocca, B l o c k i n g s e t s i n i n f i n i t e and a f f i n e s p a c e s , J. o f Geometry, 2 8 ( 1 9 8 7 ) , 111-116.
projective
[ 2 8 ] T.C. Brown, Monocromatic a f f i n e l i n e s i n f i n i t e v e c t o r s p a c e , J. Combinat o r i a l Theory (A), 39 ( 1 9 8 5 ) , 35-41.
[29] P. Cameron and F. Mazzocca, B i j e c t i o n s which p r e s e r v e b l o c k i n g a p p e a r on G e o m e t r i a e D e d i c a t a .
1303 M . J . d e R e s m i n i , On b l o c k i n g s e t s i n Geometry 18 ( 1 9 8 2 ) , 194-198.
C311 M.J. d e R e s m i n i , On 2 - b l o c k i n g sets n a t o r i a 20-B ( 1 9 8 5 ) , 59-69.
symmetric in
BIBD's
projective
with
planes,
sets,
to
I, 2 2 ,
J.
Combi-
Ars
[323 M.J. d e Resmini, On 3 - b l o c k i n g sets i n p r o j e c t i v e p l a n e s , P r e p r i n t 1987.
C331 D . A . Drake, B l o c k i n g sets i n ( 1 9 8 5 ) , 459-462.
block
designs,
J. Combin.
Theory
[341 F. E u g e n i , S u l l a e s i s t e n z a d i t - f i b r a z i o n i i n P G ( r , q ) d i B o l l . Un. Mat. 1 t a l . A l g e b r a e Geometria, 1 ( 1 9 8 4 ) , 19-43.
1351
F. E u g e n i a n d Proceedings.
E.
Mayer,
On
blocking
sets
of
index
fissato two,
In
A,
2
tipo, these
1361 F. Eugeni a n d S. I n n a m o r a t i , On f i x e d p a r i t y sets, I n t h e s e P r o c e e d i n g s .
1373 F. Eugeni and S. I n n a m o r a t i , Arcs a n d b l o c k i n g s e t s i n s y m m e t r i c d e s i g n s , P r e p r i n t 1987. C383 M.Huber, A c h a r a c t e r i z a t i o n o f Baer c o n e s i n G e o m e t r i a e D e d i c a t a . To a p p e a r .
finite
projective
spaces,
1393 F. Mazzocca, Some r e s u l t s o n b l o c k i n g s e t s , Announcements a t C o n f e r e n c e " C o m b i n a t o r i c s 84" Bari ( 1 9 8 4 ) and " F i n i t e Geometries" Oberwolfach (1985). C401 C. O'Keefe, A . V e n e z i a , B l o c k i n g sets i n A G ( r , 5 ) , Roma "La S a p i e n z a " n. 56 (1983).
Sem Geom.
Comb.
Univ.
r411 S. R a j o l a , Un esempio d i b l o c k i n g set i n PG(3,q) p e r o g n i q 2 5 , I n Proceedings.
these
c421 G. T a l l i n i , B l o c k i n g s e t s n e i sistemi d i S t e i n e r e d - b l o c k i n g P G ( r , q ) , Quaderno n. 3, Sem. Geom. Comb. 1st. Mat. Appl. Univ. (1983).
sets i n L'Aquila
45 1
Annals of Discrete Mathematics 37 (1988) 451-458 0 Elsevier Science Publishers B.V. (North-Holland)
SYMMETRIC DESIGNS WITHOUT OVALS AND EXTREMAL SELF-DUAL CODES
Vladimir D . Tonchev* I n s t i t u t e of Mathematics, S o f i a 1090, P. 0. Box 373, Bulgaria
ABSTRACT. Constructions of doubly-even s e l f - d u a l codes from symmetric d e s i g n s a r e discussed. I t i s shown t h a t t h e absence of o v a l s i n a design i s u s u a l l y necessary, and sometimes even 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 corresponding code t o be extremal.
1. INTRODUCTION The terminology and n o t a t i o n s from design and coding theory used i n t h i s paper a r e i n accordance with those from [ 3 A binary
t o r space V
1,
[ 5 1, and L8] r e s p e c t i v e l y .
(n,k) code C is a k-dimensional subspace of t h e n-dimensional vec-
1
Over GF(2). Given an ( n , k ) code C , t h e (n,n-k) code C
= {xeVn:
yx = 0 f o r each F C ) i s c a l l e d t h e orthogonal, o r dual of C. A matrix with t h e p r o p e r t y t h a t t h e l i n e a r span of i t s rows g e n e r a t e s t h e code C , i s a g e n e r a t o r
1 m a t r i x of C. The g e n e r a t o r m a t r i c e s of t h e d u a l code C a r e c a l l e d p a r i t y check matrices of C. We s h a l l o f t e n r e f f e r t o t h e elements of a code a s codewords, o r words only. The weight of a codeword i s t h e number of i t s nonzero p o s i t i o n s , and t h e minimum weight of a code i s t h e weight of a l i g h t e s t nonzero codeword. An
( n , k , d ) code i s an ( n , k ) code with minimum weight d. A code C i s self-orthogonal
(resp. self-dual)
i f CCC'
(resp. C =
2 ) . The
weights of a l l words i n a s e l f - o r t h o g o n a l code a r e even. I f i n a d d i t i o n a l l weights a r e d i v i s i b l e by 4 , t h e code i s c a l l e d doubly-even. d u a l ( n , n / 2 ) code e x i s t s i f and only if n
=
0 (mod
a),
A doubly-even
self-
and t h e minimum weight
d of such a code i s bounded by
(1)
d S 4[ 11/24] + 4 , (cf.
[a]).
A code s a t i s f y i n g t h e e q u a l i t y i n (1) is c a l l e d extremal. The code-
words of minimum weight i n an extremal doubly-even s e l f - d u a l code y i e l d o r 5-design provided t h a t
I-, 3-,
n 5 16, 8 , o r 0 (mod 2 4 ) .
The number of extremal codes i s f i n i t e ( b u t unknown), t h e s m a l l e s t open c a s e being f o r n = 7 2 . The following theorem d e s c r i b e s a c o n s t r u c t i o n of doubly-even s e l f - d u a l
CO-
d e s from symmetric d e s i g n s . THEOREM 1.1. Let A be an incidence matrix of a symmetric 2-(v,k,k)
of an odd o r d e r k-A.
Then
( i )If k E 3 (mod 4 ) , t h e code generated by t h e matrix
*Research p a r t i a l l y supported by t h e
cscMBunder
contract
37/1987-
design
V.D. Tonchev
452
i s a doubly-even s e l f - d u a l
(2v,v) code.
(ii) I f k Z 2 (mod 4 ) , then t h e code generated by t h e matrix
(3)
i s a doubly-even s e l f - d u a l
(2v+2, v + l ) code.
This statement i s a v a r i a n t of a m r e g e n e r a l c o n s t r u c t i o n ( c f . [ 21
,[
4) )
.
I n t h i s paper we a r e i n t e r e s t e d i n necessary and s u f f i c i e n t c o n d i t i o n s f o r a code c o n s t r u c t e d i n t h i s way t o be extremal.
In o t h e r words, what combinato-
r i a l p r o p e r t i e s should a design possess i n o r d e r t o produce an extremal code? A notion p l a y i n g an important r o l e i n t h e study of symmetric d e s i g n s , espe-
c i a l l y p r o j e c t i v e p l a n e s and b i p l a n e s , is t h a t of "arc" o r "oval". An a r c i n a design i s a s u b s e t S of p o i n t s such t h a t each block i n t e r s e c t s S i n a t most 2 p o i n t s . The c a r d i n a l i t y of an a r c S i n a n o n t r i v i a l symmetric 2-(v,k,h)
design
of an odd o r d e r k-h i s bounded by
IS/ S (k+h-l)/h.
(4)
An a r c with maximum number of p o i n t s i n t h e sense of bound
( 4 ) i s c a l l e d an
oval [ 2 ] . I t w i l l be seen by t h e next e x p o s i t i o n t h a t t h e absence of o v a l s i n a sym-
m e t r i c design i s u s u a l l y necessary, and sometimes even 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 corresponding s e l f - d u a l code t o be extremal. 2 . ARCS AND CODES
Let D be a symmetric 2-(v,k,h) Denoting by ni
design with an a r c S of s i z e s , i . e .
IS/ =s.
( i = 0,1,2) t h e number of blocks having e x a c t l y i common p o i n t s
w i t h S , w e have:
no
+
nl
+
*2 =
vt
Consider t h e code with g e n e r a t o r matrix ( 2 ) defined by D . C l e a r l y , t h e weight of t h e sum (over GF(2)) of s rows of t h e g e n e r a t o r matrix ( 2 ) indexed by t h e p o i n t s of an a r c of s i z e s i s s + n l = s(k+l)-(s-l)A). S i m i l a r l y , t h e sum of s
r o w s of t h e matrix ( 3 ) , indexed by t h e p o i n t s of an a r c of s i z e s has weight
n+ 1 s+nl+(l+(-l) ) /2
=
s ( k + l - ( s - l ) h ) + (1+(-1)'+l) / 2 .
Thus we have t h e following THEOREM
2.1.
Let D be a symmetric 2-(v,k,h)
design s a t i s f y i n g t h e assump-
453
Symmetric Designs without Ovals
t i o n of Theorem 1.1, and admitting an a r c S of s i z e s . Then t h e minimum weight of t h e code d e f i n e d by D a s i n Theorem 1.1 i s bounded a s follows: s ( k + l - ( s - l ) h i n c a s e (i); d S {
s (k+l- (s-1)h ) + ( l + ( - l'+') )
(6)
/2 i n case (ii).
I t t u r n s o u t t h a t i f t h e a r c S i n an o v a l , t h e i n e q u a l i t i e s (6) a r e o f t e n
s h a r p e r than ( 1 ) . Thus a code a r i s i n g from a design with o v a l s i s u s u a l l y n o t extremal. We s h a l l i l l u s t r a t e t h i s by some examples. In Table 1 t h e parameters of symmetric 2-(v,k,A)
d e s i g n s with k - h S 9 ,
y i e l d i n g doubly-even codes by t h e c o n s t r u c t i o n of Theorem 1.1. a r e l i s t e d . TABLE 1 NO.
k-h
1
1
k
v 3
h
2
1
Codes Extended Hamming ( 8 , 4 , 4 ) code; extremal.
2
3
3
Extended Golay (24,12,8) code; extremal.
3
5
19
10
5
Three extremal (40,20,8) codes.
4
7
31
10
3
Extremal (64,32,12) code from any design without o v a l s (Theorem 2 . 5 ) .
5
7
27
14
7
An extremal
6
9
35
18
9
7
9
36
15
6
8
9
40
27
18
1
1
6
Parameters N o . 2-(4t-l,2t,t). 2t-1.t-1))
(56,28,12) code from t h e q u a d r a t i c r e s i d u e design.
Is t h e r e any extremal (72,36,16) code? ( 8 0 , 4 0 , d > l 2 ) code from any design without ovals.
A
1 , 2 , 3 , 5 , 6 i n Table 1 a r e of Hadamard t y p e , i . e .
The s i z e of an o v a l i n a Hadamard 2 - ( 4 t - l , 2 t , t )
design i s 3 , t h u s t h e bound from Theorem 2.1 f o r t h e minimum weigth
of a code of type (ii) i s 4. However, a Hadamard 2 - ( 4 t - l , 2 t , t ) t
of t h e form
( o r 2-(4t-1,
design with odd
> 1 does n o t admit any o v a l s ; t h i s follows from a r e s u l t of Morgan [ 9 1 , and
can be a l s o e a s i l y seen d i r e c t l y . For, i f a 2 - ( 4 t - l , 2 t , t )
design p o s s e s s e s an
a r c of s i z e 3, then t h e sum of t h r e e rows of i t s incidence matrix A indexed by t h e p o i n t s of t h e a r c , as w e l l as t h e sum of a l l remaining rows over GF(2) is t h e z e r o v e c t o r . Consequently, t h e rank of t h e incidence matrix over GF(2) must be l e s s o r equal t o 4t-3. (I,J-A),
where J.denotes
On t h e o t h e r hand, s i n c e t i s odd, t h e matrix
t h e a l l - o n e m a t r i x , g e n e r a t e s a s e l f - d u a l code. Thus
t h e rank of J-A over G F ( 2 ) i s 4t-1, whence t h e rank of A i s 4t-2,
a contradic-
t ion. Estimating t h e weight of a sum of a t most 4 rows of a matrix of t h e form ( 3 ) , where A i s an incidence matrix of a Hadamard 2 - ( 4 t - l , 2 t , t )
odd t , say t = 2 m + l , THEOREM 2 . 2 .
design with
t h e following can be proved:
[13
1.
The minimum weight of a (16m+8,8m+4) code with genera-
V.D. Torichav
454
t o r m a t r i x ( 3 ) d e f i n e d by a Hadamard 2-(8m+3,4m+2,2m+l) d e s i g n i s e q u a l t o 4 if m=O,
and i s a t l e a s t 8 i f m
> 0.
In f a c t , t h e codes d e r i v e d from Hadamard 2-(8m+3,4m+2,2m+l) d e s i g n s f o r
m S 2 a r e a l l e x t r e m a l . I t i s worth n o t i n g t h a t Hadamard 2 - ( 4 t - l , 2 t , t ) d e s i g n s which a r e e x t e n d a b l e i n t o isomorphic Hadamard 3 - ( 4 t , 2 t , t - l )
designs, y i e l d
e q u i v a l e n t codes. For i n s t a n c e , t h e r e e x i s t e x a c t l y 6 nonisomorphic 2 - ( 1 9 , 1 0 , 5 ) d e s i g n s and 3 nonisomorphic 3 - ( 2 0 , 1 0 , 4 ) d e s i g n s , producing 3 i n e q u i v a l e n t extremal (40,20,8) codes [ 1 3 ) . The absence of o v a l s i n a 2-(8m+3,4m+2,2m+l) design with m > 2 i s n o t a 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 t r e m a l i t y of t h e r e l a t e d code.
In t h i s case
some a d d i t i o n a l c o n d i t i o n s are t o be f u l f i l e d , i n c l u d i n g t h e absence of o v a l s i n t h e complementary 2-(8m+3,4m+1,2m) d e s i g n a s w e l l . THEOREM 2.3.
[ 1 3 1 . 3 n e c e s s a r y c o n d i t i o n f o r an (16+8,8m+4) code d e f i n e d
by a Hadamard 2-(8m+3,4m+2,2m+l) d e s i g n D t o have minimum weight d 2 12 i s any t r i p l e of pOints of D t o be c o n t a i n e d i n a t l e a s t 2 b l o c k s , and a t most 2m-1 b l o c k s of D .
-
Formulated f o r t h e complementary 2-(8m+3,4m+2,2m) design D , t h i s c o n d i t i o n s t a t e s t h a t each t r i p l e of p o i n t s must occur i n a t most 2m-2 l e a s t one block of
D.
In particular,
-
b l o c k s , and a t
D cannot have gny o v a l s .
The enumeration of a l l 2-(27,14,7)
ciesigns up t o isomorphism i s n o t y e t
completed. However, t h e only primes which can d i v i d e t h e o r d e r of t h e automorphism group of a Hadamard m a t r i x of o r d e r 28 a r e 13, 7 , 3 and 2 , and t h e m a t r i c e s p o s s e s s i n g automorphisms of o r d e r 13 o r 7 a r e a l r e a d y known [ 1 4 ] , [ 1 5 ] . Among t h e d e s i g n s a r i s i n g from Hadamard m a t r i c e s of o r d e r 28 w i t h automorphisms
of o r d e r 13 o r 7 , only t h o s e r e l a t e d t o t h e Hadamard m a t r i x of q u a d r a t i c r e s i due type y i e l d an extremal (56,28,12) code. Any 2 - ( 3 5 , 1 8 , 9 ) d e s i g n d e f i n e s a doubly-even
(72,36) code with minimum
weight d 2 8. Although t h e e x i s t e n c e of an extremal (72,36,16) code i s s t i l l i n doubt, i t i s known t h a t such a code cannot be o b t a i n e d from a 2 - ( 3 5 , 1 8 , 9 ) des i g n w i t h an automorphism of o r d e r 17 [ 1 6 ] . More g e n e r a l l y , such a code cannot p o s s e s s automorphisms of o r d e r 17 or any l a r g e r prime o r d e r 161, 1111, 1121. Another approach f o r c o n s t r u c t i o n of doubly-even of symmetric 2-(36,15,6)
d e s i g n s ( s e e Theorem 1 . 1 ,
(72,36) codes i s by means
( i ) ) In . t h i s c a s e Theorem
2.1 g i v e s t h e f o l l o w i n g r e s u l t : THEOREM 2.4.
A necessary condition f o r a 2-(36,15,6)
design t o y i e l d a
( 7 2 , 3 6 , d 2 1 2 ) code i s t h e absence of any a r c s of s i z e 3 . The n e x t theorem i s an example of a better-behaved
THEOREM 2.5. A doubly-even s e l f - d u a l
situation.
(64,32) code d e f i n e d by a 2 - ( 3 1 , 1 0 , 3 )
d e s i g n D i s e x t r e m a l ( i . e . has minimum weight 12) i f and only i f D does n o t p o s s e s s any o v a l s . Proof. The f i r s t row of t h e g e n e r a t o r m a t r i x ( 3 1 , where A i s now an i n c i -
S.ririitzetric Dcsigrrs without Ovals
455
d e s i g n D , h a s w e i g h t 32, w h i l e a l l re-
dence m a t r i x of a symmetric 2 - ( 3 1 , 1 0 , 3 )
maining rows a r e of weight. 12. The w e i g h t o f a sum of t w o rows of ( 3 ) i s 24 p r o v i d e d t h a t one of t h e rows i s t h e f i r s t row of
( 3 ) , or 16 i f b o t h rows are
o t h e r t h a n t h e f i r s t row of ( 3 ) . The sum of 3 r o w s of
(3) including t h e f i r s t
row i s 20. C o n s i d e r now t h e s u m of a t r i p l e of rows n o t i n c l u d i n g t h e f i r s t
row. L e t n . d e n o t e t h e number of b l o c k s c o n t a i n i n g e x a c t l y i p o i n t s of t h e t r i p l e of p o i . n t s o f D c o r r e s p o n d i n g t o t h e choosen t r i p l e of rows. W e have:
no + n l +
n2 +
n3
=
31,
n l + 2n2 + 3n3
=
3.10,
n2 + 3n3 = 3.3. T h i s system h a s t h e f o l l o w i n g s o l u t i o n s :
no
nl
10
n2
n3 0
12
9
9 1 5
6
1
8 1 8
3
2
7 2 1
0
3
E v i d e n t l y , t h e w e i g h t o f t h e sum o f t h e c o n s i d e r e d t r i p l e o f r o w s i s 4 + n l + n 3 2 16. F u r t h e r m o r e , t h e w e i g h t of a sum of 4 rows of
(3) including t h e f i r s t r o w
i s 4+n +n +1 2 12. 0 2 Now w e e s t i m a t e t h e w e i g h t of t h e sum of a q u a d r u p l e of rows of ( 3 ) n o t i n c l u d i n g t h e f i r s t row. Denoting by n , ( 0 S i S 4 ) t h e number of b l o c k s of D c o n t a i n i n g e x a c t l y i p o i n t s from t h e q u a d r u p l e of p o i n t s c o r r e s p o n d i n g t o t h e choosen q u a d r u p l e of rows, one h a s : no + n l + n
1
n2 +
n3 +
+ 2n2 + 3n
n4 = 31,
+ 4n
= 4.10, 3 4 n 2 + 3n3 + 6n4 = 6 . 3 ,
= 3n + 8n4 + 4 . 3 The w e i g h t of t h e c o n s i d e r e d sum i s 4+n +n = 8+4n +en4. By o u r a s s u m p t i o n 1 3 3 D d o e s n o t p o s s e s s any o v a l s , hence n +n > 0 , and c o n s e q u e n t l y t h e w e i g h t of
whence n l
t h e sum of f o u r rows i s a t l e a s t 12. Thus t h e w e i g h t of any l i n e a r c o m b i n a t i o n of a t m o s t 4 rows of t h e g e n e r a -
tor m a t r i x ( 3 ) is a t l e a s t 12. A
symmetric d e s i g n of an odd o r d e r k-X c o n t a i n s a n o v a l if a n d o n l y i f t h e
d u a l d e s i g n c o n t a i n s an o v a l ( 2 1 . The f o l l o w i n g m a t r i x
1 At
1 01...1
I
(7)
V.D. Tonchev
456
i s a p a r i t y check matrix of t h e code generated by ( 3 ) , and s i n c e t h e code is s e l f - d u a l , t h e matrix ( 7 ) is a l s o a generator matrix of t h e same code. Since t . is an incidence matrix of t h e d u a l design of D , we can apply t h e same argu-
A
ments f o r t h e weight of t h e sum of a t most 4 rows of ( 7 ) . Since a codeword of weight 8 o r less must be sum of a t most 4 rows of one of t h e matrices ( 3 ) o r
( 7 1 , t h i s completes t h e proof. The 2-(31,10,3)
design l i s t e d i n H a l l book ( 7 1 possesses o v a l s . Since t h e r e
a r e no c y c l i c d i f f e r e n c e - s e t s with parameters ( 3 1 , 1 0 , 3 ) , it can be e a s i l y seen by use a r e s u l t of Aschbacher [ l ] t h a t t h e g r e a t e s t prime which can be an o r d e r of an automorphism of a 2-(31,10,3) design i s 7. We enumerated a l l such designs f i n d i n g e x a c t l y 4 nonisomorphic s o l u t i o n s , one of them being without o v a l s [17]
-
A s s u m e t h a t an automorphism 5 of o r d e r 7 a c t s on t h e p o i n t s and blocks a s follows :
B = (1,2
,...,7 ) ( 8 , 9 ,...,14) (15,16 ,..., 21) (22,23 ,...,28) (29) (30)( 3 1 ) .
Then a 2-(31,10,3)
design without o v a l s i s defined by t h e following base
blocks : B1 = (1,8,13,14,15,18,21,22,25,27),
B22 = (1,4,6,10,14,19,20,22,28,31) I
B8 = (1,6,7,8,11,16,18,24,28,29), B15
= (1,4,7,9,11,15,20,26,27,30),
B29 = (8,9,10,11,12,13,14,29,30,31), B30 = (15,16,17,18,19,20,21,29,30,31), B g l = (22,23,24,25,26,27,28,29,30,31).
The f u l l automorphism group of t h i s design i s of o r d e r 42. We do n o t know whether
t h e code obtained from t h e above design is e q u i v a l e n t t o t h e extremal (64,
32) code c o n s t r u c t e d by Pasquier 1101. F i n a l l y , l e t us consider t h e parameters 2-(40,27,18).
By arguments s i m i l a r
t o those from t h e proof of t h e preceding theorem t h e following p r o p o s i t i o n can be proved. THEOREM 2.6.
The minimum weight d of a doubly-even
(80,401
code obtained
from a 2-(40,27,18) design D i s 8 i f and only i f t h e complementary 2-(40,13,4) design D p o s s e s s e s an oval. Otherwise, d 2 1 2 . L e t us remark t h a t f o r e x t r e m a l i t y one needs d = 16. As a f i r s t c a n d i d a t e ,
w e checked t h e complement of t h e 2-(40,13,4) design formed by t h e hyperplanes i n PG(3,3). However, it t u r n s o u t t h a t t h e r e s u l t i n g code has minimum weight d = 12.
REFERENCES [ l ] M. Aschbacher, On c o l l i n e a t i o n groups of symmetric block d e s i g n s , J. Combin. Theory, A 11 (19711, 272-281. [ 2 1 E.F. Assmus, J r . , and J . H . van L i n t , Ovals i n p r o j e c t i v e d e s i g n s , J . Combin. Theory, A 27 (19791, 307-324. 31 Th. Beth, D. Jungnickel, H. Lenz, "Design Theory", B.I. Wissenschaftsverlag,
Symmetric Designs without Ovals
457
Ztirich 1 9 8 5 . Bhargava, J . M . S t e i n , ( v , k , h ) c o n f i g u r a t i o n s and s e l f - d u a l codes, Information and C o n t r o l , 28 ( 1 9 7 5 ) , 352-355. P . J . Cameron and J . H . van L i n t , "Graphs, Codes and Designs", London Math. SOC. L e c t u r e Note Ser. 4 3 , Cambridge U n i v e s s i t y P r e s s , Cambridge 1 9 8 0 . J . H . Conway, V . P l e s s , On primes d i v i d i n g t h e group o r d e r of a doubly-even ( 7 2 , 3 6 , 1 6 ) code and t h e group o r d e r of a q u a t e r n a r y ( 2 4 , 1 2 , 1 0 ) code, D i s c r e t e Math. 38 ( 1 9 8 2 ) , 143-156. M. H a l l , Jr., "Combinatorial Theory", Ginn ( b l a i s d e l l ) , Boston 1967. F.J. MacWilliams and N . J . A . Sloane, "The Theory of E r r o r - C o r r e c t i n g Codes", Noth-Holland, Amsterdam 1 9 7 7 . E . J . Morgan, Arcs i n block d e s i g n s , A r s Combinatoria 4 ( 1 9 7 7 ) , 3-16. G. P a s q u i e r , A b i n a r y extremal doubly even s e l f - d u a l code ( 6 4 , 3 2 , 1 2 ) obt a i n e d from an extended Reed-Solomon code over F 1 6 , IEEE T r a n s , Inform. Theory, 27 ( 1 9 8 1 ) , 807-808. V . P l e s s , 2 3 does n o t d i v i d e t h e o r d e r of t h e group of a ( 7 2 , 3 6 , 1 6 ) doubly even code, IEEE T r a n s . Inform. Theory, 2 8 ( 1 9 8 2 ) , 113-117. V. P l e s s , J . G . Thompson, 17 does n o t d i v i d e t h e o r d e r of t h e group of a ( 7 2 , 3 6 , 1 6 ) doubly even code, IEEE T r a n s , Inform. Theory, 28 ( 1 9 8 2 ) , 537-
[ 41 V.K.
i
51
1
61
1
91 [lo1
[ll] [ 121
541. V.D. Tonchev, Block d e s i g n s of Hadamard t y p e and s e l f - d u a l codes, Problemi p e r e d a t c h i i n f o r m a t s i i , 19 ( 1 9 8 3 ) , No. 4 , 25-30. V.D. Tonchev, Hadamard m a t r i c e s of o r d e r 2 8 w i t h automorphisms of o r d e r 1 3 , J . Combin. Theory, A 35 ( 1 9 8 3 ) , 4 3 - 5 7 . V.D. Tonchev, Hadamard m a t r i c e s of o r d e r 28 w i t h automorphisms of o r d e r 7 , J. Combin. Theory, A 40 ( 1 9 8 5 ) , 62-81. V . D . Tonchev, R.V. Raev, C y c l i c 2 - ( 1 7 , 8 , 7 ) d e s i g n s and r e l a t e d doubly-even codes, Compt. rend. Acad. bulg. S c i . , 35 ( 1 9 8 2 1 , 1367-1370. V.D. Tonchev, Symmetric 2 - ( 3 1 , 1 0 , 3 ) d e s i g n s with automorphisms of o r d e r 7 , Annals of D i s c r e t e Math. ( t o a p p e a r ) .
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Annals of Discrete Mathematics 37 (1988) 459-468 0 Elsevier Science Publishers B.V. (North-Holland)
459
GROUPS IN HYPERGROUPS Thomas VOUGIOUKLIS Democritus University of Thrace, 67100 Xanthi, Greece. We study the fundamental relation introduced by Koskas as the transitive closure of a basic relation in a given hypergroup. We use the quotient set, which is a group, in order to define a semi-direct hyperproduct of two hypergroups. We obtain an extension of hypergroups by hypergroups. Moreover we reveal some hypergroups in a given hypergroup. 1. INTRODUCTION
-
of The class of hypergroups in the sense of Marty [8] is the largest class multivalued systems that satisfies group-like axioms:
if we can find I1’.
. .,I
In H
the fudarnentai! equivaZcnce
relation
X~,X~,...,X,,-~, hi , hi ,....hi E H 1 2 H
finite sets of indices, such that
The fundamental relation introduced by Koskas in [7,p.167] was defined the transitive closure of the relation BH defined by setting
as
T. Vougiouklis
460
The fundamental r e l a t i o n
which i s a two-sided s t r o n g r e g u l a r r e l a t i o n ,
,B;
has been s t u d i e d i n s e v e r a l papers such as [ 3 ] , [ 51, 161 and many i n t e r e s t i n g r e s u l t s have been obtained. The most u s e f u l result remains that i s a group; t h a t i s why 6; i s c a l l e d t h e fundamental r e l a t i o n . W e HI84
-
H/Bi
shall call
fundumental
the
L e t us denote by
FH(x), x E H
c l a s s e s has t h e simple form
group
t h e elements
H.
of
H/Bi.
of
F H ( a )*FH(b)= FH(ab) = F H ( x ) ,
The product
of
Vx Eab, ( s e e [ 3 1 ) ,
i n s t e a d of t h e u s u a l form:
1
F H ( a ) * F H ( b ){FH(x) = xEa'b',
a'Bga,
b'Bib}
fulzdamentaZ property.
it i s c a l l e d t h e
An element x of H w i l l be c a l l e d a fundamental element w e have a b = {XI or FH(x)={x}; i.e f o r any a,bEH
iff x$ab.
If
H i s a group then a l l elements a r e fundamental; t h e r e f o r e i n t h i s case the H. The fundamental elements are t h e unique fundamental group c o i n c i d e s w i t h images of t h e same p a i r s f o r e v e r y fundamental map needed f o r t h e d e f i n i t i o n of a hypergroup a l g e b r a ; s e e [13]. A l l u n i t elements of
H,
i f they e x i s t
rp:H
H/BC
Remark 2 . 1
If
is called t h e h
nucleus
Let h relation
H
of
i s a subhypergroup of Fhfx)CFH(x),
H/Bi
belong t o t h e i d e n t i t y of
because i f , for example, e is a l e f t unit i n H F H ( e ) - F H ( x=) FH(ex)= F H ( x ) . The k e r n e l of and so
H
then the
x €ex, VxEH, canonical map
w
and denoted by
H'
then
VxEh.
be a subhypergroup of H; then we do not n e c e s s a r i l y have Example: I n [ 4 1 a g e n e r a l i z a t i o n of Ih/B;
I<
I.
the the
canonical hypergroups, t h e f e e b l y canonical hypergroups, have been introduced and proved t o be j o i n spaces [ll]. A n example of t h o s e i s t h e following one [ L I : Let G be an a b e l i a n group and S a s e t . One supposes that IGI>l
Y(s1,s2)ES
hypergroup.
2
,
PROPOSITION 1 Let
w e have
s1-s2=G.
IGI > 2
For
Then
be a hypergroup and
j E {l,.. ..s}.
becomes a
feebly
canonical
we o b t a i n
hl,...yhs,kl,...,ks€
Then, f o r every
clE hl.. .h
H
such and
clBic2.
proof It i s enough t o prove t h i s p r o p o s i t i o n f o r
h, = k , , j E {2,.
.. ,s).
that
c 2 € k l'. .ks'
Groups in Hypergroups
n
Therefore ( hlh2.
. hs ) u ( z
(zx-l h 2 . . .hs) U(klh2.. .hs)
so i f we choose
c
n
ixE
I
and
h2.. .h
pi
h
A
r E {l,.
y I' E z I'h2 . . .hs,
c l € h l h 2...h
..,
I ui h2. . . h s , . 1 1
h 2.. . h s ) C ilE
46 1
c2Eklh 2...h
.. ,?,-l}
o b t a i n t h a t for every
we
c16Ec2.
we have
P-hypergroups introduced i n [12] and g e n e r a l i z e d (G,.) and a i n [I&], [151. They a r e defined as follows. Given a group non-empty subset P of G then G becomes a hypergroup w i t h hyperP* such t h a t xP*y=xPy ; we denote t h i s by
L e t u s now consider t h e
THEOREM 1 Let Z be t h e c e n t e r of t h e group
Proof Let us denote by in
(G,-).
t h e r e exist
aB:b
1
a =x
*11 Then
...
1
n-1
then
2 G/Z.
t h e hyperproduct i n
xi,hi
n
i E I H
H-l.b
.II
and
t h e product
such t h a t
1
~ z ,~ ,,x ~ = bzx 1
G ;
-
a=bz xH-l,b
h H
1 H
.
...za , x l
Z
4
-
a=bz'-
a 5 bmod 2 .
A polygroup
[2]
i s a system A=<M,.,e,-'>
u n i t a r y o p e r a t i o n on M, .maps M2 x,y.z following axioms hold for all (x*y).z=x*(y.z) , x ~ y * z implies
where
eEM,
i n t o nonempty s u b s e t s of in M:
e*x=x=x.e y E x . 2 -1
,
and
-1
My
is and
a the
and z~y-lx.
In [l] an extension of polygroups by polygroups have been introduced i n t h e following way: that Suppose @ and g a r e polygroups whose elements have been renamed so AnB=Ce) where e i s t h e i d e n t i t y of both and g. A new system
46 2
T. Vougioukiis
% [ g =] <M,*,e,’>,
called the extension of
3,
% by
is formed in
the
following way: Set M = {xEA:x#e}U{xEB:x# e}u{e} and let el= e, I -1 x =x , e*x=x*e=x for all xEM, and for all x,yE{xE M:x#e} , x,yE A
if
x.yUA The extension
if
xEB, Y E A
if
xEA, ~
if
x , y ~ ~y#x-’ ,
if
x,yEB
E
B
-1 y=x
and
%[31 is a polygroup which preserves being chromatic.
THEOREM 2
Proof -1 -1 We have by definition: x*x = x - x uA belong to the fundamental class we have
bf3&[g,bl
”5;
iff
therefore all the elements of A Moreover f o r the elements of B
bl ;
that is because two elements of B can be contained only in the hyperproducts of elements of B. Therefore the extension has in the unit fundamental class all the elements of A and the other classes have exactly the elements of B/Bi
‘&[%?I
3.
THE FUNDAMENTAL HYPERGROUPS
DEFINITION 2 Let
aLib {a,z,} Then
.
be a hypergroup.
LEI
R i - fundamental
us denote by
Li(a)
[lo1
fundamental
such
x
of
H.
One
that
can a l s o
using right multiplication by a,
then
x.,Let
the hyperproduct
is given in the same way as in any
H/Li
relation
equivalence
i.e.
LE( a) *Li(b = {Li(c) The quotient
relation
the equivalence class of
in the quotient set relation
-
. . . , {~,,-~,b} C Xh,, .
,
{z1,z2}Cxh2
is an equivalence relation for every
define the
Li
zl,z2 ,...,~ ~ - ,h ~ , h ,h E H 1 2’”. n
if we can find C x h1’
We define the
H/Li
fundamental hypergroup
1
cea‘b’
where
a’Lia,
becomes a hypergroup which
of
H
we
shall
corresponding to the element
Remark 3 . 1 G(h)cFH(h),
b’Lib 1
Vx,hEH.
x.
call
the
Groups in Hypergroups
463
Remark 3.2 Given a reproductive generalized permutation
f:H
-
q(H),
uH
with
f(h)=H,
see
[13],
we obtain a similar definition as for the fundamental hypergroups by afb if we can find zl,...,zH-l, hl,...,hH€H such that
setting
)C {a,zljcf(hl) ,..., { Z ~ - ~ , ~ f(hH).
LX
- fundamental relations correspond to H generalized permutations of H. In this sense the
Remark 3.3 For every left scalar element
THEOREM 3 For every hypergroup
H
x
H
of
and every
x
2 (H/Lz)/%filL;
H/B:
we have of
H
the
inner
H/Li 2 H.
we have
.
Pro0 f We consider the mapping
s:H/BC >-
(H/Li)/BiiL;
We have for
:
hl,h2EH
FH(h)
c--3
and for every
FHlLx (L;(h)) H z Ehlh2
x s(FH(hl)-FH(h2))= s(FH(hlh2)) = s ( F ~ ( z ) ) = F ~ / ~(Li(z)). H On the other hand we have
From the first inclusion we obtain
but by remark 3.1 SO we can take
L; ( z , ) = L E we obtain similarly
ki L; kil 1
c1,c2
such that
(c2)CFH(c2). clbg c2
-
ki 5* k 1 il
LE (hl)=
.
$ (cl)CFH(cl)
and
From the construction of c1 and c2 and propositionl,
and
FH(~H-l)= FH(h2);
FH(cl) = FH(c2). so
Therefore FH(hl) = FH(zl)
FH(hl)=FH(h2)
and
which is a contradiction.
T. Vougiouklis
464
COROLLARY For every element
h
of a hypergroup
H
we have
4. THE SEMIDIRECT HYPERPRODUCT DEFINITION 3 Let A,B be hypergroups. We consider the group and the fundamental group B/f3; Let
.
":B/B;( >-
AutA:FB(b) >-
be a homomorphism.
Then in
(a,b)(al,bl) = {(x,y)
AxB
[16])
(see WALL
denote
FB (b) = i; A
we can define a hyperproduct as follows:
xEat(a 1) , yEbblj= (ac(al),bbl)
~
and we shall call this the
of
semidirect hyperproduct
the fundamental property we have for every x,y E B The semidirect hyperproduct is associative, since
=I
Aut A
J {(z,w) lzEa6(x),
A
and
x^y= G o $ =
that
B.
2,
From
siz Exy.
w~by}=
x E a1L1(a2) Y E blb2
={(z,w)
={(z,w)
I I
z
Eat(altl(a2)), A
w E b b 1b 2}={(z,w)lzEat(al)
A
A
b(bl(a2)),
wEbblb2j=
h
zEa6(al)bob 1(a2),wEbblb2}={(z,w)/zEat(al) $b 1(a2) , wEbblb2}.
It is also easy to see that the reproduction axiom is valid in AxB ; 'therefore AxB equipped with the semidirect hyperproduct becomes a group which we denote AQB.
hyper-
Remark 4.1 Using the fundamental group one can define the wreath product as well. Remark 4.2 In case that in
A
there exists an absolute unit element
ea
then
for
Groups in Hypergroups every eb
b
of
B
which
6(ea)=ea
i s any u n i t element of
B
465
we o b t a i n
{e }gBZB.
e e = eb ,
such t h a t
Similarly i f
Ac{e
then
This remark can b e a p p l i e d , f o r example, f o r r e v e r s i b l e hypergroups [lo], and polygroups [1,2]. n o n i c a l hypergroups Now l e t us consider i n
b = b 1.
( a , b ) d (al,bl) by
AxB
( A + b ) ={ ( a , b )
1 aEA}.
A
and
[ 9 ] , ca-
such t h a t
Let us denote t h e equivalence c l a s s of Then w e o b t a i n t h a t t h e map
i s a hyperisomorphism between t h e hypergroups Therefore we have t h e following. THEOREM 4 For every hypergroup
d
t h e equivalence r e l a t i o n
Z A.
b
B,
bEB
s:(A,b) -b
(AGB)/d
and
B.
B.
(A;B)/SLIZ
Remark 4.3 The above theorem allows us t o say t h a t group A by t h e hypergroup B. LEMMA 4 . 1 Let A,B
be hypergroups and
i s an extension of t h e
A h
a€A,
bEB;
hyper-
6 ( F A ( a ) )= F A ( $ ( & ) ) .
then
Proof Let
xE$(FA(a));
x=k(c).
i.e.
then t h e r e e x i s t s
c
in
6(c)
Conversely, l e t
(a).
x E FA(%( a ) )
Therefore
x€FA(k(a))
or
xBi$(a).
I f w e now apply t h e i n v e r s e autornorphism
so
f~-’(x)B;a
LEMMA 4.2 Let A,B
Pro0f W e have
or
6-l(x)€FA(a),
be hypergroups and
{(a,bf,(al,bl)} {a,al)Ch;
FA(&)
such t h a t
so
...h ’
C (hl,kl)
and
a,al€A,
...(hv,kv)
Then
of
$,
we o b t a i n
and t h e r e f o r e
b,blEB.
f o r some
{b,bl) C kl...kv
.
xE6(FA(a)).
Then hiEA,
f o r some
k.EB h i € A , ki€B.
-
T. Vougiouklis
466
(hl,kl).
.. (hv,kv)= (hl*f$(h2). ..klk2.. .kv-l(hv), kl.. . .
Therefore 11
<=
TI
" is clear. is also clear because for given
kV)
"
{a,a,)
h:
such that
C hi.. .h'
we can take the elements
iff
aB*a A
and
bB$bl
REFERENCES [l] Comer,S. , Extension of polygroups by polygroups and their representations Algebra using color schemes, Lecture notes in Math., no 1004, Universal and Lattice Theory, p.p. 91-103 (1984). [2) Comer,S., Polygroups Derived from Cogroups, Journal of Algebra, V . 8 9 , No 4, P.P. 397-405 (1984). [ 31 Corsini ,P. , Contributo alla teoria degli ipergruppi , Atti SOC. Pelor. Sc. Mat. Fis. Nat., Messina, (1980). [4] Corsini,P., Feebly Canonical and 1-Hypergroups, Acta Universitatis Carolinae - Mathematica et Physica, V. 24(2), p.p. 49-56 (1983). [5] De Salvo, M., Sugli ipergruppi completi finiti, Riv. Mat. Univ. Parma (4)8, P.P. 269-280 (1982). [6i De Salvo,M.-Freni , D . , Ipergruppi finitamente generati, Bolletino U.M.I.
(1984).
46 7
Groups in Hypergroups
[ 71 Koskas ,M., Groupoides, demi-hypergroupes et hypergroupes, J. Math. pures et appl., V. 49, p.p. 155-192 (1970).
[8i Marty,F., Sur une g&&alisation
de la notion de groupe, Huitisme congrss de mathgmaticiens Scandinaves, Stockholm, p.p. 45-49 (1934). [ 93 McMullen,J.R.-Price,J.F., Reversible hypergroups, Rendiconti del Seminark Matematico e Fisico di Milano, V.XLVII (1977), p . p . 67-85 (1979). [lo] Mittas,J., Hypergroupes canoniques, Mathematica Balkanica, V.2, p.p. 165-
179 (1972). [ll] Prenowitz,W.-Jantosciak,J., Join Geometries, U.T.M., Springer ( 1979) .
-
Verlag
[12] Vougiouklis,T., Cyclicity in a special class of hypergroups, Acta Univ. Carolinae Mathematica et Physica, V. 22(1), p.p. 3-6 (1981). [13] Vougiouklis ,T., Representations o f hypergroups. Hypergroup algebra, Proceedings in: Convegno su: Ipergruppi, altre strutture multivoche eloro appl. Udine (1985) p.p. 59-73. [14] Vougiouklis ,T., Generalization of P-Hypergroups , to appear in Rend. Circolo Mat. di Palermo. [IF] Vougiouklis ,T.-Konguetsof,L., P-Hypergroupes, to appear in Acta Univ. Carolinae Math. et Physica. [16] Wall,H., Hypergroups, American Journal of Mathematics, V. 59, p.p. 77-98
-
-
(1937).
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Annals of Discrete Mathematics 37 (1988) 469-478 0 Elsevier Science Publishers B.V. (North-Holland)
469
THE PERRON-FROBENIUS PROJECT ON I N THE THEORY OF GRAPHS, DIGRAPHS, DESIGNS AND STOCHASTIC PROCE SES K a r l E r i c h WOLFF Fachhochschule Darmstadt, F a c h b e r e i c h Mathematik und Naturwissens c h a f t e n , S c h o f f e r s t r . 3, D-6100 Darmstadt, FRG. Problem 5 i n " S p e c t r a o f graphs" by CVETKOVIC, DOOB, SACHS [2,p.266] says: " F i n d t h e r e l a t i o n between t h e t h e o r y o f Markov c h a i n s and t h e t h e o r y o f graph s p e c t r a . " We show t h a t t h e i n v e s t i g a t i o n o f s p e c t r a l p r o p e r t i e s o f graphs, d i g r a p h s , d e s i g n s and s t o c h a s t i c m a t r i c e s a r e r u l e d by t h e PERRONFROBENIUS p r o j e c t i o n and t h r e e m a t r i x r e p r e s e n t a t i o n s o f it, w h i c h y i e l d a common g e n e r a l i z a t i o n o f t h e HOFFMAN theorems f o r graphs [ 4 ] , d i r e c t e d graphs [ 5 ] and p o i n t s t a b l e d e s i g n s [ 9 ] as w e l l as t h e e r g o d i c theorem [ 3 ] f o r s t o c h a s t i c m a t r i c e s and t h e r a n k i n g procedure o f WE1 [ 8 ] and KENDALL [ 6 ] f o r tournaments.
1. THE PERRON-FROBENIUS PROJECTION I n t h e HOFFMAN theorems
[4,5,9],
i n t h e e r g o d i c theorem and i n t h e r a n k i n g
procedure o f WE1 [ 8 ] and KENDALL [ 6 ] t h e r e appears a n o n - n e g a t i v e
a and a p r o j e c t i o n m a t r i x
i t s PERRON-FROBENIUS e i g e n v a l u e
MP = aP = PM
P
n x n - m a t r i x M,
such t h a t
.
F o r a more g e n e r a l d e s c r i p t i o n o f t h i s s i t u a t i o n we need t h e f o l l o w i n g n o t a t i o n s : Let
(F),
denote t h e s e t o f
For
E specM
plicity of
nxn-matrices over a f i e l d
M E (F),
m a t r i x i n (F)n. For
let
we denote by
a. For
z
,
E F.
E Fn
a ( a ) t h e a l g e b r a i c , by let
d > :=
F , E
the identity
be t h e s e t o f e i g e n v a l u e s o f M.
speeM {
tz I
g(a) t h e geometric m u l t i -
t E F }.
Def in i t i on : Let
M E (F),
if
ker(M
(hence
ker(M
-
M
i s called
a-quasi-opthogonat
aE) n Im(M
-
aE) =
aE) IB Im(M
-
BE) =
{
5
,
}
Fn ) .
Lemma 1: Let
M E (F),
(0)
and
(0)
M
,
E specM.
Then f o r any
P
E (F),
the following conditions
(1) a r e e q u i v a l e n t :
is
a - q u a s i - o r t h o g o n a l and
P
i s the matrix o f the projection
o n t o ker(M - aE) a l o n g Im(M - aE), 2 (1) P = P , MP = aP = PM , r a n k ( P ) 2 g ( a ) . The p r o o f i s obvious.
If
( 1 ) h o l d s , we say: P
i s the (matrix of the)
(~,a)-projection.
470
K.E. Wolff
Lemma 2:
,
M E (F),
For
E specM
the following properties are equivalent:
,
a)
M
b) c)
a(.) = g(a) 9 ker(M - aE) = ker(M
d)
t h e r e e x i s t s a polynomial
is
a-quasi-orthogonal
-
aE)
minimal polynomial o f
2
, S(X) E F[X] and
M
4
S(a)
T h i s u n i q u e l y determined polynomial
such t h a t
S(X)
(X
-
a)S(X)
i s the
.
0
i s called the
IM,~l-poZynorniaZ
.
Proof: Let x(X)
,a
M E (F),
a ) ==> b ) :
M
of
,
E specM
Since
is
M
(2
f ( z ) := MZ
E Fn).
a-quasi-orthogonal,
satisfies
x(X) = (X
-
t h e c h a r a c t e r i s t i c polynomial
+6
s i n c e o t h e r w i s e t h e r e would e x i s t a v e c t o r E ker(M
- d) n
b ) ==> c ) : f/El,
where
xl(X)
= (X
Im(M
-
-
, hence x (X) a(.)
equivalent t o c). Clearly ker(M
-
(a)
+0,
a(.)
= g(a).
of the r e s t r i c t i o n
, satisfies
decomposes i n t o l i n e a r f a c t o r s . Therefore
= g ( a ) . Hence
c)
The main examples o f
-
= ker(M
-
aE) , which i s
+ {z}2 i s e q u i v a l e n t t o
aE)
aE) < ker(M
-
a-quasi-orthogonal
Any symmetric m a t r i x
El
i s equivalent t o d).
aE) n I m ( M
{ 6 } < ker(M -
Ex.1:
2
i s s i m i l a r t o a Jordan m a t r i x , and t h e r e f o r e s i m i l a r t o a
diagonal m a t r i x , s i n c e c ) ==> a ) :
x
aE). Then
x 1 (X)
i s t h e g e n e r a l i z e d eigenspace o f a
t h e m a t r i x of f l E l
i s the
such t h a t
aE), c o n t r a d i c t i n g a). Hence
C l e a r l y t h e c h a r a c t e r i s t i c polynomial El
-
f IIm(M
c h a r a c t e r i s t i c polynomial of t h e r e s t r i c t i o n
2
, where x,(X)
a)g(a)~,(X)
M
is
aE)
.
r e a l m a t r i c e s are:
a-quasi-orthogonal
Ex.2:
Any s t o c h a s t i c m a t r i x i s
Ex.3:
Any i r r e d u c i b l e non-negative m a t r i x i s
1-quasi-orthogonal
PERRON-FROBENIUS eigenvalue
a
(since
f o r any eigenvalue a o f M.
(cf.[2],2.6.d).
a-quasi-orthogonal = 1 = g(a),
a(.)
for i t s
cf.[71).
Definition: Let
M
be
a-quasi-orthogonal.
PE&?ON-F8OBENIUS
Then t h e (M,a)-projection
projection ofM , i f
M
i s c a l l e d the
i s non-negative and
a
i s the
PERRON-FROBENIUS eigenvalue o f M. 2. THE POLYNOMIAL REPRESENTATION OF THE (M,a)-PROJECTION THEOREM 1: Let Then
M E (F),
, a
E specM
P :=
8
,M
a-quasi-orthogonal
i s the
and
(M,a)-projection.
S(X) t h e ( M a ) - p o l y n o m i a l .
The Perron-Frobenius Projection
47 1
Main case:
If
S(X) =
-
(X
(e.g.
F = t ), then
for
A+a
AEspecM
AispecM Proof: Let
P = S(M)/S(a)
(3)
P? =
hence
2
.
From
2
for a l l
ker(M
-
(M
aE) 6 I m ( P ) .
Im(P) = Im(S(M)) s ker(M
-
P
i s the
aE)S(M) = 0
MP = aP = PM. C l e a r l y
we g e t
- aE),
On t h e o t h e r s i d e we o b t a i n from
(M
-
aE)S(M) = 0
aE), hence
( 4 ) Im(P) = ker(M - aE). From ( 4 ) and ( 3 ) we g e t Hence
-
E ker(M
P
2
and t h e r e f o r e
= P
(1) because o f
(4).
according t o Lemma 1.
(M,a)-projection
3. THE DYADIC REPRESENTATION OF THE (Mya)-PROJECTION
,
Let
M E (F),
R(a) L(a)
:= {
;E
:= {
2
a E specM
Fn
1
M2
=
E Fn IMTx' =
,
a; }
t h e righteigenspace o f a
ax'
the lefteigenspace o f
}
a
.
THEOREM 2: Let
M E (F),
,
Then t h e r e e x i s t such t h a t
R(a) =
E specM
and
a(.)
= 1.
f = (rl ,..., r n ) T , tf> , L ( a ) =
1=
(R1
,..., Rn) T
E
and f o r any such
Fn
,
FYI
i s t h e (M,a)-projection. Proof: Since Let
= 1
a(.)
P
we have
a ( a ) = g(a)
M
is
denote the m a t r i x o f t h e ( M y @ ) - p r o j e c t i o n . From
hence the columns o f
P
,...,
6 9
xT = l T P
,
(1) we have MP = aP,
R(a) = c h , s i n c e g(a) = 1. Hence n cn)T E F such t h a t P = ( r . c . ) = ? tT
.
1 J
dimL(a) = dimR(a) = 1. Hence from
0 f R . = xTc.F , hence 3 J t h e r e f o r e ( 5 ) holds.
a-quasi-orthogonal.
a r e elements o f
there e x i s t s a vector = (c Since PM = aP , t h e rows o f P obtain
hence
a r e elements o f L(a) = P2 = P
0
(by
,...
,
K.E. Wolff
472
P
4. WHEN I S
Let
Jn,l
A MULTIPLE OF
T ( 1 ,..., 1 )
:=
J
,
E Fn
?
J := J
I f i n t h e dyadic representation
(5)
( t h e all-one-matrix). 1 , then P = J
J
n , l ;,l = II = Jn,l
The f o l l o w i n g simple, b u t u s e f u l theorem answers t h e q u e s t i o n :
,
J ?"
a multiple of
i.e.
P E tJ> :=
{tJ
1
"
.
When i s
P
.
t E F}
THEOREM 3: M E (F), , a E specM, (M,a)- p r o j e c t i o n . Then
Let
M
a-quasi-orthogonal,
P
the matrix o f the
a)
P E
t ~ ~ , ~ , fao r~ some > a
E F"
iff
~ ( a = ) < J ~ , ~ ,>
b) c)
P E
?
E Fn
iff
L ( a ) = tJn,l>
- JnT1>
f o r some
P E <J>
iff
<Jn,l>
R(a) =
,
= L(a).
Proof: Since some
a E Fn
, then
, hence
rank(P) 2 1
s i n c e t h e columns o f
a
r a n k ( P ) = g ( a ) L 1. I f
i s the (M,a)-projection,
P
P
generate
r a n k ( P ) = 1 and
P E tJn,ld
-1 > f o r
R(a) = CJ,,~>
R ( a ) . F o r t h e converse t a k e
a E L(a)
, ,
and use Theorem 2. T h i s proves a ) .
f 5
b ) The rows o f
P
generate
L(a), since
PM = aP
and
rank(P) = g(a) = dimL(a).
Hence t h e " l e f t v e r s i o n " o f t h e p r o o f o f a ) shows b ) . c ) f o l l o w s f r o m a ) and b ) . 5. APPLICATION OF THE POLYNOMIAL AND D Y A D I C REPRESENTATION I n t h i s s e c t i o n we i n t r o d u c e t h e HOFFMAN e q u a t i o n f o r connected d e s i g n s and show how t h r e e well-known theorems o f HOFFMAN, HOFFMAN-McANDREW and t h e a u t h o r can be d e r i v e d e a s i l y f r o m o u r p r e v i o u s r e s u l t s . 5.1. The HOFFMAN e q u a t i o n o f a connected d e s i g n Let
D
d e n o t e a connected d e s i g n ( = i n c i d e n c e s t r u c t u r e ) ,
dence m a t r i x , N := AAT FROBENIUS e i g e n v a l u e o f (since
N
E (R),
i t s connection m a t r i x
N. Then t h e r i g h t e i g e n s p a c e
i s i r r e d u c i b l e , because
L ( a ) = R(a) =
th
f o r some
E
pv
and R(a)
A
its
a
t h e PERRON-
i s one-dimensional
D i s connected). From N
.
vxb-inci-
= NT
we have
Hence we o b t a i n f r o m Theorem 1 and
Theorem 2 t h e f o l l o w i n g HOFFMAN e q u a t i o n f o r connected d e s i g n s
XEspecN s i n c e b o t h s i d e s o f t h i s e q u a t i o n r e p r e s e n t t h e PERRON-FROBENIUS p r o j e c t i o n T N = N .
of
The Perron -Frobenius Projection
473
5.2. POINT STABLE DESIGNS From Theorem 3c) we o b t a i n COROLLARY 1: Let
N
denote t h e connection m a t r i x o f a design
N
N I U S eigenvalue. Then E
a-quasi-orthogonal
and
i t s PERRON-FROBE-
and t h e PERRON-FROBENIUS
P satisfies
projection
P
is
D
<J>
NJ = aJ
iff
and
D
i s connected.
Proof: Because o f t h e symmetry o f
N, N i s
Theorem 3c) i s e q u i v a l e n t t o connected i f f a(.)
=
a-quasi-orthogonal
"NJ=aJ
and
D
and t h e r i g h t s i d e o f
i s connected",
since
D
is
1.
The f o l l o w i n g d e f i n i t i o n was i n t r o d u c e d by t h e a u t h o r i n 191. Def in i t i o n A design w i t h connection m a t r i x f o r some
a
E
N
i s called
p o i n t stubZe
,
NJ = aJ
R.
The most i m p o r t a n t examples o f p o i n t s t a b l e designs a r e t h e the
if
(r,X)-designs,
I-designs and t h e r e f o r e a l s o t h e r e g u l a r graphs.
The d i r e c t g e n e r a l i z a t i o n o f t h e HOFFMAN theorem f o r connected r e g u l a r graphs t o connected p o i n t s t a b l e designs i s t h e f o l l o w i n g (cf.191) THEOREM 4: a) A design
D w i t h connection m a t r i x
t h e r e e x i s t s a polynomial
N
i s connected and p o i n t s t a b l e
f ( X ) E R[X] such t h a t
f(N) = J
iff
.
D i s connected and p o i n t s t a b l e , then t h e r e e x i s t s e x a c t l y one p o l y nomial h(X) o f minimal degree such t h a t h(N) = J ,
b) I f
namely t h e HOFFMAN polynomial
EspecN Remark: The HOFFMAN e q u a t i o n s p e c i a l case o f
h(N) = J
(6), where
of a connected p o i n t s t a b l e d e s i g n i s j u s t t h e = Jv,l
. Note
t h a t C o r o l l a r y 1 g i v e s a charac-
t e r i z a t i o n o f t h e connected p o i n t s t a b l e designs i n terms o f t h e PERRON-FROBEP = v- 1h(N). T h e r e f o r e we c a l l C o r o l l a r y 1 t h e " p r o j e c t i o n
NIUS p r o j e c t i o n
v e r s i o n " o f Theorem 4.
5.3. GRAPHS AND DIGRAPHS Now we g i v e t h e p r o j e c t i o n v e r s i o n s o f t h e HOFFMAN theorem f o r connected r e g u l a r graphs
[4] and o f t h e HOFFMAN-McANDREW theorem f o r s t r o n g l y connected
r e g u l a r digraphs [ 5 ] .
K.E. Wouj
474
__ COROLLARY ..__ 2: ( , p r o j e c t i o n v e r s i o r . o f t h e HOFFMAN theorem)
P
Let
eigenvalue. Then
F
G
denoce t h e adjacency m a t r i x o f a graph
A
is
AJ
=
and
d
i t s PERRON-FROBENIUS
d - q u a s i - o r t h o g o n a l and t h e PERRON-FROBENIUS p r o j e c t i o n
satisfies
P E <J>
iff
dJ
and
i s connected ( i . e . G i s r e g u l a r and connected).
G
The p r o o f i s t h e "same" a s t h e p r o o f o f C o r o l l a r y 1. COROLLARY 3: ( p r o j e c t i o n -_ ~-
A E (R),
Let
v e r s i o n of t h e HOFFMAN-McANDREW theorem)
denote t h e adjacency m a t r i x o f a d i g r a p h
G,
d
i t s PERRON-FRO-
BENIUS e i g e n v a l u e . Then t h e f o l l o w i n g c o n d i t i o n s a ) and b ) a r e e q u i v a l e n t :
A
a)
is
o f t h e PEKRON-FROBENIUS p r o -
P E d> ,
jection satisfies bl
P
d - q u a s i - o r t h o g o n a l and t h e m a t r i x
i s s t r o n g l y connected and r e g u l a r .
G
Proof.
7
G
, hence G i s r e g u l a r .
AJ = d J = A J
a ) ==> b ) : By Theorem 3 c ) we o b t a i n
a ( d ) = g ( d ) = r a n k ( P ) = 1 and
i s s t r o n g l y connected, s i n c e
AJ
=
T dJ = A J
(cf.[Z],p.lS,Th.0.4). b ) ==> a ) : S i n c e g(d)
G
i s s t r o n g l y connected, A i s i r r e d u c i b l e , hence a ( d ) = 1 =
by t h e PERRON-FROBENIUS theorem [ 7 ] ,
From t h e r e g u l a r i t y we o b t a i n R(d) If
<Jn,l>
=
G
= L(d),
since
,
A
is
d-quasi-orthogonal.
d , hence d
f o r some
, hence
g(d) = 1
i s a s t r o n g l y connected and
A E (R),
hence
AJ = a J = AT J
P E <J>
= d
and
by Theorem 3 c ) .
d - r e g u l a r d i g r a p h w i t h adjacency m a t r i x
t h e n we o b t a i n f r o m Theorem 1 and Theorem 2 t h e
HOFFMAN-McANDREW e q u a t i o n
where
i s the (A,d)-polynomial.
S(X)
6. THE LIMIT REPRESENTATION OF THE (M,a)-PROJECTION
The e r g o d i c theorem ( c f . [ 3 ] ) m a i n l y says t h a t t h e l i m i t matrix
S
i s a special dyadic m a t r i x o f the form ( 5 )
LigMk
More g e n e r a l l y we c o n s i d e r t h e l i m i t
for
$i!Sk
with
M E (t),
o f a stochastic = Jn,l.
and use t h e
f o l 1owing we1 1 -known Lemma 3:
M E (t)n
For any m a t r i x !A/ < 1
X
for all
the l i m i t
t specM\{lI
,llgMk and
M
exists i f f is
1-quasi-orthogonal.
THEOREM 5: Let P
M E (E)n i s the
.
If
P : = $',iMk
(M,a)-projection
e x i s t s , then (with
CY
=
1 )
or
P = 0
.
The Perron-Frobenius Projection
475
Proof: Let
k
P := J&M
f
P 2 = P , MP = P = PM
0. Then
E x a c t l y as i n [ 3 , p.17,371 Lemma 1 we o b t a i n t h a t
therefore
1 E specM.
one can p r o v e t h a t r a n k ( P ) = g ( 1 ) . Hence f r o m
P
i s the (M,1)-projection.
and Theorem 5
From Theorem 2
, and
we g e t
THEOREM 6:
.
M 6 (a),
Let
eigenvalue
1 of limM k-
BIMk
If
k
e x i s t s and t h e g e o m e t r i c m u l t i p l i c i t y o f t h e g(1)
=
1
,
then
7 IT
E '
=
where
0
satisfies
M
& ,
R(l) =
L(l) =
Remarks : (i)
S i n c e any s t o c h a s t i c m a t r i x
is
S
1-quasi-orthogonal,
we o b t a i n f r o m
t h e e r g o d i c theorem and Lemma 1 t h a t t h e PERRON-FROBENIUS p r o j e c t i o n of
F o r s t o c h a s t i c m a t r i c e s we o b t a i n f r o m
(ii)
P
satisfies
S
(8)
;. =
with
Jn,l
t h e main
s t a t e m e n t o f t h e e r g o d i c theorem. T h i s c e n t r a l p a r t o f t h e e r g o d i c theorem i s g e n e r a l i z e d by t h e f o l l o w i n g c o r o l l a r y . COROLLARY 4: Let
M E (E)n
, 1E
A E specM\{l}.
limM
k
J
spec#,
M
1 - q u a s i - o r t h o g o n a l and
1x1 <
a
aT
f o r some E En i f f MJ = J and a ( 1 ) = 1 n,l The p r o o f r e s u l t s f r o m Lemma 3 and Theorem 3 a ) .
k+m
=
1 for all
Then
.
F i n a l l y we m e n t i o n as a consequence of Theorem 1 and Theorem 5 a n o t h e r a l g e b r a i c k p o s s i b i l i t y t o c a l c u l a t e t h e l i m i t Jj&M . THEOREM 7: Let
M E (E)n
k
.
I f $j&M
S(M)
k
40
e x i s t s , then
-
( MI-h AE)
-
( n o t a t i o n s as i n ( 2 ) ) .
kspecM 7. THE PERRON-FROBENIUS PROJECTION I N THE RANKING PROCEDURE OF WE1 AND KENDALL For t h e c a l c u l a t i o n o f a "meaningful" ranking o f t h e p a r t i c i p a n t s o f a tournament f i r s t d e t e r m i n e (cf.[1,6,8]) o f t h e tournament d i g r a p h
t h e l i n e a r o r d e r o f t h e s t r o n g components
T. I n each s t r o n g component o f
p a r t i c i p a n t s i n t h e f o l l o w i n g way:
T
rank t h e
m (#)
K.E. Wolff
416
If If
m
, ,
= 3 m > 3
a l l t h r e e p a r t i c i p a n t s g e t t h e same r a n k . t h e n t h e adjacency m a t r i x A o f t h e g i v e n s t r o n g component
primitiv (cf.[ll).
a is t h e PERRON-FROBENIUS e i g e n v a l u e o f
If
A1 :=
C
is
A, t h e n f o r
aA
k
P := l i m A1 0 e x i s t s a c c o r d i n g t o Lemma 3, s i n c e / A ] < 1 f o r kX E specA1\{13 and a ( 1 ) = 1 = g ( 1 ) by t h e PERRON-FROBENIUS theorem
the l i m i t a1 1
F u r t h e r , f r o m Theorem 5
f o r p r i m i t i v e matrices (cf.[7]).
t h e PERRON-FROBENIUS p r o j e c t i o n o n t o t h e one-dimensional
-
ker(A1
E)
= ker(A
- aE).
follows that
P
is
kernel
Hence t h e r e e x i s t s e x a c t l y one p o s i t i v e v e c t o r
such t h a t
<&
= ker(A
-
aE)
is t a k e n as
This vector
t r i = l . i=1
and
"ranking vector":
j , if ri > r j ' The c o m b i n a t o r i a l meaning o f t h i s r a n k i n g procedure o f WE1 and KENDALL can be
i i s ranked h i g h e r t h a n p a r t i c i p a n t
Participant
e x p l a i n e d e a s i l y u s i n g Theorem 2: k ) . be t h e i - c o o r d i n a t e o f A Jm,l , hence ski is t h e m,l 1 number o f p a t h s o f l e n g t h k s t a r t i n g a t v e r t e x i ( i n t h e g i v e n s t r o n g Let
ski
: = (AkJ
C ) . Then
component
Proof:
lim
Clearly
k-
lT/(xTF)
P = S
lim
k-r-
'ki
a
ki
=
:m)il, = ( l i m AJ
,
(PJ, l)i
hence =
cri
= c':
PJml,
,
. According
= (PJm,l)i
k-
hence
,
where
(10)
Remark: The c o r r e s p o n d i n g r e s u l t o f WE1
c:=
t o Theorem 2
we have
> 0 . Therefore f T Jm , l /(IT;)
holds.
[81 f o r graphs ( c f . [ Z ] , p.104)
can be proved w i t h Theorem 2 i n t h e same way. ACKNOWLEDGEMENT I w i s h t o thank
A.W.M.
DRESS
( B i e l e f e l d ) f o r h i s v a l u a b l e comments on
quasi-orthogonal matrices. REFERENCES
[l]Bondy, J.A. and Murty, U.R.S., Graph t h e o r y w i t h a p p l i c a t i o n s (Amer. E l s e v i e r Publ. Co. , I n c . , New York, 1976) [ 2 ] CvetkoviE, D.M. and Doob, M. and Sachs, H., S p e c t r a o f graphs (Academic Press, New York, 1979) [33 F r i t z , F.J. and Huppert, B . and Willems, W., S t o c h a s t i s c h e M a t r i z e n ( S p r i n g e r Verlag, B e r l i n , 1979) [ 4 ] Hoffman, A.J., On t h e p o l y n o m i a l o f a graph, h e r . Math. Monthley 70 (1963) 30-36.
The Perron-Frobenius Projection
477
[51 Hoffman, A.J. and McAndrew, M.H., The polynomial o f a directed graph, Proc. Amer. Math. SOC. 16 (1965) 303-309. [61 Kendall, M.G., Further contributions to the theory of paired comparisons, Biometrics 11 (1955) 43-62. [71 Seneta, E., Non-negative matrices (George Allen and Unwin Ltd., London, 1973) [83 Wei, T.H., The algebraic foundations of ranking theory (Thesis, Cambridge, 1952) [91 Wolff, K.E., Punkt-stabile und s e m i - p a r t i a l - g e o m e t r i s c h e Inzidenzstrukturen Mitt. math. Sem. GieBen 121 (1976). [I01 Wolff, K.E., Punkt-stabile Inzidenzstrukturen und stochastische Matrizen in: Grabmeier, J. and Kerber, A., (eds.), Shinaire lotharingien de combinatoire (Publ. IRMA, Strasbourg, 1985) pp. 124-126. [ll] Wolff, K.E., Zur kombinatorischen Bedeutung der PERRON-FROBENIUS-Projektion in: Strehl, V . , (ed.), SGminaire lotharingien de combinatoire (Publ. IRMA, Strasbourg, 1986) pp. 153-162.
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Annals of Discrete Mathematics 37 (1988) 479-484 Elsevier Science Publishers B.V. (North-Holland)
479
ON THE NON-EXISTENCE OF CERTAIN DIFFERENCE SETS
N. Zagaglia S a l v i D i p a r t i m e n t o d i Matematica P o l i t e c n i c o d i Milano P.za L. d a V i n c i 3 2 , M i l a n o , I t a l y *
SUMMARY- I n t h i s p a p e r we d e t e r m i n e some p r o p e r t i e s c o n c e r n i n g a ( v , k , A ) - d i f f e r e n c e s e t when v i s even. I n p a r t i c u l a r i t i s proved i n c e r t a i n c a s e s t h e n o n - e x i s t e n c e of c i r c u l a n t Hadamard m a t r i c e s .
INTRODUCTION
Recall t h a t a subset
of Z
i s a (v,k,X)-difference
set if
\@I k =
and t h e
d i f f e r e n c e d.-d ( di,djem , d . # d . ) t a k e s each non-zero v a l u e i n Z A l j 1 J times. A m a t r i x i s c i r c u l a n t i f e a c h row d i f f e r i n g from t h e f i r s t i s d e r i v e d from i t s p r e v i o u s row by s h i f t i n g i t c y c l i c a l l y o n e p o s i t i o n t o t h e r i g h t . An Hadamard m a t r i x h a s e v e r y e n t r y 1 and i t s rows are o r t h o g o n a l t o each other. 2 It i s proved t h a t a c i r c u l a n t Hada a r d a t r i x b a s o r d e r 4N and c o r r e s p o n d s t o a d i f f e r e n c e s e t of p a r a m e t e r s (4N', 2N-' N , N - N). I t h a s been c o n j e c t u r e d t h e n o n - e x i s t e n c e of c i r c u l a n t Hadamard m a t r i c e s of o r d e r v > 4. Up t o now Turyn [6] proved t h e c o n j e c t u r e f o r N even; moreover i t i s proved f o r e v e r y odd N < 55. was a b l e t o p r o v e t h e n o n - e x i s t e n c e of symmetric c i r c u l a n t HadaB r u a l d i [3] mard m a t r i x . I n i t s p a p e r it w a s i m p l i c i t e l y proved t h e c o n j e c t u r e f o r r e f l e c t i v e c i r c u l a n t , where we c a l l a c i r c u l a n t ( 0 , l ) m a t r i x r e f l e c t i v e i f , a r ranged t h e e l e m e n t s of t h e f i r s t row r e g u l a r l y on a c i r c l e , t h e r e e x i s t s a d i a m e t e r of t h e c i r c l e w i t h r e s p e c t t o which 1 ' s a r e symmetric. I n t h i s p a p e r we d e t e r m i n e some p r o p e r t i e s c o n c e r n i n g a ( v , k , X ) - d i f f e r e n c e s e t , when v i s even. I n p a r t i c u l a r w e d e t e r m i n e i n c e r t a i n c a s e s t h e n o n - e x i s t e n c e of a 2 2 (4N2, 2N -N, N - N ) - d i f f e r e n c e s e t . F i n a l l y w e g i v e a c o n s t r u c t i o n of a s u b s e t of Z s a t i s f y i n g t h e p r o p e r t y of a d i f f e r e n c e s e t o n l y f o r odd v a l u e s .
1 . F i r s t w e d e t e r m i n e t h e f o l l o w i n g p r o p e r t y of a d i f f e r e n c e s e t . PROPOSITION 1 . 1
is even and or
-
Let 8
k-A= N
2
= {dl ,d2,.
. Then
. ., d d
be a (v,k,),)-
difference s e t where v
the number of the odd elements of
1
is Z(k + N)
i(k-N).
T h i s r e s e a r c h was s u p p o r t e d by t h e M i n i s t e r 0 d e l l a P u b b l i c a I s t r u z i o n e .
4 80
N. Zagaglia Salvi
Proof. Let r and s be the numbers of the even and odd elements of A s v i s even, the d i f f e r e n c e a = d . - d mod v , where d , d . e@ 1 j i J and only i f d i s even and d odd o r conversely.
i
@
,
. i s odd i f
j
Moreover i f a i s odd, a l s o -a i s odd. V So the number of odd d i f f e r e n c e s i s 2rs and i t has t o c o i n c i d e with 2 A. Then t h e parameters r , s s a t i s f y t h e r e l a t i o n s r+s = k and r s = v / 4 . 1 1 So we o b t a i n t h a t r = -(k + N) and s = -(k - N) o r conversely.<> 2 2 2 We n o t e t h a t i n the case of a (4N2, 2 N - N , N 2 7 N) - d i f f e r e n c e s e t we have t h a t t h e numbers of odd and even elements a r e N and N - N o r conversely.
2 . 1 - Let
2. DEFINITION
2
=k
c2
... cvJ
a sequence of v numbers
, with
v = hk. We s a i d g has period h i f t h e fikst h elements of g a r e repeated consecutively k times. I f h i s t h e minimum i n t e g e r t h a t s a t i s f i e s t h i s r e l a t i o n , we c a l l h real period.
-If L , 1 5 i 5 v , are the rows of a circuZunt mutrix
PROPOSITION 2 . 2
order v= hk, then the vector r -1
Proof. Denoted
1 5 i 5 k-1. If 5 =(a1 a 2 a1 .
=
-1r +-h+l r +. . +-(k-l)h+l r
= ~ l c 2 . . . c v ] , we have r -ih+l
... av]
c1 . + cv-h+i+
.
has period
i s the vector
...+ cv-(k-l)h+i,
=p
v-ih+l 'v-ih+Z..
... ah
.C?ih+v
]
'
' '
...= a l + ( k - l ) h and
2 5 j 5 h . Then the sequence a l a2
of
I - ~ + & + ~ + . . . + L - ( ~ - ~ ) ~ we + ~ ,have v'
So we o b t a i n a 1+rh = Cl+rh+cv-h+l+rh+"'+Cv - ( k - l ) h + l + r h , f o r 0 It follows a =a
C
h'
,
analogously, a j = a j + h =
is repeated k t i m e s and
0~
5 r 5 k-1.
...=
a
j + (k-I ) h '
has period h . < >
3 . Denoted y H a c i r c u l a n t Hadamard m a t r i x , we consider t h e ( 0 , l ) - c i r c u l a n t matrix K = - ( H + J ) , where J i s a l l one matrix. It is well t h a t the p o s i t i o n s of t h e e l y e n t ? 1 on f i r s t row of K determine a d i f f e r e n c e s e t of parameters (4N , 2 N + N , N 2 N). Since the d i f f e r e n t values of k and A correspond t o Fomplementary d i f f e r e n c e s e t s , t h e r e i s n o t l o s s of g e n e r a l i t y i n assuming t h a t (v,k,A)= 2 2 (4N2, 2% - N, N -N)
9 hewn
odd
tp
. even
We c a l l or an element of a sequence if i t i s i n odd o r even p o s i t i o n . Let C be t h e l i n e a r code over Z 2 generated by t h e rows of K. Let r .
-1'
1 5 i 6 v and r
-1
=El c 2 ... c v ] ,
Consider t h e word e = r +r +...+ r -1 -1 -3 Y-1By Prop. 2.1 has period 2. The f i r s t
el
be the rows of K.
Pecond]
element c o i n c i d e s with t h e
sum of elements of r i n odd P v e n J p o s i t i o n . -1 4s k i s odd, one number i s odd and t h e o t h e r even. So -1e is obtained by repeao r [OI] Suppose e = [ I O I O IOJ ting -1 Then t h e word = r +r +. . . + r coincides w i t h 0101 011 -2 -4 Y
001
e2
.
...
.
I: ...
.
On the Non-Existence of Certain Difference Sets
Ll
A s , by Prop. 2 . 1 ,
+
48 1
fl+v/2 = a has period 2N2, consider t h e p o s s i b i l i t y
t h a t a has r e a l period 2 , t h a t i s a = e - -1
or a = e - -2'
a = -1 e Without loss of g e n e r a l i t y , we can suppose REMARK 3.1 - The condition
rl + rl+v12
=
-le
-
conmdicts the existence of K.
Proof. A s detK = k(k-A) ( v - 1 ) / 2 5 1 mod 2 , t h e rows of K a r e l i n e a r l y independent i n Z is e f o r odd i and e The condigion f1 + v12 = e implies -i r + r -i + v / 2 -1 -2 -1 f o r even i. 1 ,C j g v / 2 , depent on rj, 1 6 j < v / 2 , and e2. So the rows r . -J + v / 2 ' Then t h e number of l i n e a r l y independent rows of K i s L V/2 + 2 , a contra-
.
rl+
el,
diction.
<>
= 1 f o r odd i and ci + ci+ v12 = o The c o n d i t i o n s = e implies c + c -1 i i+v/2 f o r even i. [even3 elements a t d i s t a n c e v / 2 a r e d i s t i n c t F o i n c i d e n t ] It follows odd
NL
.
So half of t h e odd elements a r e equal t o 1 , t h a t i s i s t h e number of c . = l f o r odd i. = 1. So t h i s element t h e c o n t r a r y , i f i i s even and c . = l , a l s o c . 1 + v/2 determines a d i f f e r e n c e equal t o v/2.
d
As t h e s e d i f f e r e n c e s a r e
A
= N
2
-
i t follows t h e number of t h e even e l e -
N,
ments equal t o 1 i s N2 - N.These numbers agree with those of Prop. 1 . 1 . However t h e following Theorem 3.5 proves t h i s s i t u a t i o n i s impossible. LEMMA 3.2 - Let
a
c4 c6
=
... c4q]
be, where c i E
Z2,
mod 2 , f o r 2 5 i & 4q. Moreover, l e t 8 = positions of 1 ' s i n a. mod bq, where d i , d . E @ Then every difference a= d.-d. 1 J J ber of times.
c i + ci+2q = -0
be the s e t of the
, occurs
an even n m -
0 mod 2 , we o b t a i n t h a t , i f Proof. From t h e condibion ci + ci+2 c d . = cdj = 1 , a l s o t h e elements cogresponding t o di' = di+2q and d 1
coincide t o 1 . Denoted a =d.-d
a l s o t h e d i f f e r e n c e di'- d ' c o i n c i d e s with j' j So every d i f f e r e n c e occurs an even number of t i m e s . <> 1
LEMMA 3.3 -
ci + c
Let
=
[c1c3c5.. .c4q-l]
be, where
q is odd, c . e
'
j~
=d.+2q
a mod 4q.
z2 and
~ 2 1+ mod~ 2 , ~1 L i $ 4q-1.
Moreover l e t @ =id d2...dk] be t h e s e t of the positions of 1's i n b. Then the number 4 occurs An odd nwnber of times as difference di-d. J where d G d3. t @ .
=kl
. ..
Proof. Let v=4q be, hl c5 c 9 cv-3] We n o t e t h a t d.-d. E 4 mod v , where d . , d j e 1
3
and h2 @ , if
=p3 c7
mod v , t h a t i s c d . = c d . = 1 a r e two a d j a c e n t elements i n -1 h As it i s c
to h
-1'
+
1
ci+
... o r -2 h -
c11 Cv-lI * d.+4 and only i f d . 1
J
J
v12s 1 mod
2 and q i s odd, i t follows t h a t , i f c . = l belongs
then c i+ v12 = 0 belongs t o h
-2
and conversely.
N. Zugugliu Sulvi
482
,
[1,1]
L e t r , s r e s p e c t i v e l y t h e numbers of o r d e r e d c o u p l e s i n hl. number of o r d e r e d c o u p l e s [l ,O] o r [O,lJ
I t i s c l e a r t h a t whenever t h e r e i s an o r d e r e d c o u p l e [ l , O ] an o r d e r e d c o u p l e LO,l]. So m i s even. Denote by r ' , s ' , m ' t h e analogous v a l u e s f o r -2' h We n o t e t h a t f o r every c o u p l e
ciJ
=
of
,l]
[O,OIof h2 [hi]
couple
[ c .l+4+v/2' ~ i + ~ / 2 J =
[O,Oj
,
and m t h e
t h e r e is a l s o
hl [h2]
there is the
.
S o r l = s and s l = r , w h i l e c l e a r l y i t i s m'=m. We n o t e t h a t t h e number of o r d e r e d c o u p l e s [ 1 , 1 ] , L O , O J , [l ,O] and [ O , l ] i n h c o i n c i d e s w i t h t h e number of elements of h t h a t i s r+s+m=q. -1 -1 ' i s r + r l = r+s Moreover t h e number of c o u p l e s c i ] =[1,1] of hl and -2 h
pi+4,
and t h i s number i s odd b e c a u s e q i s odd and m e v e n . L >
THEOREM 3.4 - L e t
v
=
4N
2
z n t h odd
p o s i t i o n s of
r
-1
1's i n r -1.
z1 s a t i s f i e s rl
Then, Lf
( O , l ) - r n a t r i x of order
be t h e J'irst roij of a c i r c u l a n t
N. Moreoiier, l e t m = i d l , d 2 ,
+
..., dk)
-,, where t h e s m
r-l+v12= e
Is
he t h e set of
mod 2 ,
I ; not
a difference set.
rl+v,2
at distance v/2 a r e = e , t h e n t h e odd e l e m e n t s of r Proof. If r + -1 -1 d i f f e r e n t , while t h e evenlelements coincide. Suppose Q i s a d i f f e r e n c e s e t w i t h t h e p a r a m e t e r s ( v , k , h ) . A s k(k-1) = (v-l)X , A is even. Denoted by D and P t h e s e t s of odd and even e l e m e n t s , w e n o t e t h a t a d i f f e rence a = di - d where d i , d . & @ , i s even i f and o n l y i f b o t h d i , d . b e l o n g j' 1 J t o D o r t o P. P , a o c c u r s an even number of t i m e s . By Lemma 3 . 2 , i f di,d j By Lemma 3 . 3 , i f d . , d . e D , we c a n n o t e t h a t t h e number 4 o c c u r s an odd number 1 J of t i m e s . So 4 o c c u r s a n odd number of t i m e s , w h i l e h i s e v e n . < >
4. I f 2 i s a ( 0 , l ) - s e q u e n c e
, we d e n o t e by -h a
t h e sequence o b t a i n e d by s h i f -
t i n g c y c l i c a l l y e v e r y element of h p o s i t i o n s t o t h e r i g h t .
THEOREM 4 . 1 - L e t
a b e a (0,l)-sequence of l e n g t h
the s e t of t h e p o s i t i o n s o f 5 satisfies a + 5iI2 =
in
1's i n
el, where
If
Z
occurs
v=4q and @ = [ d l , d 2 , . .
5. t k e sum is mod
h = k - q times as d i f f e r e n c e
2 , then e v e r y odd value
d . - d . , where d . , d . t 1 J 1 3
a.
P r o o f . L e t D and P be t h e s e t s of odd and even elements of As v i s e v e n , t h e d i f f e r e n c e d i - d . , where d i , d j 6 @ , i s odd i f and only J i f an i n t e g e r belongs t o D and t h e o t h e r t o P. Let
a =
~ l c 2 . . . c v ] be
.
.
On the Non-Existence of Certain Difference Sets If 5 + a
lr/2
=
e
the odd even]
-1'
483
elements of 5 at distance v/2 are different
.
[coincident)
This implies the odd elements are v/4
=
q and hence k
2
q.
Let c. be, where i is odd, an element of a. We have two possibilities: 1)
ci = 1
2)
ci = 0.
In the first case, let d. be an element of P; so also d J
Denoted c1 = i - d
we have also
c1' =
i
-
+ v/2 t P. i (d.+v/2) = a +v/2.
j' J = 1 , we obtain the differences = i+v/2 - d = In the second case, as c. l+V/2 j = c1 + v/2 and 6' = i+v/2 -(d + v/2) = a . j Hence, for every d. & P, we have the same differences in both of the cases. 1 J
and 2. Then, in order to calculate all the odd differences, we can suppose ci for every
odd i
e
,
[1,v/2]
that is
D =i1,3
=
1
,...,
Now, consider an element d; 6 P. J
We determine the differences
a. 1
=
d.
where i 6 D and also
1
-
a. + v/2
i =
(d. + v/2) J
-
i.
(2)
So for every d. t P, the differences ( 1 ) and ( 2 ) give every odd number in
[I ,v]
J
exactly once.
As there are also the differences -ai, - ( ai+v/2), that determine again every odd number of C1,vJ exactly once, we note that for every couple
d., d . + v/2 J
J
belonging to P, we obtain every odd number exactly twice. 1
As the couples d., d.+v/2 of P are -(k - q ) , we have proved that every odd J I 2 number in Z occurs exactly k - q times as difference di - d., where di,d. t J
0
J
.<>
REFERENCES
p]
Baumert, L.D., Cyclic Difference Sets, Lecture Notes in Mathematics n. 182 (Springer Verlag, 1971).
[2]
Biggs, N., Discrete Mathematics (Clarendon Press, 1985).
D]
Brualdi, R.A., A note
on
multipliers of difference sets, J. of Res. Nat.
B. of Standards, vol. 69 B ( 1 9 6 5 ) , pp. 87-89. Davis, P.J.,
61
Circulant matrices ( A Wiley-Interscience Publication, 1979).
Van Lint, J . H . ,
Coding Theory, Lecture Notes in Mathematics n.201
(Springer Verlag, 1971).
N . Zagaglia Salvi
484
[6]
Turyn, R., Character Sums and Difference
Sets, Pacific J. Math. 15 (1965)
pp. 319-346.
P]
Zagaglia Salvi, ti., Combinatorial structures corresponding to reflective circulant (O,l)-matrices, Annals of Discrete Mathematics 30 (1986), pp. 363-372.
Annals of Discrete Mathematics 37 (1988) 485-492 0 Elsevier Science Publishers B.V. (North-Holland)
ON COMPLETE 12-ARCS
485
I N PROJECTIVE PLANES OF ORDER 1 2
Corrado Zanella Dipartimento d i Matematica, UniversitB d i Roma "La Sapienza", 1-00185 Roma, I t a l y .
Let K be a complete 12-arc i n a p r o j e c t i v e plane 8 1 2 of Extending a r e s u l t i n [61 we show t h a t ~ 1 1 2has no point Moreover, there a r e a t most two p o i n t s of index 8. This i n v e s t i g a t e a l l p o s s i b l e c o n f i g u r a t i o n s of the tangents may occur e x a c t l y 1 7 c a s e s . 1. ON q-ARCS
IN
T
9'
order 12. of index 10. enables u s t o to K ; there
q EVEN
In a p r o j e c t i v e plane TI^ of even o r d e r q , l e t K be a q-arc, i . e . a s e t of q p o i n t s no t h r e e of which are c o l l i n e a r . On any p o i n t of K there a r e e x a c t l y two tangents. Thus K has e x a c t l y 2q tangents and q(q-1)/2 s e c a n t s . Furthermore, through any p o i n t o f nq there i s an even number of tangents. A p o i n t P E rq i s s a i d to be of index j , b r i e f l y a j - p d n t , i f t h e r e are e x a c t l y j tangents t o K through i t . L e t t j ( j = O , l , . . . , q ) be the number of j - p o i n t s i n TI^. Then t2h+l= = 0 f o r every h. Moreover [ 61 ,
Obviously, = 0 iff K i s complete. (1.2) q If TI i s Desarguesian, then K i s never complete [ 1 , 4 ] . In what follows, K i s 9 always supposed t o be complete; t h e r e f o r e , rq i s n o t Desarguesian. Whenever K i s incomplete, i t i s contained i n a (q+Z)-arc. Hence, i f TI c o n t a i n s no (q+2)a r c ( a s i t happens f o r q = 10, [ 21) then any q-arc i s c m p q e t e . Therefore
t
q > 10. Let
R
(1.3)
be a non-tangent l i n e and denote by u j the number of j - p o i n t s
412
1u
j=o
. 'J
on it. Then
412 =q+1,
1juZj =q.
j =O
I f t i s a tangent t o K , and v j i s the number o f j - p o
n t s on t, then
In [ 61 the following r e s u l t s a r e proved. I. Any conplete q-arc admits a t most one ( q - 2 ) p o i n t .
11. If t i s a tangent to K through a ( q - 2 ) p o i n t P then there e x i s t s a 4 p o i n t Q (PP) on t and any p o i n t on t , o t h e r than P m d Q , is of i n & x 2.CmseqwntZy
C. Zanella
486
t
q-2
=
1, t
> q-2,
t2
4 -
1. ( q - l ) ( q - 2 ) .
111. If t i s a tangent to K through a ( q - 4 ) p o i n t P , then e i t h e r a 6-point Q ( # P ) e x i s t s on t and m y o t h e r p o i n t on t i s of index 2, or t h e r e are two d i s t i n c t 4-points on t m d any o t h c r p o i n t on t is of index 2 .
I V . Assme K
is a
complete q-arc a d n i t t i n g a ( q - 2 ) p o i n i . P . %en,
Moreover, on m y tangent through P a 4-point l i e s , the remaining points being o f index 2; any tangent n o t un P &s (q-2)/2 4-points and ( q + 4 ) / 2 2-points. On any n m - t a n g e n t not on P there l i e v4 < 2 4-points and vo = l+v4 0-points. The r e m i n i n g p o i n t s have i n & x 2. V.
If a corrplete q-arc a h i t s a (q-Z)-point,
Therefore, the case q deal with it.
=
then q
=
12.
1 2 seems to be i n t e r e s t i n g . The aim o f t h i s n o t e i s t o
2 . ON COMPLETE 12-ARCS I N n12 V I . k t K be a complete 12-arc
Proof.
in n12. T?en t h e m is no l o p o i n t .
Suppose t h a t P i s a 1 0 - p o i n t . Then, by I V , t10 = 1,
t
t6 = t = O ,
8
0
=25,
t 2 =111,
t
4
=20.
Again by I V , on any of t h e t e n t a n g e n t s on P t h e r e i s e x a c t l y one 4-point, t h e remaining ones b e i n g 2 - p o i n t s . L e t r be t h e unique s e c a n t t h r o u g h P . From ( 1 . 4 ) , s i n c e ul0 = 1, u4 = uo - 5
(2.1)
f o l l o w s . Since u2 > 2 , we have ~0 + u4 < 10. T h i s , t o g e t h e r w i t h (2.1), g i v e s u4 5 2. On t h e o t h F r hand, t 4 = 20 and o n any t a n g e n t through P t h e r e i s exa c t l y one 4-point. T h e r e f o r e , on t h e remaining ( e x t e r n a l ) l i n e s t h r o u g h P t h e r e l i e a t l e a s t e i g h t 4 - p o i n t s . Thus, t h e r e i s a l i n e R on P which i s e x t e r n a l t o K and c o n t a i n s a t l e a s t f o u r 4 - p o i n t s , a c o n t r a d i c t i o n ( u s e ( 1 . 4 ) ) . 0 From V and V I i t f o l l o w s t h a t V I I . ."tocomplete q-arc i n
VIII.
nq, m y even q , admits a (q-Z)-point. 0
A corrphtte 12-arc K in n12 admits a t most tuo 8-points.
proof. Suppose A, B , C a r e t h r e e d i s t i n c t 8 - p o i n t s . F i r s t l y , assume t h a t A, B, and C a r e n o n - c o l l i n e a r . By 111, t h e l i n e s AB, AC, BC a r e non-tangents. Thus t h e 8 t a n g e n t s through C meet t h e l i n e AB i n 8 p o i n t s o t h e r than A and B.Hence, e a c h of them i s o f p o s i t i v e index. S i n c e U8 2 2, ( 1 . 4 ) ~i m p l i e s t h a t t h e r e a r e a t most f o u r p o i n t s d i s t i n c t from A and B, which a r e of p o s i t i v e i n d e x , a cont r a d i c t ion. Next, assume t h a t A, B, C l i e on the l i n e R. ( 1 . 4 ) h o l d ; hence, u =3, 8 Consider a p o i n t Q
Again
u = O for j # 0,8. (2.2) j AB o f i n d e x a t l e a s t 4 . Then t h e t a n g e n t s t h r o u g h Q meet
uo=lO,
4
By 111, R i s non-tangent.
On Complete 12-Arcs in Projective Planes of Order 12
487
AB i n f o u r d i s t i n c t p o i n t s of p o s i t i v e index. T h i s c o n t r a d i c t s ( 2 . 2 ) .
E l
L e t K be a complete l z - a r c , and suppose t 8 = 2. The l i n e j o i n i n g A w i t h B i s non-tangent (by 1 1 1 ) . From (1.1) we g e t t0=25-t6 , I X . I f t8 = 2 ,
then t
t 2 = 112+3t6
, t4
=
18-3t6,
t 8 = 2.
(2.3)
> 23.
0 -
Proof. L e t A and B be t h e two % p o i n t s , and r one of t h e f o u r non-tangents through A and n o t through B. Then U 8 = 1, and, by ( 1 . 4 ) , uo
=
4+u4+2u
=+
uo
2 4.
The e i g h t t a n g e n t s through B m e e t r i n e i g h t p o i n t s (#A) of p o s i t i v e i n d e x , hence uo = 4.
(2.4)
L e t n b e t h e number of 0 - p o i n t s on AB. By ( 1 . 4 ) , n h a s a 0 - p o i n t , by ( 2 . 4 ) , co = 16+n,
7. S i n c e no t a n g e n t on A
n > 7. 0
(2.5)
X . I f t 8 = 2, then e i t h e r t o = 25, and the Line joining the two 8-points i s external t o K, or to = 23.
to 5 25. I f to 7 24, t h e n ( 2 . 5 ) i m p l i e s n = 8 (n i s d e f i n e d i n t h e proof t o IX), and ( 2 . 3 ) 1 i m p l i e s t 6 = 1. C a l l P t h e unique 6-point. I f P E AB, t h e n n = 9. I f P 4 AB, t h e n t h e t a n g e n t s on P meet AB i n s i x d i s t i n c t p o i n t s of p o s i t i v e index; s o , n 5 7. Ia b o t h c a s e s we g e t a c o n t r a d i c t i o n . T h e r e f o r e , to # 24.
Proof. By ( 2 . 3 ) and I X , 23 <
I f t o = 25, t h e n from ( 2 . 3 ) t 6 = 0 f o l l o w s . By ( 2 . 5 ) , n = 9 , and ( 1 . 4 ) g i v e u2 = 0 f o r t h e l i n e AB. Since t h e p o i n t s on K have index 2 , AB i s e x t e r n a l t o K. 0 X I . I f t 8 = 2 and t 23, then, denoting by A mrd B the 8-points, any nontangent through A, ~ L L ; not through B, has e i g h t 2-points and four 0-points ( t h i s 7:s a l s o true i f to = 2 5 ) . The line AB has seven 0-points and four 2-
points. Through A there are eight tangents, two o f them have m e 6-point each, whereas the remaining Ones are o f index 2. f i e remaining s i x t m g e n t s have two 4-points each, a l l other points being o f index 2 . Proof. The f i r s t a s s e r t i o n immediately f o l l o w s from t h e proof t o I X and ( 1 . 4 ) . I f to = 23, t h e n n = 7 and t 6 = 2. The remaining s t a t e m e n t s can b e o b t a i n e d w i t h t h e h e l p of e q u a t i o n s ( 1 . 4 ) and ( 1 . 5 ) . 0
I f t 8 = 2 and tangent to K.
XII.
to
=
23, then the line through the two 6-points
is non-
Proof. L e t A and B be t h e 8 - p o i n t s , and C and D t h e 6 - p o i n t s . By X I , P = A B n CD i s d i s t i n c t from A, B, C , D. I f CP i s a t a n g e n t , t h e n P , a g a i n by X I , i s of i n d e x 2 . By (1.5), v2 = 10,
v 4 = 1,
V6
=
2.
B e s i d e s P , on t h e l i n e AB t h e r e are f i v e o t h e r p o i n t s of p o s i t i v e index.Through C t h e r e a r e f i v e t a n g e n t s b e s i d e s CD; e a c h of them i n t e r s e c t s AB i n a p o i n t of p o s i t i v e index. T h e r e f o r e , t h e l i n e s j o i n i n g e a c h of t h e s e p o i n t s w i t h C are t a n g e n t . The same argument a p p l i e s t o D. Consequently, t h e t a n g e n t s through t h e
C. Zanelta
488
2-points o t h e r than P on AB, meet CD i n C and D. CD h a s one 4-point Q . The t h r e e t a n g e n t s t h r o u g h Q, o t h e r t h a n CD, must i n t e r s e c t AB i n t h r e e p o i n t s of p o s i t i v e i n d e x d i f f e r e n t from 2 , by t h e p r e v i o u s argument. Since j u s t two such points e x i s t , we get a contradiction. 0 Next, a more d e t a i l e d i n v e s t i g a t i o n i s c a r r i e d o u t i n c a s e to = 23. L e t A, B be t h e 8 - p o i n t s , and C , D t h e 6 - p o i n t s . L e t S be t h e s e t c o n s i s t i n g o f t h e f o u r 2-points on AB. W.1.o.g. w e may suppose t h a t A # P = AB n CD. L e t u ' be the number of j - p o i n t s on CD. By X I , CD c o n t a i n s n e i t h e r A n o r B; t h u s , U8'= 0 . The e i g h t t a n g e n t s on A m e e t CD i n p o i n t s of p o s i t i v e index. Thus, u2 + u
> 6. 4 -
(2.6)
With t h e h e l p of ( 1 . 4 ) we o b t a i n uo = 5 ,
u 2 = 6,
u4 = 0 , u6 = 2 ,
u8 = 0.
(2.7)
Notice t h a t CD h a s e i g h t p o i n t s o f p o s i t i v e index, namely t h e i n t e r s e c t i o n s w i t h CD o f t h e t a n g e n t s t h r o u g h A. Hence, P = AB n CD i s of index z e r o .
(2.8)
The s i x t a n g e n t s through C i n t e r s e c t AB i n p o i n t s of p o s i t i v e index. Since on AB t h e r e l i e e x a c t l y s i x p o i n t s of p o s i t i v e index, t h e l i n e j o i n i n g each of them w i t h C i s a t a n g e n t . The same argument a p p l i e s t o D. I n p a r t i c u l a r , The t a n g e n t s through any 2-point on AB are t h e l i n e s j o i n i n g i t w i t h C and D. L e t t be any of such t a n g e n t s . By ( l . 5 ) , v2 = 9,
v4
3,
=
V6
=
1.
(2.10)
Consequently, a l l t h e twelve 4 - p o i n t s l i e on t h e union of t h e t a n g e n t s on C. Since t h e same argument a p p l i e s a l s o t o D , w e conclude t h a t any 4-point i s t h e i n t e r s e c t i o n of two t a n g e n t s , CSi and DSj, S = { S 1 , S 2 , S ,S$, i # j . On t h e o t h e r hand, t h e e i g h t l i n e s CSj, DSj, meet o f f S U {C,D? i n p r e c i s e l y twelve po i n t s . The r e f o re, X I I I . Suppose t o = 23; k t A and B be t h e two 8-points, C and D the tw o 6p o i n t s , S the s e t of t h e four 2 p o i n t s on AB. Then t h e r e are p r e c i s e l y twelve 4 p o i n t s , namely the p o i n t s i n t h e set X = i C Q n DR
I
Q , R E S, Q
# R}.
0
3. THE CASE t8 = 1 Throughout t h i s s e c t i o n we denote by K a complete 12-arc i n "12, By (1.1), to = 28-t6,
t2 = 104+3t6,
t4 = 24-3t6.
such t h a t t a = l .
(3.1)
X I V . The 6 - p o i n t s form a t 6 - a r c H .
p r o o f . L e t R be a non-tangent; by ( 1 . 4 ) 2 , U6 < 4. Moreover, i f U6 = 4, then u j = 0 f o r j # 0,6. Denote by P t h e unique 8-point. Then P 4 R and t h e e i g h t t a n g e n t s on P m e e t R i n p o i n t s o f p o s i t i v e index, a c o n t r a d i c t i o n . I f U6 = 3 , t h e n a g a i n P 4 R. Indeed, by eq. ( 1 . 4 ) 2 , 4u8 (12-3u6. T h e r e f o r e R has a t most
On Complete 12-Arcs in Projective Planes of Order I2
489
s i x points of p o s i t i v e index, a c o n t r a d i c t i o n . Thus, II has a t most two 6 - p o i n t s . Next, assume t h a t t i s a tangent. By ( 1 . 5 ) , v6
xv.
If t 8
=
1, then t6
5 6.
5
2. 0
( P r o p o s i t i o n X I X y i e l d s a better r e s u l t . )
P r o o f . Suppose Q i s any 6-point. Then rhere a r e a t l e a s t f i v e tangents through Q which a r e n o t on the 8-point P. By (1.5), according to the p o s s i b l e values f o r V6 (v6 = 1 or v6 = 2 ) , we have v4 = 3 or v4 = 1. In any case, a tangent t , on Q and not on P, contains a &-point; t h e r e f o r e , t 4 2 5. The statement follows from ( 3 . 1 ) . 0 X V I . Let P be the Lmique 8-point m d H the s e t of a l l 6 p o i n t s . H u {PI i s a (t6+1)-arc.
If t 6 f 2 , then
proof. In case t 6
3, we prove t h a t on any l i n e on P one 6-point l i e s a t m s t . Let A be a 6-point. I f PA i s a tangent, then 111 implies t h a t A i s the unique 6-point on PA. I f PA i s non-tangent, by ( 1 . 4 ) , U6 5 2 . Suppose u6 = 2 , then (1.4) imply (3.2) u 0 -> 8 .
Let B be a 6-point n o t on PA. The s i x tangents through B must meet P A i n p o i n t s of p o s i t i v e index. T h i s c o n t r a d i c t s (3.2). C i We were n o t able to prove X V I when t6 = 2 . Thus, we i n v e s t i g a t e the case i n which P and the two 6-points A, B l i e on a l i n e , say R . F i r s t l y , R i s nontangent (by 111). Next, from (1.4) e i t h e r
9,
u2 = 0,
u4 = 1,
u6 = 2 ,
u0 = 8,
u2 = 2,
u4 = 0,
U6
u0
=
=
2,
u8 = 1, o r U8
=
1
(3.3) (3.4)
follow. By (1.5) A tangent t without 6-points and 8-points has e x a c t l y f i v e 4-points,
(3.5)
t h e remaining ones being of index 2.
I f (3.3) holds, then I? has f o u r p o i n t s of p o s i t i v e index. Denote by C the 4p o i n t on R. By (3.5), t h e union of the tangents through C c o n t a i n s 1 7 4-points. From ( 3 . 1 ) t 4 = 18 follows; thus another 4-point e x i s t s , say Q. The tangents on Q meet R i n f o u r p o i n t s of p o s i t i v e index o t h e r than C , a c o n t r a d i c t i o n . Theref o r e , (3.4) m u s t hold. Denote by C and D the two 2-points on R, and by S the s e t of t h e 18 4-points. I f Q E S, then the tangents through Q i n t e r s e c t II i n p o i n t s of p o s i t i v e index. Consequently, a t l e a s t one tangent through Q contains e i t h e r C or D. By ( 3 . 5 ) , on t h e union of the tangents on C ( o r D) t h e r e a r e ten 4p o i n t s ; hence, t h e r e a r e e x a c t l y two p o i n t s U,V E S such t h a t U C , U D , V C , V D a r e tangents. W e c l a i m that
The tangents on U ( o r V) meet
L
in A,B,C,D.
If
(3.6)
Q E S-{U,V}, then PQ i s a tangent.
I f t i s a tangent through A ( o r B ) , then by (1.5) t has t h r e e 4-points. Thus S is contained i n t h e union of the s i x tangents through A (or B) and the f i r s t statement follows. Next, by 111, on any of the e i g h t tangents on P t h e r e l i e two 4-points. Therefore, t h e remaining two 4-points must coincide w i t h U andV. Next, we i n v e s t i g a t e , i n m r e d e t a i l , t h e (t6+1)-arc
M = {P E IT 4
I
P of index a t l e a s t 6 ) .
C. Zanellu
490
Unless otherwise s t a t e d , t 6 > 3 up t o t h e end o f t h i s s e c t i o n .
X V I I . k t A be a 6-point.
If t 6 > 3 , then t h e m i s no &-point on PA.
8 . Then PA i s non-tangent and ( 1 . 4 ) h o l d . Since t h e t a n g e n t s on any 6 - p o i n t C ( # A) i n t e r s e c t PA i n p o i n t s o f p o s i t i v e i n d e x , e q s . ( 1 . 4 ) have t h e unique s o l u t i o n uo = 7 , u2 = 3, u4 = 1, u 6 = 1 , u g = 1 . There a r e e x a c t l y s i x p o i n t s of p o s i t i v e i n d e x on PA. T h e r e f o r e ,
Proof. Suppose t h a t PA h a s a 4-point
I f C i s any 6 - p o i n t ,
then AC i s a t a n g e n t .
(3.7)
N o t i c e t h a t , by (1.5), I f t i s a t a n g e n t t h r o u g h two 6 - p o i n t s , t h e n t h a s e x a c t l y one 4-point. I f on a t a n g e n t s t h e r e i s j u s t one 6 - p o i n t and no 8 - p o i n t , then s h a s e x a c t l y t h r e e 4 - p o i n t s .
(3.8)
On A t h e r e a r e t6-1 t a n g e n t s , e a c h o f them j o i n i n g i t w i t h a 6 - p o i n t . None of t h e remaining 7-t6 t a n g e n t s c o n t a i n s a p o i n t ( # A) of index 8 o r 6 . S i n c e B i s > (th-l)+ a &-point t h a t does n o t l i e on a t a n g e n t through A, (3.8) i m p l i e s t 4 < 3, a c o n t r a d i c t i o n . + 3 ( 7 - t 6 ) + 1 = 2 1 - 2 t 6 . Eq. ( 3 . 1 ) 3 g i v e s t 6 I f t i s a t a n g e n t on P w i t h o u t 6 - p o i n t s , t h e n by 111 t h a s two & - p o i n t s . I f any t a n g e n t on P h a s no 6 - p o i n t , t h e n t 4 > 1 6 and ( 3 . 1 ) 3 y i e l d s t 6 2 . S i n c e , by assumption, t 6 L 3,
5
A 6-point
A e x i s t s such t h a t PA i s a t a n g e n t .
L e t A and B b e two 6 - p o i n t s . Then, e i t h e r AB i s a t a n g e n t , i n which c a s e AB c o n t a i n s e x a c t l y one 4-point, o r it i s a non-tangent and c o n t a i n s no 4-point.
(3.9) (3.10)
The f i r s t s t a t e m e n t i s a consequence of ( 3 . 8 ) . I f AE i s non-tangent, t h e n P F A B (by XVI) and t h e t a n g e n t s on P m e e t AB i n p o i n t s of p o s i t i v e i n d e x . Hence, e q s . ( 1 . 4 ) have t h e unique s o l u t i o n uo = 5, u2 = 6, u4 = 0 , U6 = 2, u8 = 0 . Now assume t 6 > 3 , and c o n s i d e r a 6 - p o i n t A. I f t i s a t a n g e n t on A w i t h n e i t h e r 6 - p o i n t s ( o t h e r than A) n o r tl-points, t h e n , by ( 3 . 8 ) , t h a s t h r e e 4 - p o i n t s . I f any t a n g e n t on A h a s no o t h e r 6 - p o i n t , t h e n t 4 > 15 a s a t l e a s t f i v e o f such t a n g e n t s a r e n o t on P. Thus eq. ( 3 . 1 ) 3 g i v e s t6-5 3. Hence, I f t 6 > 3 and A i s a 6 - p o i n t , t h a t AB i s a t a n g e n t .
X V I I I . If t 6
=
6, then each secant t o M
t h e n a 6 - p o i n t B e x i s t s such
(3.11)
is a t a n g en t t o K .
Proof. By ( 3 . 1 ) , t 4
= 6. F i x any 6 - p o i n t A , and denote by x t h e number o f tang e n t s t o K t h r o u g h A e a c h o f which c o n t a i n s a n o t h e r 6 - p o i n t . Thus, x < 5. By ( 3 . 8 ) , on t h e union o f t h e s e t a n g e n t s t h e r e a r e x 4 - p o i n t s . On A t h e r e a r e a t l e a s t 5-x t a n g e n t s t o K h a v i n g no p o i n t of index 8 o r 6 o t h e r t h a n A . By ( 3 . 8 ) ,
6 = t4
2 x+3(5-x)
=
15-2x
=)
x = 5.
T h e r e f o r e , a l l l i n e s which j o i n A w i t h any o t h e r 6 - p o i n t a r e t a n g e n t t o K . Now suppose PA i s non-tangent. Thus t h r o u g h A t h e r e e x i s t s a t a n g e n t t t o K, on which A i s t h e unique p o i n t of i n d e x > 4. by ( 3 . 8 ) , t has t h r e e 4 - p o i n t s . S i n c e on t h e union o f t h e above c o n s i d e r e d t a n g e n t s t o K on A t h e r e a r e f i v e 4-points, w e g e t t 4 2 8, a c o n t r a d i c t i o n . 0 XIX.
If t 8
=
1, then t 6
5
5.
Proof. Suppose t 6 = 6. A s a consequence of t h e proof t o X V I I I , A t a n g e n t j o i n i n g a 4 - p o i n t w i t h a 6-point c o n t a i n s a n o t h e r 6 - p o i n t , t h e remaining o n e s b e i n g of i n d e x 2 .
(3.12)
On Complete 12-Arcs in Projective Planes of Order 12
49 1
Indeed, i f t i s a t a n g e n t through a 6-point A, then e i t h e r t c o n t a i n s P, i n which c a s e i t h a s no 4-point, o r t h a s a n o t h e r 6-point and p r e c i s e l y one 4 - p o i n t Now f i x a 6 - p o i n t A. There a r e f i v e t a n g e n t s t h r o u g h i t and n o t t h r o u g h P, e a c h of them h a s one 4 - p o i n t . There a r e e x a c t l y two t a n g e n t s on P c o n t a i n i n g 4-points; namely, t h o s e t a n g e n t t o M t o o . Each of such t a n g e n t s h a s two 4 - p o i n t s (by X V I I I and 1 1 1 ) . T h e r e f o r e , a 4-point Q e x i s t s , such t h a t AQ i s t a n g e n t and PQ i s n o t . Suppose t h a t a t a n g e n t t e x i s t s t h r o u g h Q. such t h a t on t t h e r e i s no p o i n t of i n d e x > 4 . By ( 3 . 5 ) , t h a s f i v e 4 - p o i n t s . There a r e two t a n g e n t s on P c o n t a i n i n g 4 - p o i n t s . Each o f them h a s a t l e a s t one 4-point n o t on t . Hence, t 4 7 , a cont r a d i c t i o n . Consequently, s i n c e (3.12) h o l d s , any t a n g e n t t h r o u g h Q h a s e x a c t l y two 6 - p o i n t s . T h e r e f o r e , on t h e union of t h e t a n g e n t s on Q t h e r e a r e e i g h t 6p o i n t s . T h i s c o n t r a d i c t i o n and XV prove t h e s t a t e m e n t . 0
4 . THE CASE t 8 = 0 In t h i s s e c t i o n K d e n o t e s a complete 12-arc i n 7 ~ 1 2w i t h t 8 = 0 . By (l.l), to = 31-t6,
t 2 = 96+3t6,
t 4 = 30-3t6.
(4.1)
Under t h e s e assumptions, by ( 1 . 5 ) , Suppose t h a t t i s a t a n g e n t through a 6 - p o i n t A. Then e i t h e r t h a s e x a c t l y two 6 - p o i n t s and one 4 - p o i n t , t h e remaining ones b e i n g of i n d e x 2 , o r t h a s one 6 - p o i n t (A) and t h r e e 4 - p o i n t s , t h e remaining p o i n t s b e i n g of i n d e x 2 .
u.If
t g = 0, then t 6
5 8.
Proof. By ( 4 . 2 ) , on any t a n g e n t t h r o u g h a 6 - p o i n t A t h e r e i s a t l e a s t one 4point. Therefore, t 4
2 6.
The s t a t e m e n t f o l l o w s ( t a k e i n t o a c c o u n t ( 4 . 1 ) 3 ) .
XXI. Denote by H t h e s e t of al?, 6-points.
If t 6
2
7 , then H i s an incomp&?t&
t6-al"C.
Proof. Assume t h a t A , B and C a r e t h r e e c o l l i n e a r 6 - p o i n t s . L e t x be t h e number of t a n g e n t s t h r o u g h A c o n t a i n i n g no f u r t h e r 6-point. Since AB i s non-tangent and t h e 6 - p o i n t s , d i s t i n c t from A,B,C, number t6-3, we have x 2 6-(t6-3) = + t 6 . 24-2t6 4 - p o i n t s . Thus, on t h e t a n g e n t s through A t h e r e are 3x+(6-x) = 2x+6 By (4.1)3, 24-2t6 5 30-3t6 * t 6 5 6. To prove t h a t H i s i n c o m p l e t e , we d i s t i n g u i s h two cases, a c c o r d i n g t o t h e v a l u e of t 6 . t 6-= 8. Let A be a 6 - p o i n t . By ( 4 . 2 ) , on t h e t a n g e n t s through A t h e r e a r e 3x+(6-x) = 6+2x 4 - p o i n t s . A has another 6-point.
By ( 4 . 1 ) , t 4 = 6; hence, x = 0, i . e .
any t a n g e n t on
N o t i c e t h a t t h e number of t a n g e n t s t o K h a v i n g two 6 - p o i n t s e q u a l s the h a l f of t h e c a r d i n a l i t y of t h e s e t {(P,Q)
1
P,Q E H and PQ i s t a n g e n t t o K}.
Indeed, l e t t be t h e t a n g e n t t o K t h r o u g h P,Q E H. With t t h e two o r d e r e d p a i r s (P,Q) and (Q,P) are a s s o c i a t e d . S i n c e t h e r e are e i g h t p o s s i b l e c h o i c e s f o r P and j u s t s i x f o r Q, t h e t a n g e n t s t o K number 8 * 6 / 2 = 24 which a r e a l s o s e c a n t t o H . Consequently, t h e y e x h a u s t all t a n g e n t s t o K. Hence, Any t a n g e n t t o K i s s e c a n t t o H. As a consequence- of ( 4 . 3 ) , t a n g e n t t o K.
(4.3)
t h e r e a r e p r e c i s e l y f o u r s e c a n t s t o H which are non-
L e t Q be one o f the s i x t a n g e n t s t o H through a n a r b i t r a r y 6 - p o i n t , s a y A. By ( 4 . 3 ) , R i s non-tangent t o K. Hence, by ( 1 . 4 ) , uo = 3, u2 = 9, u6 = 1. Denote by S t h e s e t o f t h o s e 0 - p o i n t s which l i e on t h e t a n g e n t s t o H t h r o u g h A; t h u s ,
492
C. Zunellu
I S \ = 18. L e t r be a s e c a n t t o H which i s non-tangent t o K . There a r e f o u r such l i n e s . By ( 1 . 4 ) w i t h u4 = 0 , we g e t uo = 5 , u2 = 6 , U6 = 2 . I f A € r , t h e n r n S = 0. On t h e o t h e r hand, i f A @ r, t h e n r n S I 5 5 . Since t h e r e are prec i s e l y t h r e e l i n e s which are s e c a n t t o H, non-tangent t o K and n o t on A, a p o i n t P E S e x i s t s , such t h a t t h e r e i s no s e c a n t t o H through i t . Hence, H i s incomplete .
I
t 6- = 7. F i r s t of a l l , w e prove t h a t There e x i s t a t most t h r e e l i n e s which a r e s e c a n t t o H and non-tangent t o K.(4.4) F i x a p o i n t A E H, and d e n o t e by x t h e number of l i n e s through A and t a n g e n t t o b o t h H and K. By ( 4 . 2 ) , on t h e union of t h e t a n g e n t s t o K on A t h e r e a r e 6+2x 4 - p o i n t s . Eqs. ( 4 . 1 ) imply t 4 = 9; t h u s , x < 1. Obviously, x i s a l s o t h e number of those l i n e s on A which are s e c a n t t o H &d non-tangent t o K . T h e r e f o r e , through any p o i n t of H a t most one l i n e p a s s e s s e c a n t t o H and non-tangent t o K . Hence, ( 4 . 4 ) f o l l o w s . A s a consequence of ( 4 . 4 ) , a p o i n t Q e x i s t s i n H, such t h a t any s e c a n t t o H on Q i s a t a n g e n t t o K, and v i c e - v e r s a . L e t r be a t a n g e n t t o H a t Q . There are seven s u c h t a n g e n t s . Since r i s non-tangent t o K, by ( l . 4 ) , u2+u4 5 u2+2u4 = 9 which, t o g e t h e r w i t h eq. (1.4)1, i m p l i e s u O L 3 . u2+2u4 = 9 Next, denote by S ' t h e s e t o f t h o s e 0-points l y i n g on t h e t a n g e n t s t o H a t Q . Thus, S'I 21. Assume a s e c a n t , s a y p, e x i s t s t o H, which i s non-tangent t o K . T a k e X E p n H, t h e n X Q . F i v e o f t h e s i x t a n g e n t s t o K on X a r e s e c a n t t o H, whereas t h e remaining one i s t a n g e n t t o H. Hence on t h e union of t h e t a n g e n t s t o K through X t h e r e are e i g h t 4 - p o i n t s . Since t 4 = 9, t h e r e i s a t most one 4-point on p. T h e r e f o r e , by ( 1 . 4 ) , e i t h e r uo = 5 o r uo = 6 , which i m p l i e s Ip n S ' / '6. Thus, t h e same argument a s i n case t 6 = 8 y i e l d s t h e e x i s t e n c e of a p o i n t P E S such t h a t H U { P I i s an a r c .
1
5. CONCLUSION X X I I . E q u a t i o n s (1.1) have a t most 1 7 s o l u t i o n s w i t h
a s s o c i a t e d w i t h a c o n p k t e 12-arc Proof.
in ~ 1 2 ) .
a geometrical rneming-(i.e.
The s t a t e m e n t f o l l o w s from p r o p s . VI,VIII,X,XIX,
and XX. 0
REFERENCES H i r s c h f e l d , J.W.P., P r o j e c t i v e geometry o v e r f i n i t e f i e l d s , Clarendon P r e s s , Oxford 1979. Lam, C.W.H., T h i e l , L . , Swiercz, S . , M c Kay, J . , The non-existence o f o v a l s i n a p r o j e c t i v e p l a n e of o r d e r 10, Discrete Math. 45 (1983), 319-321. M a r t i n , G.E., On a r c s i n a f i n i t e p r o j e c t i v e p l a n e , Canad. J . Math. 19 (1967), 376-393. T a l l i n i , G . , S u i q - a r c h i d i un piano l i n e a r e f i n i t o d i c a r a t t e r i s t i c a p=2, Rend. Accad. L i n c e i ( 8 ) 23 (1957), 242-245. T a l l i n i , C., L e z i o n i d i Geometria 111, anno ace. 1983-84, Dip. Mat. Univ. Roma "La Sapienza", 1984. T a l l i n i , G . , Sui q-archi c o m p l e t i d i un p i a n o p r o i e t t i v o non d e s a r g u e s i a n o d i o r d i n e q p a r i , Quaderni Sem. Geom. Comb. 54, Marzo 1985, Dip. M a t . Univ. Roma "La Sapienza". Z a n e l l a , C . , S u i 1 2 - a r c h i c o m p l e t i d i un piano p r o i e t t i v o d i o r d i n e 1 2 , Quaderni Sem. Geom. Comb. 58, Dicembre 1985, Dip. Mat. Lhiv. Roma "La Sapienza".
Annals of Discrete Mathematics 37 (1988) 493-496 0 Elsevier Science Publishers B.V. (North-Holland)
493
BLOCK DESIGNS ADMITTING FLAG TRANSITIVE GROUPS OF AUTOMOFWHISMS
Paul-Hermann Zieschang Mathematisches Seminar der Universitlt Kiel OlshausenstraRe 40 D-2300 Kiel 1 West Germany
In recent years, a large number of papers (e. g. [ 2 1 , [ 31, [S], [lo]) has been investigating finite block designs with 1 = 1 admitting flag transitive groups of automorphisms. Most of these papers consider additional assumptions which restrict either the structure of the design or that of the acting group. In the following we will discuss finite block designs with )r = 1 admitting a flag transitive group G of automorphisms such that [S],
(*) each two point stabilizer Gxy unique block determined by x and y.
acts trivially on the
The following proposition is the main result of
[9].
PROPOSITION 1. Let D be a block design with A = 1 and point set x and let G be a flag transitive group of automorphisme of D that satisfies ( * ) . Let x E x and let A be a non trivial normal p-subgroup of G , . Then each one point stabilizer Gr contains exactly one conjugate Ar of A and the family of fixed point sets {Fix(Arn Gs) i r,s E X I rPs) forms a block design with A = 1 on X on which G acts as a group of automorphisms.
P.-H. Zieschang
494
Proposition 1
is needed to prove the following result of
191.
THEOREM 2. Let D be a block design with A = 1 and point set X and let G be a flag transitive group of automorphisms of D that satisfies ( * ) . Let x E X and assume that Gx has an abelian normal subgroup which is not semiregular on X - {x}. Then, for some integer n 2 3 and some prime power q, we PrL(n,q). have PSL(n,q) 5 G
Theorem 2 generalizes Theorem A of O'Nan's paper [6]. It is crucial in the proof of the following theorem [lo].
THEOREM 3 . Let D be a block design with A = 1 and point set X and let G be a flag transitive group of automorphisms of D that satisfies ( * ) . Let x E X and assume that Gx has a normal subgroup which is a T.I. set in G and not semiregular on X - (x}. Then the conclusion of Theorem 2 holds.
Theorem 3 is a generalization of Theorem A of [ 7 ] . The following result 181 is similar to Bender's theorem [l]. Its proof uses complex character theory and a lot of results due to Hiller [ 4 ] .
THEOREM 4. Let D be a block design with X = 1 and point set X and let G be a flag transitive group of automorphisms of D that satisfies ( * ) . Assume that the one point stabilizers of G have odd order and that the two point stabilizers are cyclic. Then either G is solvable or G acts doubly transitively
on X.
Block Designs Admitting Flag Transitive Groups
49 5
REFERENCES H. Bender, Endliche zweifach transitive Permutationsgruppen, deren Involutionen keine Fixpunkte haben, Math. Z. 104 (1968), 175 - 204. F. Buekenhout, A . Delandtsheer, J. Doyen, Finite linear spaces with flag-transitive and locally primitive groups. To appear. A. R. Camina, Groups acting flag-transitively on designs, Arch. Math. 32 (1979), 424 - 430. W. Hiller, 7/4-transitive Permutationsgruppen, Dissertation, Math. Sem. der Univ. Kiel, 1980.
W. M. Kantor, Primitive permutation groups of odd degree, and an application to finite projective planes, J. Algebra 106 (1987), 15 - 45.
M. O'Nan, A characterization of Ln(q) group, Math. Z. 127 (1972), 426 - 439.
as a permutation
M. OrNan, Normal structure of the one-point stabilizer of a doubly-transitive permutation group. I., Trans. Amer. Math. SOC. 214 (1975), 1 - 42. P.-H. Zieschang, Uber eine Klasse von Permutationsgruppen, Dissertation, Math. Sem. der Univ. Xiel, 1983.
P.-H. Zieschang, Fahnentransitive Automorphismengruppen von Blockplanen, Geom. Dedicata 18 (1985), 173 - 180. Zieschang, A spaces. To appear.
[lo] P.-H.
theorem of
O'Nan
for finite
linear
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Annals of Discrete Mathematics 37 (1988) 497-502 0 Elsevier Science Publishers B.V. (North-Holland)
497
AN INDEPENDENCE THEOREM ON THE CONDITIONS FOR INCIDENCE LOOPS
Elena ZIZIOLI Dipartimento di Matematica, Universith Cattolica via Trieste 17 25121 Brescia. Italia *
1.
INTRODUCTION
A fibered Zoop ( P , p , - ) is a loop (P..) together with a set 9 (191>1 of ) subloops of (P,.) such that for any atP\(ll there is exactly an element F t s with atF. In the recent paper [51 ,we have stated algebraic conditions in order to provide a fibered loop (P,@,.) with a geometric structure either of an incidence loop or of an incidence space with parallelism or of a kinematic loop. In this note we prove (Theorem 1) that the conditions stated there are orderly independent.
2. PRELIMINARY NOTIONS First of all we recall some definitions and properties. An inci.dence space is a pair ( P , g ) where: is a set of subset of P i) ii) v a , b o P , a f b g1R09 such that a,bcR iii) V R E . ~ 1~122 In an incidence space ( P , g ) let us assume the set P to be a loop (P,.) and let us consider the following properties: (L) V a t P the map a , : P - P ; x -ax is an element of Aut(P,.%); (R) V a s P the map a, : P-P; x -xa is an element of Aut(P,W); (K) VXoW with 1 t X (X,.) is a subloop of (P,.); then we shall say (cf. [2] $ 7 ) that (P,w,.) is: an incidence Zoop if (L) holds ; a 2-s7:ded incidence Zoop if (L) and (R) hold ; a kinematic loop if (L),(R) and ( K ) hold
a
.
.
fibered loop (P,@,-) is a loop (P,.) together with a subset 9 of the powerset of P such that: i) V F E ~ (F,.) is a subloop of (P,.) and I p l > 1 , ii) VaoP\111 3 , F ~ s such that aoF ; the set .Fwill be called a fibration of (P,.) A
.
By definition a kinematic loop ( P , g ,- ) to the fibration $:= 1x09 I lox) .
is always a fibered loop with respect
An incidence space ( P , g ) is called an
incidence space with paralZelism
-
* Research supported by the Italian Ministry of Education (M.P.I.
40%)
.
E. Zizioli
498
(P.2,
11)
if there is a relation
" / I t ' on 3
(E) ( t / I ' ' is an equivalence relation (P) V X E P , V Y E B glZE 98 with In an incidence space with parallelism que element ZE.%? such that XEZ and
such that:
XEZ IIY . (P,B,
Z IIY
.
11)
we denote by
{xllY ithe uni-
From now on let ( P , F , . ) be a fibered loop and let %:=tax 1 a E P , X E , 1~ Moreover we define the relation ' l / l r l on .% in the following way: aX /IbY : t) 3 a ' , b ' ~ P, 3 Z ~ 2such that aX = a'Z , bY =b'Z . In [5] we introduced the following conditions for a fibered loop ( P , 9 , * ) : (Fl) !ax 1 aEP , X E F ) = iYb I beP , YE 9 I ; (F2) ~ B E F: x(yA) = (xy)B ; ~ A F E VX,YEP AX)^ = C(xy) ; (F2') VAEF \ d x , y ~ P ~ C E S: (F3) VAEF , VaEA , V X E P (xa)A = xA and we proved (see [ 5 ] (9)c),(16),(10)): (1) (P,!%,.) is an incidence loop if and only if ( F 2 ) holds. ( 2 ) ( P , g , - ) is a kinematic loop if and only if (Fl,2,2') hold. ( 3 ) (P,.%, I / ) is an incidence space with parallelism if and only if ( F 3 ) holds. the foll.owing condition as well: We now consider for the loop (P, g,.) (F3') VAEF , VaEA , V X E P A(ax) = Ax ; and we define on 2 the following relation (right-parallelism): aX /I,bY 3 a ' , b ' E P , 3 2 ~ 9such that aX = Za' , bY = Zb' , then we have: (4) (P,2,11, 11, ) is an incidence space with parallelism if for the fibered loop ( P , F , . ) the conditions ( F l ) and ( F 3 ' ) hold. Proof. By (F1) % = IYb 1 YE 9 , bEP 1 , hence the proof follows from prop. (10) of is] .
:*
Therefore from a fibered loop (P,9, verifying (F1,3,3') we can obtain an incidence space with two parallelisms (P,.G@, 11, 11, 1 . A doubZe space (cf. [ 3 ] ) is an incidence space (P,,%) with two parallelisms 11 , 1 1 , such that the following condition holds: (D) let A,BE%', let asA , bEB and let B':=tal/B1 , A':AblI,AI then A'?B' f if AnB f @ . But our fibered loop ( P , F , - )fulfilling (F1,3,3') is not a double space because the condition D) is only proved if (P,.) is a fibered group such that for any aEP a 9 a c 9 (cf. [3] ) . Otherwise,for a proper fibered loop (i.e. a loop which is not a group),we have only the specialization of (D) where 1 EAQB . a )
-4
Let tpl,p2,...,p 1 be a set of conditions in a given structure.We recall the following definitfons: i ) the conditions p1,p2, ...pn are abso1uteZy independent if for any iE{l,2,. ..,n 1 the condition p . cannot be proved by means of the remaining p l , ..., 1 P i - l p P i + l , - .'Pn . I ii) the conditions P1,P2,. . .pn are o r d e r l y independent i f for any idl,2,.. .,ni the condition p.cannot be proved by means of the previous p . . ,Pi-l 1 ones. The definition i ) implies ii).Viceversa B.Levi proved '1 that if ; p ,p 1 2"'
A n Independence Theorem for Incidence Loops
499
.,pm} are orderly independent it is always possible to find a set of condiabsolutely independent and equivalent to the set tioEs { p;,p;, . . . ,p; } tP1'P 2*...9Pn 1 . In $4 we shall prove the following main Theorem: Theorem 1 . There exist proper fibered loops (P,@,-) fulfilling the conditions (Fl,2,3,3');moreoverthese conditions are orderly independent.
3. FIBERED LOOPS AND TACTICAL CONFIGURATIONS
In order to prove Theorem 1 we recall the following notions
(cf. [ 11):
A tactical COnfi@YatiOn is a pair ( p . 3 ) where: i) is a subset of the powerset of P , ii) V X , Y E ~ 3 k E N : IXI=IYI=k, iii) 3 r E K such that for all PEP r = I { XE 9
}
I
PEX
I .
We now assume that the fibered loop ( P , y , - ) has all the fibers of the same cardinality and we introduce the following numerical constants.Let v:=[PI , n:=191 , k:=IXI for XE@. Then v = n(k-l)+l and n 2 3 . If we set Y : = [al 1 aEP 1 and &':=
.
.
4. PROOF OF THEOREM 1
In this section we shall prove Theorem 1 in four steps by showing at each step that there exist fibered loops fulfilling the first n but not the last (4-n) conditions of Theorem 1 (with n = 1,2,3,4 ) .
I Case
n
=
1
.
We consider the set P = l,a,b,c,ab,bc,ca,(ab)c,(bc)a,(ca)bl with the composition law defined by the following multiplication table: b ab
ab c ca b bc a 1 1 (ab)c bc a (ab)c 1 C b (ca)b (ca)b a (bc)a (bc)a ab c (ab)c (ca)b ca ca (bc)a bc
bc ca (bc)a c c (ca)b b a (ca)b (bc)a 1 (ab)c (ab)c 1 ca bc a ab ab b
E. Zizioli
5 00
It is easy to verify that ( P , . ) is a commutative loop of order 10 with the fibration g:={{l,a,b,abl , {l,c,(bc)a,(ca)b) , {l,bc,ca,(ab)c)l and k = 4. Then this loop fulfils (Fl) but since 10/4 and (10*3)/4 are not integers ( P , p , .) fulfils neither (F2) nor (F3) nor ( F 3 ' ) In a subsequent note we shall introduce a constructive method to obtain finite fibered loops (in general non commutative) with all the fibers of the same cardinality such that only the condition (Fl) holds for k>2.
.
I1
Case
n = 2
.
To obtain fibered loops verifying ( F 1 ) and (F2) but not the remaining conditions, it is enough to choose k z 2 and v odd.The smallest example is the set P = {l,a,b,c,d} where the multiplication table is given by:
d b c
l d a
a l b
c a 1
(tab. 2)
Moreover we have also other examples with k > 2 :e.g. the remark 6 of theorem (18) in (51 gives us a class of examples of kinematic loops ( P , g , - ) where the fibration 9 violates the laws (F3) and (F3') and where ( P , . % ) is an affine space of dimension 5. I11
Case
n = 3
Now we show that there are examples of fibered loops which are not groups but where (Fl),(F2) and (F3) are valid. Since v/k has to be integer and since any loop with four elements is a group the smallest order f o r such a possible loop is at least 6 .The following multiplication table gives us such an example:
e
a
b
l
c
Here p:=I{l,a) ,tl,b) ,tl,c) , {l,dl ,tl,ell and one has (xy)y = x for if Y = t 1 , y ) all x,y~P.Therefore (xy)Y = txy,x) = XY But il,aF-(ab) = id,cl f (b,d) = i1,aI.b , hence (F3') is not valid.
.
IV Case n
=
4
.
A large class of examples of proper fibered loops with (F1,2,3,3')can be obtained in the following way.By remark 3 of $6 in [5] (see also [2]) we know that there are proper fibered loops ( P , F , - ) for any k = pn , p f 2 a prime, 7 nEM and IPI = k such that all the four conditions of theorem 1 hold and ( P , $ % ) is an affine space of dimension 7.
This achieves the proof of Theorem 1.
An Independence Theorem for Incidence Loops
50 1
We may investigate the case of the smallest cardinality for the loop (P,$F,.) 2 fulfilling the conditions (F1,2,3,3').Hence let k = 2 (i.e. x = 1 for all xeP ).Then we have: Let (P,$F,.) be a fibered loop with k = 2.Then (F3) is equivalent to : t/a,bpP (ab)b = a; a(ab) = b ; (F3') is equivalent to : v a , b c P the conditions (F3) and (F3') hold if and only if for all a,beP\{l) ,a f b the subloop
a) b) c)
.
.
With these observations one recognizes that the smallest proper loop with (F3) and (F3') and k = 2 (hence with (Fl) and (F2)) is the set P = {l,a,b,c,ab, bc,ca,(ab)c,(bc)a,(ca)b} with the fibration y*:={ {l,x) I xEP\{l) } and the multiplication table of (tab.1). Hence IPI = 10
.
If we set now k = 3 ,the smallest cardinality for a fibered loop ( P , g , * )with (F1,2,3,3') can be only 9.But for v = 9 the corresponding incidence loop with two parallelisms (P,.@, ) I , I I r ) can be only the affine plane (cf. [4] ) and the loop (P,.) is the abelian group n 3 x n3 . Therefore we have the smallest proper fibered loop ( P , g , - ) fulfilling (F1,2,3, 3') for k = 2 and IPI = 10
.
Remark.
.
In this note we did not consider the independence of the condition (F2') Actually we don't possess examples of loops verifying the conditions ( F 1 , 2 ) but not (F2') . In this context we can only prove the absolute independence of (F3) and (F3') from (Fl),(FZ),(FZ'),as the following examples show. 2 2 Let P:= {l,a,a2,b,b2,c,c,d,d ).We consider the composition law 'i.l' defined on P by the following multiplication table: 1 l a a' a Ia21 2 l a a c2 dZ c2
C C
d
2
1
d d b
d2
b b c2
b c c2 bZ 1 d d2 a
bZ
d d2 1 b a a'
c
c b b', d d c 1 a'
c2 d2 d a a2 1 c bZ
d d2 b2c2 b c 2 c a 2 c a 2 a b a b2 d2 1
Then (P,- ) is a proper loop with a fibration 9:= { {l,a,a2) , {l,b,b2} , {l,c, c2) , {l,d,d2}} of 4 subgroup of order 3.It is easy to verify that in this fibered loop (Fl),(F2),(FZt) and (F3') are not valid but (F3) is fulfilled. If we consider now the new operation ' t o ' ' defined on the same set P as follows: xoy:= yx for all ' x,ycP
502
E. Zizioli
the multiplication table of (P,o) is given by transposing the matrix of (tab.4); hence ( P , ~ , o )is a fibered loop and in this case only the condition ( F 3 ' ) is fulfilled.
NOTES AND REFERENCES Atti dell'Accademia delle Scienze - Torino. ( 2 ) 54 (1904) p.284 DEMBOWSK1,H.P. : Finite geometries . Berlin-Heidelberg-New York 1968 KARZEL,H. and KIST,G.P. : Kinematic algebras and their geometries. Rings and geometry (Kaya et al. eds.) NATO AS1 series - C (1985), 437-509 ,KROLL,H.J. and SORENSEN,K. : Invariante Gruppenpartitionen und Doppelraume. J. reine angew. Math. 2 6 2 / 2 6 3 (1973) 153-157 KARZEL,H. and PIEPER.1. : Bericht iiber geschlitzte Inzidenzgruppen. Jahresber. Deutsch. Math. Verein 3 (1970) 70-114 ZIZIOL1,E. : Fibered incidence loops and kinematic loops. J.of Geometry (to appear)
