Finite Groups

PROCEEDINGS OF A SYMPOSIUM IN PURE MATHEMATICS

OF THE AMERICAN MATHEMATICAL SOCIETY

Held in New York April23-24, 1959

Cosponsored by

THE INSTITUTE FOR DEFENSE ANALYSIS under contract Nonr 2631(00) with the Office of Naval Research

iinitorial Committee A. A. Albert Irving Kaplansky

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Prepared by the American Mathematical Society under Contract Nonr 2631(00) with the Ofrce of Naval Reeearch

International Standard Serial Number 0082-0717 International Standard Book Number 0-8218-1401-X Library of Congress Cataiog Number 50-1183 Copyright O 1959 by the American Mathematical Society Third printing,1979 Printed in the United States of America reserved except those granted to the United States Government May not be reproduced in any form without permission of the publishers

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contained in

II. It then foll.ows that N*(H) S NX(V). If we further

suppose

P, is a p-Sylow subgroup of Nr(H), then unless Pt is a p-Sylow subgroup of X, it follows that Nr(H) < NX(V), since N*(Pt) S N*(V) and

that

NX(P1)

{

w*(H). lntuitively, playing off Theorem B against suitably cho-

sen subgroups of P1, there is a possibility of finding p-subgroups of X whose normalizers have prescribed properties and which involve the

prime p to a considerable amount. This explains the underlying idea of the proof, which otherwise becomes obscured by details.

It might

be well to mention some ideas which don't seem to work,

though superficially they seem relevant. The

first

of these concerns the

maximal p-Sylow intersections P OPx of a group X. One might think that a knowledge of these groups, with

their normalizers, would lead to very

precise information regarding the way in which P is embedded in X. This may in fact be the case, but these groups seem very difficult to manage, and my impression

is that knowledge of them provides only a crude ap-

proximation to what is really going on. A second idea would be to keep an eye on the terms of the ascending (descending) central series of P, and to study

their normalizers. This idea broke down, apparently because

if A is a term in the ascending (descending) central series of P, and A . Pl < P, there is no reason why A should be

of the following possibility:

a

term in the ascending (descending) central series of Pt. ln any case, after considerable floundering in the central series, it

occurred to me that as regards Theorem A, the relevant p-groups are the maximal abelian normal subgroups of P. Here, abelian normal subgroup of P, and mal abelian normal subgroup of

if A is a maximal

A ' Pl < P, then A is also a maxi-

Pt.

Even more fortunate, however, is

It is this last property which, at a very critical point in the proof, makes it possible to assert that a parthat A is a p-Sylow subgroup ol Ct(A).

ticular p-element, about which information is sorely needed, actually lies in A.

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BURNSIDE GROUPS AND ENGEL RINGS by R. C. LYndon

$1. Introduction

If G is a discrete group with descending central series Gl Gn+1 = (Gn, G), then the formation of commutators (u,v) =

"-1

= G,

u-1

rru

L on the direct sum of the abelian quotient groups Gn/Gn*l. If G has prime exponent P, uP = 1 for all u in G, then L has characteristic P, PU = 0 for all U in L, and moreover L induces the structure of a Lie ring

satisfies the (p-1)-st Engel condition:

(E-,): 'p-r'

[u,vn-r1=tt... [u,v],...1,v]=o

for all u and v in L. we are concerned with the question of whether, in general, L is subject to any further identities that are not consequences of characteristic p together with the Engel condition. Investigations by Sanov and Kostri.kin strongly indicate that there are no such further iden-

tities, but are not fully conclusive. we approach this problem by a method introduced by Grirn, which we are able to simplify for the purposes of the present context. Although we are far from proving that the identities

of Grii'n are consequences of the Engel condition, or even of recovering the results of sanov and Kostrikin, we think it is worthwhile to present the problem in a direct and elementary form.*

The connection between a discrete group G and a Lie ring L was

first

investigated by Magnus, and by

witt,

Zassenhaus and others, nota-

bly, in a recent paper, by Lazard. This connection has been fundamental to much work on the problem of Burnside. Recently Kostrikin has shown

'I

error, due to faulty arithme1957, p. 309.

take this occasion to acknowledge an

tic, which was pointed out by Kostrikin,

I

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e

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1zq1

G(.) =nG1n) = 1. In any case G = G/G1.; is faithfully represented in T. A/ t' , with the induced map of G into f, = : Ln/Ln+l mapping =

Glrr;/G1rr*r; homomorphically onto

1n

I a'\+L and

car rying commutation

into commutation and p-th powers into p-th powers'

If w = uv, writing w = 1+W, u = 1+U, v = 1+V, from W=w

-

1 = uv

-

1 = (1+U)(1+V)-1 = U + V + UV

it follows that A is in fact generated by the elements X = x-1 and X' = x-l -1 for any set of generators x for G; if G is a torsion group n, any n, it suffices to restrict the X' are superfluous. (Indeed , f.or A/ A the x to a set of coset representatives of a set of generators for G/G2.) In general, fromO = X + X' + XX', iterating the substitution X' = -X - )O(' shows that

X' agrees modulo An, for any n, with the n-th partial sum of

the series (1

+x)-1

-1=-X+x2-

which leads to a representation of G in A, ti

"

obtained by completing E according to powers shown by Magnus and Witt that

if

Lu

= 1,

= 0. whence

G(r)3G(r)

We are concerned here with

nent p: G =

r/FP,

power series ring in the X

of ,l . By this

means

it was

G is a free group, on generators x, then

andthatL is free

on generatorsX'

a'relatively free' group of prime expo-

F free, FP the normal subgroup generated by

all p-th

powers uP of elements u from F. Since O(n)O = 1, the dimension series reduces to the descending central series: G(n) = Grr. Since the quotients

of the descending central series have exponent p, L has characteristic p' and we may as well take K to be the prime

of Novikov, we will not in general have

G,

iield Zn. In view of the result = 1,

nor a'

0; but, since

G is a torsion group, we have no occasion to introduce power series, and

it is optional whether we reduce A modulo ao . In any case' if G is relatively free on generators x, it is easy to see that each ull in G = G/G .

L

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anblrm e seq

where 0(iL, .. . , js) is evidently the number of ways that the sequence o = (jf . . . j") can be written as a product of p non-empty strictly increasing sequences o(1), . . . , o(p). More explicitly, let a = &(i1...is)

in c, j, . ii*1. Then d is the number ofways,afterinsertingabreakateachofthe F =s -1 - a non-increasing steps, of choosing the remaining b-1) - (s-1- c ) = p-s + o breaks

be the number of increasing steps

amongthe a increasingsteps: a =(ois1 u=

1+

=("10). tttns,

") ("(:t "' j"))x

: t pssstp j1,...,j"\

s-P- /\,"'1"'

If F is any polynomial in a set T of indeterminates X, which need not commute, then F = : . _ Fsu where Ft is the sum of those terms

^ o<s
that contain precisely the indeterminates in the set S, while inversely,

> " o
Fa

=

(-l)cardinal (s-R) r(n), where F(R) is the result of sub-

stituting, in F, O for all X not in R. It follows that, instead of the generic u = 1 + U considered above, we may consider the 1 + U*, where U* con-

sists of alt those terms from U that contain all of X1, . . . ,

\.

The lead-

ing term of such a U* is

st = xt "[:o)l

*o,

. . . x'.,, t ZP,

where the summation is over all permutations

Proposition

1. For all t, P ( t

< 2p,

o of the indices

1,2, . . . ,t.

E satisfies the identical relation

st(Ul, .' ., Ut) =0, all U1, . . ., U,inE. This was established for U1 = X1, . .

'relatively free' on Xl, . .

.,

Ut = X,, and E is evidently

.,Xt, . . . . More explicitly,

choosing u*

=

1+ U* modulo ot + 1 and u, with leading term Ui, the substitution of u, for x, in u* evidently yields an expression with leading term

st(ul,...,ut). Let ,1," denote left multiplication by Y, and rt denote right multiplication; then the inner derivation dt: X + [X,y] = XY - YX is given Uy dy =

6

'a8ue.r s-rql

ul 'acuaqn 'd >

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1 .rog

'V Bur.r aqt ut

O=(ln""'In){'lS ,"I .r.l ln,...,Inpue,4+d ? lIIErod .arnlcoluoC aq1 slsaE3ns srql

1^

I

"ox'"'lxt111,;r

={'1g

uolllulJap le.raua8 arotu aql Sulryt{

':f 3ut-r aqt ut o=

'f

l'n'''ton tT

a

= (1o

"''

'Io)o'ls

ln ' ' ' ' 'Ifl 'd ? I ile "o.{

'Z uor.llsodord

-I tuo.r1 In "ql "oy saqsluel 'JIaslI T ol ueql raq}"r

'!x

sluaurala Jo uorlnlrlsqns raprm.{11ecr1uepr

y 8ur.r Surdola^ua aql o1 s8uolaq le.reua8 ul qclq^\ ,O'1g luatuala aq1 I *l > I > I + 'd ? 1II"ro''acuaq,tt'I"1I k'''''l*'''''IX)0'1S 0'I + 1g d1.rea1c,1' x' . . to" K = o'1, Burlr.r,tr,lce] uI =

T

o=

Jo

ds

^fir1uapr

'uorlrpuoc 1a5ug aq1 ...

aql puB'[1-a{'x] ;(1-d)rsan13 u, u,

dS^

=

\

=

=

sa11dru1

k

= zx

= 'X' '' ''* a = r"ql ra}ur 0 = [1_AI'X1 uolt-rpuoc la8ug eq1 uror; pu" 'dx ' ' ' ' 'Zx o1 lcadsa.r qlrrn ,{laarssacans ozrr"aull deru an'I-d' 'I'0 = l'r'.reln8uls-uou s1 (r1) apuoruJapue1 aqlaculs'I-d' 'I'0=l\ 'dXdY * "'*ZXZ\ =I'IX=XBuJllas Eu111as',{1as.ranuo3'0

r-d=r;0 g-1-6trX1l 3 r-r-dtr" r^ r (r1d)r(r-)

r-d

t I t o*

=

=

1-6(Ia - ^")* = r-olP.x = [r-dl'x] 'd atlsr.ralcereqa seq y pw 'e1nw*oc tr" pua LT aculg 'tr T - L,

the conjecture is equivalent to

>o(o)kuor. We pause momentarily

.

Uor=0, allp+k ( t < 2p.

for an historical digression. The method of

Griln, restricted to the present conte:rt, is better paraphrased in a slightly different fashion. Embed B = F/FP, relativelv free on generators x, in B'

relatively free on the x together with an additional 'generic' generator y.

If C is the normal subgroup of B' generated by all conjugates of y,

then

B = B'/C naturally, and Co = C/ (C,C) becomes a cyclic B-module. Writ-

ing cO = c(C,C), the coset in CO of c in C, one has in particular that

.o'X1 . .. Xt= (c,x1,.. .,

*t)0.

Grirn exploits this relation by substituting

for y elements from various

compound commutator subgroups of B. Under this representation of on the module CO, one has the additional relation Xp-1

of our relation St = St,t-p = 0 for small

-

E

0, and instead

t Z p, one has the relations

St,t-p*1 = 0. If B is infinitely generated, the restriction of this representation to

f, annihilates at most the centralizer

of.

f,

in tr.

S4. Relations among the identities of Griicr We conclude with some fragmentary results that exemplify such methods as we have found

Proposition

3.

For

allt Z p + 1, and U1, . . .,

st,1(U1,,.., in the ring

for investigating the conjecture stated

Ut)

above.

U, in L,

= !o(.)u al,..uo,=o

f.

We shall give two pro<-rfs of this proposition, which have different

generalizations. Our

first proof depends

on the following

10

II

't' * "' to"

t1,Jo

)s = (Ix'""u";{'1, leql alou

Sursea.reap Jo raqumu eql

'(, )" -I-t

=

(t)d

o,r\

, < Io

(I+l

sdals

8ur1r.ur ure8e acug

.@ tI4aIIrur{F € si >i'ls [aqt-]Tlleer.tuapl g = I-)i'I-it

II

.9 uor.llsodord 'SrrrwaI

aql Jo acuanbasuoc InJasn arour E u-relqo

aA\

srallal luacefpe Sursodsuerl

dlaarssaccns dq paurelqo aq uBc uorlelnru.rad ,{"ra,ra

.(1x,...,I+lx,*rx,

_ru* snoeua8ornoq

1eq1 1cB}

aql iuory

...,I, x,Ix,...,Ix)* {'ur+ls, e 3 _ (lx '" ''I+lx'iA'lx ( " "Ix)I+{(I+ls

'rlfeT5failffi'rua@ " X " c a = A\ loT .? uorJ.sodord E aq " X "' I

'uorlecrldde ou

prmoJ

o^€r{ eA\ qalq^\ roJ '11nsa.r 3urino11o1 aq1 saa13 Brnural s-rql Io uorl€rall

(n\-('\=rr-{\ \l - (")"/ \r - t')" \t"1"

/

)

olNIxr+lxH= oy.ro;pue

,( n \ -('\ =/'-n \ \r - t"1"7 \1.)"/ \r - (.)"/ uorlEI

-ar aro 01 sacnpor sluercrJlaoc

-

aq1 yo,{111enba

'yl+l*l*t

to =

t'* t ...

Jod 'uorlenbe aql Jo Joqwaur Jaqlla ur lualclJJaoc oJaz-uou 'y qlrAr rncco lxl+lx ro I+lxlx 1.red e qlr^{ suJel dluo leql lsrl} aloN

'(k'"

ru.re1 B

'

'!x'I+lx " "'Ix){'lS - (1x" " 'I+lx'lx '.. .,Ix){'ls =

(lX,..',Jl+l*rl*],'..,IX)I-{'I-lS '€uIuraT

From Proposition 5 it follows that

Ir"\j-t /t-t-"\llf sJr k /) l\x'-\

x^o1 ...x ot

=0.

- (t-i-") is a polynomial with leading term < k,weinferthat Io hx-c =0forall 2 ,k/xl. If S..=0forallh t,h h < k,and,from :f(")X o =0,that2 lnkx., =0. SupposingP > 2, kx we have also that : o o = 0, whence St,k = 0. This establishes If k is odd, f (o ) = (;)

6. Ifk is odd, St-1,k-1 = 0, St,0='..= St,k-l

Proposition

p > 2, !irg! St,k

= 0, and

= 0.

This suggests trying to establish by induction on k the condition Ck' St,k = 0 for

C, ,

allt I

p +k. Proposition 5 then shows that Cg, . .

.,

implies C,- provided that k is odd. Since Proposition 2 gives C^,

^-l-s

from Proposition 5 we infer Ct. Our second prooi of Proposition 3, that is, of C1, can be generalized, at the price of some complication, to establish C2, whence, by Proposition 5, C, follows; but the complications seem to prohibit an extension beyond

this point. Summing a set of substitution instances of St-1,0 = 0 gives

o=

Xo

n?

o

t,,o("l,..., *h,..., *k,...,

[xn,xuJ) = o.

coef o.t '., x ot - arises once in the expansion of A with ficient +1 for each part *or* or*, *t,n o, . oia1, an4 once with coefficient -1 for each part X oiX oi*l with c, > oi+1. Therefore the total coefficientof Xo inAis "-p= " -(t-1-")=2 a -(t-1),andA=

A term

2S,,1

=X

- (t-1)St,0.

Supposing again that

P > 2, it follows that S,,1 = 0'

To establish C, we begin with the identity

A=

r St-2,o(Xl,... , *i,..., *r,... ' *p, '..,

ik,.. . xt, [xixj], [xnxol) t2

=

o,

TI

ul€lqo aa 0 = O'Z-]q urord ',O- = O-

=

=,q+,J,0 =,CI* O,0 = q +,C suorlelaJ,(*) qllrn Z tq1 ur'urelqo a^r., yo1q31.rpu"lJaI (f ...1) * (l ...I)

,C = C acuaq^l'0

d.rlauurds

I€sralor aq18urd1ddy.0 =C+ C a^€qa^\puy',e _ CI _ ,C+ C = zv u"qJ'[zre] * [slz] =,c'[rsz] + [zsr] = c'[rzt] =,c,[tzr] = c tar 'o = [i8z] + [zsr] + [ezr]

(,r)

'snqJ

.[ozr] = txlx(lx,.'.,tT,...,I",.'.,Ix)o'z-t, aAEq a^\ g =

'[ozr] +[rez] +[zer] '1eu1ru.re1

'asr^\roqro 0 pue

'ox

t

t

i

=o

o'z-1, ruo.r;,3 + d !

*[szl] = ox,i

1 acurg

-I'1r=s

sr .ro a1dr.r1 e sur3aq raqlta .rred Bursea.rJul r{cea acurs

lo I l-ln

Jr

I = {lort}

"""u^

'x

{,tozt} .

=

[ozr] rer '[rrz] - [zre] - [rez] - [zrr] - [rze] + [rzr]

zv u"qr = ,lrB o ,E o

s saldr-r1 a^nnJosuoD Jo req.,,nu aql aq {'ntr} t"t'e'z'r;o uorlelnru.rad due 4'f'1 rod 'I+" = q qll/r\ ,s! leql ,1u1ots1p lou are leql q > qll^\ (t+q,q) " '(1+e'e) s.rled uro.r; uorlnqrrluoc aql sI ZV aJeq/t\ ZV * V = Iy .1ae; u1 {./rfr} :

= [:rtr]

lar

'>{'['! ot crqd.rowosr .rep.ro Z+€

.g

o*rr

= z,lg ecuaq

3 aa13pFo.{\0=Iyacuaq.Tr'D ulcllerpenb'3 < pu€0= roJ'srt4crr4t^'Z/.fi - t- rd=Z/Z(d - ")--Z/ed + tl"Zsureluoc luarcrJJaoc aql Jo r.ural crle.rpunb oel

:'x { t!u'I + do - t9l } I I

=-

Iy

a.,\r8 plno^r

, o _I_l = €/

-

I+Qo

)u8s .1Bo

Jo rrrns aql eroJaror{l

r"y

acws

stql'uorlrpuor ssaurrurl

-slp eql ruo.r;1redy'lcurlslp ar" I+q'q ,f+e,e ler{l qcns ,q > 1Qr,

d

ra^o " IIe

- I+" o )u3s

sI v uI

'x

Jo luarcrJ;aoc

aqI'q > r pue'rl > q'[ > I lsql qans >1,q,[,1 lau]lsp IIB ralo pauruns

o=,.i.0t,-r,o (X1,''', ir''' " *i''' " lk'

.

', Xt. [x,x,xoJ) = [rzs] - [zrs] - [stz] + [szr]

=C-D'+C' = 3C. Supposing

p>3, we conclude that C = C'

= 0, D =

D' = 0' hence A, = 0'

SinceA=0,itlollowsthatAr=A+A2=0,whichcompletestheproof'

?. If t Z p +2, andP > 3, litelSt,z = 0' Corollary. Ut >p+3,andP > 3,!hg!St,g=0'

Proposition

UniversitY of Michigan

BIBLIOGRAPHY

W.Burnside,onanunsettledquestioniq.thetheoryofdiscontinuous "' (1e02) 230'

?ilp;;b"i.i.

o. Griin.

J' Math.

35

Zusammenhang zwischen Potenzbildung und Kommutatorbildung'

Crelle 182 (1940) 158. between periodic groups and Lie rings, A' I. Kostrikin, on the relation (1957) 289' 21 mat. Izvestia, ser'

A.I.Kostrikin,OntheBurnsideproblem,Izvestia,ser'mat'23(1959)3' M.Lazard.SurlesgroupesnilpotentsetlesanneauxdeLie,Ann.Ecole ---

\or-.'sup. (3) 71 (1954) 101. Magnus, Beziehungen zwischen Glgppen und Idealen W. "' -;;Ai;ilen Ring, ilIath. Ann. 111 (1935) 259'

in einem

Crelle W. Magnus, Uber Gruppen und zugeordnete Liesche Ringe'

182

(1940) 142.

P. S. Novikov, On periodic groups, Doklady Akad' Nauk

SSSR 12? (1959)

149.

I.N.Sanov,Theconnectionbetweenperiodicgroupswithprimeperiod ^' ^";"d"Lib i;;;;ti;' ser' mat' 16 (le52) 23' "ing", 1?7 (193?) 152' E. Witt, Treue Darstellung Liescher Ringe' Crelle endlichen p-Gruppe' einem LieH. Zassenhaus' Ein Verfahren,..jeder Univ' Rine mit Oer Crraiitleristiti p zuzuordnen' Abh' Math' Sem' (1940) 200. 13 Hadburg t4

9I IIB ruorJ lurolsrp sl V

-VgV luapuedapul

Jo

ltq] u^\oqs

d11sea

sl ll C dno.r8 e qans uI .sdnor8

arnlcnrls alalduroc aql paururralap s"q roqln" aql

Jo IEuJnof u"rp"u"C aql ur par?adde qcrqrrr Jad€d € rq ='scrlerneqlery v

'uralsraH qlrirr dpulot paul€lqo ora^\ srar{lo pue .{lluapuadapur paurelqo ara/h sllnsor asar{l

atuos 'uo 1.rod3r 01 qslt\

I

qclq/r\

{ro^\ slql

s-l

ll pu" 'sdnot8-ygv

Jo

ul lsa

-ralur rorprnJ Io I"ap poo8 e palelnw-rls s€q >lro^\ d.reurturla.rd s1q;,

'g

1o dno.r8qns IBurIxEur B

sMeql

qcns Jo arnlcnrls aql paururralap pue

uorldrunsse aql raplm C sdnorg

'p ;o sdno.r8qns are g

pue V araq^\

'g ul I t' q'v uI ,e'€ ',€q? tu.lo; aql Jo uoll"luasarda.r anbrrm € aAEq ro v ur raqlra ar" qctq/t\ Jo sluawala oql 'c sdnot8 'sr 1eq1-sdno.r5-ygy tuapumput Jo arnlenrls arrres aq] 1y'c11c.{c

ar" g

aq1 pa1e811salur

roqlne arn pu€ rarrruaZ arnll

sl dnorg VgV u" 1eq1 e.rnlcaf g pue V qllrrr ,gVgV inroJ aql Io

pu? V Jt alq"^Ios

-uoc ol uralsraH pal {ro^t s1q; 'c11ada

uo-rlzluasa.rdar € llrup" plp lnq .VgV ruroJ aql Jo lou sezn (d,g)TSd l€ql pa^\oqs osle .,t4sue1dey

g

puB

'rapro arurtd

Jo

sl

auo lseal 18 pue arladc a.re

V JI alqe^Ios s! dnor8 E qcns leql pa./(oqs uralsraH pue d4sueldey

'VgV rrrroJ

aq1 yo sdno"r8

;o a.rnlcn-r1s aq1 Sulu.raeuoc paultlqo ara^\ sllns

-a;1e11red awos acuaraJuoc aql 3u1.rnq'lg6I jo rerrrurns aq1 ug e3a11o3 urop/t\og 13 plaq sdnorc ;o d;oaq1 aql uo aJuaraluoc un8aq a.rern uollearlp slql

q

acroJ rtv u? 13

suorleSrlseaul owos '€I pue

y

sdno.r8qns ;o

sIrIJal ur uorl€zrJolce; palecrlduroc aJorrr B lrtupt qcrq,rn sdno.r$ uo auop uaaq p€q >lro.r

anlll ,{.ral ,{11uaaa.r ,{.ral l1lun 'gV

rrrro} aql 1o sdno.r8

Jo oJnlcn.rls aql lnoq€ u,roro{ sr lunorue olq€raplsuoc e q3noqlly

uIalsuaJoC 1a1utq ,{q

sdoouc sTsv^Tos Nrvrusc do sunJcouJ,s

UHJ, No

group' If M is its regits conjugates, and consequently G is a Frobenius h' then O(G) = hm and h 1 m - 1' ular subgroup and O(M) = m, while o(A) = et'l I hm' This on G imply in addition that (1 . Our special assumptions

M of a Frobenius group is additional condition on the regular subgroup abelian group of type (p'p' sufficient to imply that M is an elementary

nei-

(with two possible exceptions . . . . , p) and that A is maximal in G ABA-groups)' In parther ol which can arise in the case of independent is solvable' The argument is alticuLar, it lollows that G is solvable if A

require the fact that the regular most entirely arithmetical, but it does (but only in the case when subgroup M of a Frobenius group is nilpotent M is known to be solvable)' is the group G of oneThe prototype of an independent ABA-group field F' If a affine transformations xt = cx + d over a finite

dimensional

denotes the transformation

x'

= wX, where w

is a primitive element of F'

andbthetransformationx,=-x+l,thenGisanindependentABA-group B (b). In this case the reguwith respect to the subgroups e = (a) and = group over F' This examlar subgroup M of G is simply the translation plealsoServestoindicatethekindofstructuretheoremswhichholdfor cyclic' ABA-groups in the case that A and B are non-independent

Herstein and In a paper in the Canadian Journal of Mathematics'6 G in which A and B are cyclic the author have shown that an ABA-group

ofrelativelyprimeorderisnecessarilysolvable'Itisnotdifficultto group always contains a normal subdeduce from our results that such a

n T = 1' where T is either group T such that C = N(A)' T withN(A) group of odd order and the abelian or the direct product of an abelian of G are either quaternion group. In particular' the Sylow subgroups group' abelian or isomorphic to the quaternion and the theorem of Grnn.13 The main tool in the proof is the transfer transfer of G into A maps G onto A; When N(A) = A' it is shown that the 16

LI

-VgV

roJ uorl"zrrolc€J ^ue

"

Surrunsse 'swalqo.rd aseql Jo lsrlJ aql

learl

01

u" leq1 ,{1a>1r1 ^{.rea pawaas 1l aculs .I dno-r8qns lerrrrou Surpuodsa-r:oc aql Jo arnlcnJls aql aururJalap o1 ,puocas

papaou aq plnol\ luarnnS-re uorlcnpur

pue lasec etur.rd ^{1aa11e1ar aql roJ

l"ql ol sno5eleue C roJ waroaql uorl€z -rrolce; Jo pul{ auros qsrlqelse ol 'lsrr} :s1rrd o^tl olur sapl^.lp rualqo.rd aql l€ql seas ,{14crnb euo 'g puE V Jo srapro aql uo uolldrunsse ou qlld\ 1nq 'cr1a,lc g pu€ V qll,t\ 'C sdnorS-ygy Jo ornlcnrls aq1 Bur>1ce11e u1 'dno.r8 uoru.ralenb aq1 o1 crqdlouosr d

pue

'C t (dg)N 'Z = d asec

;o asodsrp

o1 pa.rrnbor

sr luaum3.re pa,rloa .C -ur raqlEr V 'uorlcnpur ,tq snolloy osle ;oo.rd aql = (dg)N JI .uollcnpur aq1

ruroJ aql yo y dno.r8qns .}Iasl, oluo dg sderu 6 olul c I"rurou e sr detu raJsu€rJ ar{l Jo laura{ aqJ ,C (dg)t{ pu€ ppo s.l d Jo raJsuerl or{l pue uorlcnpul dq uerlaqe sl d ,{q speaco.rd ;oo.rd aq1 .rou pu€

'g>Ig qllltl VISV

=

'g;o

dno.r8qns,noldg-d e st (dg)N yo

6 dno.r3qns moldg-d

:(dg)N Jo roluac

lv jo dv dno.r8qns

owos roJ oo"oo

JI

e

dg 'z

.r,

"q1 ""11 urroJ ar{l ;o dno.r8 e sl (dg)tt

lBql

dg dno-r8qns no1.{g-d e raplsuoc a^\ 'V €I U (V)N yo

<

't .I

^\oqs

pu€

(V)N ,raqm

'rapJo ppo Jo rusrqd"rowolne ue Furllrru -p€ put Z rapro;o dnotBqns anbrun e Surleq dno-rE-g uerlaq"-uou dluo

aq1

aql 'Z = dy1 apqn lcrladc sl d leql sagldrul slql ,ppo sl d II 'd ,rap.ro 1o dno"r8qns anbrun sBr.J 1l uaql ,aldurs d11ecr1sr.ra1ce

s1 dno.r8 uor"ralenb

"

-r"qc lou sl J Jo 6 dno.r8qns mo1,{g-d B JI ler{l s.roqs luarnn8"re lecrlaun{lr -rB uV 'paqsllqelsa dlrsra sl J Jo ,{cuplodlru aq1 'sluaruale paxl} I"rArrl -uou lnoqJr^l sl p y1 'uorlcnpur dq snolloy J Jo arnlcnrls aq1 puz ,;, ;o raluac aql ur adE asaql 'sluauala paxrJ IErArrl-uou seq g aql Jo snql sl

J

y1

.(fq)

Jo luaurala ,t.rena 'e dq uorle8nluoc dq pacnpul

,f

ru.roJ

J Io ruslqd

4 n 'r_ B[qrB urroJ aql Jo C Jo sluetuala oql ar€ I JosluotualaaqJ'Cu! I"wrouJpuEI = I U Vqll^\J.v=Cacuoqpu" -rouroln? aql solouap

the second of group of lower order than G, the author decided to study these Problems

first.

If we assume then that

an ABA-group G contains a normal subgroup

TsuchthatG=ATandAnT=l,andifweletdbetheautomorphism easy to show of T induced by conjugation by a generator of A' it is very thattheremustbeafixedelementginTandafixedintegerrsuchthat (g) " ' ,"(j-l)t(g)) every element of r is of the form 'trt (*) dt (g) d'Zt for some i

and

j. If r = 0, T reduces

to the form studied in the relatively

prime case.

anci

a d -group' We have called a group T admitting such a representation only the identity have called the integer r the index of T. If / leaves

element of T fixed, we say that T is a groups is easily reduced to the study

regular {-group' The study of 4'of regular l'-groups' The main re-

sultforregular{-groupsisthattheyareeitherabelianor2-stepnilpotent.Theproblemisverysimilartothatofgroupswhichadmitanauassumption is made tomorphism of prime order. However, in our case no dealing with a very speon the order of d; but on the other hand we are

cial class oi grouPs'

lf

vr is the least integer such that

g a r (s) ' ' ' 6(w-1)r (g)=1'

we

forms the clearly have O( d )w >' O(G). This inequality on the order of G

L:;isisofmanyofcurarguments.ourproolisdividedintotwoparts: lir.stshowingthataregular.l,-groupisnilpotent'andthenshowingthat

aregular{-groupofprimepowerorderiseitherabelianorofclass2' unknown to us' we were Slnce Thompson's resrllts on p-normalitylz were as well as the soto treat the general case of regular

Iiirced

'6-groups general case is no longer called non-exceptional case (the argument in the necessary) '

S.rrthenon-exceptionalcasethe''leastcriminal''isasusualagroup (p'p'''''p)' Q is abelian of the form T = PQ, where P is abelian of type 1B

6I

'JV aql sI

J

pue

'J *gV = f

'J

*g ,I = J U (V) *N aJar{.t\

qlr.{\ asr./r\luawalo salnruuroc

(V) *N = C leql qcns

;

Jo dno.r3qns rolelnrutuoc

dno.r8qns

l"tnrou allc,{c € suleluoe

C uaqJ '(V)rtt Jo punoq.raddn eq1 = *BV = (V) *tt las pue ((V) (V)rN uollcnpul ,{q aurJac 'c11c,tc are

g

pue

v

araq/n

'gy

,_rp)n

=

= C leT

g:paulBl

-qo

d11uaca"r

a^"q a.r qclq^\ uraroaql

aql ul papnlcur ar€ sllnsar

8ur.,rro11oJ

asaq;,

'S'ntr'V aq1 ur .reedde III^\ V = (V)N as"c aql w sllns ,'sFurpaeeord -ar aqJ 'c11c,tc g pue V qlln sdno-r3 gV Jo asec lelcads aq1 ur uollsanb Surpuodse.r.roc aql pa1e8r1saau1 ror{lne aql pue uralsroH ,sdno;8-ygy

roJ ruaJoaql uoll€zlrolcBJ

" Jo

luro; lc"xa aql JaAocslp o1 Bur,{r1

rq

g'IEurnof urrpsu€c aql Jo anssl ralBl E ur .readde III1rl sllnsar asaqJ 'u€rlaq€ sl

,paxr; Jo dno.r8qns .ro1dg aruos uaql

I

luatualo ,,!r1uep1 aq1 ,{1uo sa^€al , C ;1 pue

'1ua1od1ru

lou

;,

s1 r1ctql\

rapro alqrssod lsarrrol ]o r xapur yo dno.r5-p .re1n8a.r E sI J

'I uar{l '.r

/

ur leurrou

sl p

rapun luerreAul

,{q paxry asr.rluawala lJaI

'.rap.ro .ramod atur.rd Jo pue

r

J

J

II

/

sl d

.re1n8a.r

"

Jr pue

sl J JI .I :

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ue uodn paseq sl as?c l€rauo8 aq1 ur

;o sluernala paxlJ

slr{J pue

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sdno.rS-

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p

;o srs,{1eue

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pus ru

b ol palcrrlsar uaqlr f

aq1

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;, yo dauelodllu oql yo;oo.rd aq;

'lsrxa uec urroJ aloqt

ou

.Z

Jo dno.r8qns d.raaa

Jo dno.r8qns aql

xopur ;o dnor8-

yo

aq1

lp'.{1aweu-d }o ropro aql ur ru luauodxe aql

Jo

p rapro aql 'J uo

+

Jo r{

rapro aql uaa&laq

slsrxa uorl€Iar Surrrrolloy aq1 '(dnot8-p e d11;essaeau lou) ruroJ slr{l Jo J dno.r8 ,{ue Joprsuoo oir\ ;1

'I

ur lerrrJou

sI p

raprm ]uerre^ur 6 yo dno.r8qns

'luelreaur g yo dnot8qns .rado.rd ou lnq b sal"al p ,.re1uac . . . ,b,b) adfi yo IBIAIT1 E seq J, pu" 'tr ur lerurou sI d 'ub rapro pu" (b'

.rado.rd ou prre

Certainly the results on AB-groups and on ABA-groups with A, B cyclic of relatively prime order suggest the factorization theorem which

for arbitrary ABA-groups in which A,B are cyclic. However, there exist ABA-groups (all sotvable) which do not one would hope to prove holds

admit such a factorization. This class of "exceptional" ABA-groups appears to be very small, but its existence complicates the problem. At the present time a large number of partial results have been obtained,

but many details remain. As in the relatively prime case, the transfer is an essential tool. Theorem B of Higman-gattl0 also plays an important role in these results. Although the theorems on ABA-groups

rely in general

on

their spe-

cial properties, many of the results needed are of a general nature. It is hoped that the results and techniques involved in these problems

gest a method

for attacking a conjecture which Herstein has

group which contains an abelian

will

sug-

made that a

or even a cyclic subgroup which is its

own normalizer cannot be simPle.

BibliograPhY

J. On Finite Group-s-wit\?-q4gpqlaunt Generators I, F.oE. Nit. ecad. Sci. USA 3? 604-610 (1951). On Finite Groups with 2 Inde.pendent Generators IV, Proc' 2. NaL-Tcad. Sci. USA 37 808-813 (1951). 3. Feit. W. On the Structure of Frobenius Groups, Canadian Journal of Math. VoI. 9, 58?-596 (195?). 4. Gorenstein, D. A Class of Frobenius Groups, canadian Journal of Math. Vol.' 11. 39-4? (1959). Finite Groups which Admit an Automorphism with Few Or5. in Canadian Journal of Math. appear Eits;lo 1.

Douglas,

6.

and I. N. Herstein. A Class of Solvable Groups, to appear in eanTd-ian Journal of Math.

7.

andl.N.Herstein.ontheStructureofCertainFactorizable

Croups I, to appear in Proc. of Amer. Math' Soc'

20

lz

'(g96l)

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.r,

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.0,

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lrj '0t 'IoA

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tr

Jo

lrr${1r^\ sruslqd.rouolnv

l"urnof uopuor ,sluarualg

ltupv qalqlr s3u1g

aIqEzrrolc"J ur€lraJ Jo arnlcnrls aql

uo

o*r"

puu sdno"rS

,Ji^lu"ll,l"tt .5 ,ueru3r11 '6

'"readde o1 '11 sdno.ro

'uralsraH 'N 'I pu€

'8

ON GROUPS WHICH CONTAIN FROBENIUS GROUPS AS SUBGROUPS

_t

by Walter Feit'

Let G be a finite group and suppose that G contains a proper subgroup M with the following properties.

(i)

No element in M distinct from the identitv element commutes

with any element not in M.

(ii)

For everv element x in G. either xMx-1 = M or xMx-1 0 M

={+

If

a subgroup M of G which satisfies conditions (i) and

(ii) is normal in

G, thenby definition G is a Frobenius group' and M is called the Frobenius kernel of G or the regular subgroup of G. Let N(M) denote the

normalizer of G. If N(M) = M, then a classical result of Frobenius states that G is a Frobenius group and the Frobenius kernel K of G has the property that G/K is isomorphic to M. In order to avoid the case that

G

is a Frobenius group, we will also assume that

(iii)

rvr*l_

NIMLl_s.

For groups G containing a subgroup M that satisfies conditions (i), (ii), (iii), we will state a result which under certain conditions yields a relationship between the irreducible characters of N(M) and the irreducible characters of G. This result has several applications and the remainder of the paper is devoted to a discussion of these applications.

Let M be a subgroup that satisfies conditions (i), (ii), (iii). The following notation will be used. The order of M, N(M) wiII be denoted by rn, qm respectively. The assumptions imply that 22

q > 1 and q divides

(m-1)'

8U

slr{l 5uo1e:1;orn reqi..r:tg 'sdno;F el.lrxls }o spL:l)i snoIrBA ;o uorle5ilsalur a.r{l roJ lool pStanod B sr I irraJoar{,1 }s asea Telaaels ler:.; osie seq

au.1l

uo^a lt\oq pe.roqs osl€ oqtl ranerg

'g

u

plre

I

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ar{1

rapun

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I

{ITJi -

W

ol X

Frlv

Jo

uollclrlsar aql uaql'v/ttx(' .''tX

sl

ruorl lculls.rp c

>-r> o,l*x

@xi'b/\

r

'trl EI x .(G roi

clz+(x))n =1x;Ix lertlqrns'c.r@ a pue C jo-I .TIm}oJ -

'b/tl (fx

lob/x

x,

!' - Ix [')'

,Ix zbv

'

,

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>

1 ', - *'1 lz l€qiqens | + = : u51s e puuS ,1 < b71 u-eul = *l

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'@

o"r6J

+b

(rr)

(at)

rar{lrn} al'unssv .(11!) .(II)

p yomolEqns e aq W iaT

.I uraroaqJ '1.! ruorJ sAloIIoJ

d11sea 1eq1 .6.re11o.roc B pue

(.

llnsar pouolluau a^oqe aql alels

..'*It

^lou

uec

arlA

'rcurrsrp a"e *b/'l 1 r"ql os uorl'lou aql asooqc .0 < I qlr^r *1 t aq1 Euorue srol?Br"qr lcullsrp b/{ dncexa are arar{l pue 4 sapyalp b 1eq1 sa11dru1 luatuale1s

lsBI

slqyf I

-raqrrr\jJ '(nr)u ro rarc'rtsr{c

=

t: ;r dluo pue JI *[t ue

I.7

,g < r .ro;

=

*l)

arorrl

01 reql "ro1io""j "1 ^roqs lInclJJIp lou sl lI ',{1aal1cadsa.r \ '*l: dq palouap aq III/n I7 ,{q pacnp -q (n)N 'g ;o .ra1ce.r€qc aq;, '1" = (f )l ? lal pu€ ,.ra1ce-reqc Ier^rrl aql s.l o; e"aq^'11 ;o s.ralcereqc alqrcnparrr ar.{l aq {: ' ' ' '1 ,o) IIB 1 laT

been done by M. Suzuki and G. E. Wall.3

It follows flom conditions (i), (ii), (iii), that N(M) is a Frobenius group with Frobenius kernel M. Hence the fundamental result recently proved by J. Thompson4 implies that M is nilpotent. Starting with this

fact, it is possible tomake some estimates of the values of the charac-

ters of M, and hence of the characters of G induced by the characters of M. The details are rather technical and will be published elsewhere.S The annoying hypothesis (v) is needed to show that M has sufficiently many characters of degree one, There seems to be no reason why the conclusion of Theorem 1 should not Le valid without hypothesis (v).

If

this hypothesis could be removed, it would considerably strengthen some of the applications of Theorem

1.

The methods used to deduce the next two results from Theorem

1

are similar to those used by R. Brauer, M. Suzuki and G. E.WaIl6 to de-

rive analogous results from the special case of Theorem 1 in which M is assumed to be abelian. Let G be a permutation group which satisiies the condition, (*) G is a doubly transitive permutation group on m+1 letters in which no non-trivial permutation leaves three letters f

ixed.

It is an easy consequence of condition

('l') that the order of G must

be qm(m+l), where q divides (m-1).

Theorem

F)

2.

and assume

Let G be a permutation group which satisfies condition

further that G contain@

(m+1).? Then m = pe

for some prime p, md iuo'?n,l . 4q2,where

is the Svlow p-group of G. Furthermore if S'o

= 11j

sn

, there exists an exactlyE triplv transitive permutation group Go with [Co : C] < Z. o

The latter part of this Theorem is known, more generally 24

it

has

9Z

aq louuec dno.r8 aldrurs e yo dno.r8- g rrro1,{g aql

.I < pue rapro ppo Jo A

E aruos roJ

l€ql u^\ou{ ila,rr

("2 ,Z)tS @

@

s1

tI

1ffioT8

e

9 dno,rB aqg, (11)

'p

ur.

p yodno.r5--g

leru.rou s1

'rq-tfc st p

go

^\oiXS

aqJ

(II)

ano"r5-Z rrroTfS-T"raag (I)

.@

.ZS

@ razne.rluac aq1 'Zg

Jo

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Zfno.rll-g

n luatuata

D laT .r ruaroaqJ

'4

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pue

t

ol r€IIurIs poqloru

"

,{q

snoSoleue ue Surprao.rd ur l{nzns

I

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ruaroaqJ Jo asec lerceds z dllerluassa sI qclqrA aolaq g uraroar.{I

;o ;oo"rd aql ur asrr€ 1ou saop dlpar;;rp Erlxa srqJ .g dnor8 aql Jo sral -cEreqc utElraa yo sarl.rado.rd lecglarnqlrre ar{l ;o srs,{1eue pallqap raqler e olur oE

o1 d"ressacau

sI

lt I

rnaroar{J Jo asn Fur4eru soprsag

.t pu" Z

suraJoaqJ uuql aao"rd ol lIncrJJIp arour sr uorluaru aA\ llnsar 1xau aqJ

'I <

€ euros roJ (oz

'z)ts

o1 crqd-rowosr s1

c l'ql 't

uraroar{J,

;o suorldwnssE aql ruoJJ 1rloIIoJ plnod 1l '1 ua.roaql rtroJJ pa^oural eq ppoc (n) slsaqloddq tr . € awos roJ ("2 ,Z).IS ol arqd.rourosr aq lsntu 6.I C uaql 'ue11aqe aq ol paunss€ sl t uraroaql u.1 IAI Jl ]Bql u,r\ornl sl lI '1W

aqfst

t-

6S

=

: (ft)U] = b araq,r\ ,rbr, [,lrI : ru] pue C j6-fnolB._Z /,\olfS

I{

W dnorbqns e sur€luoc qcrq,n dno"r8

a1du1.- aq C laT .t

t

aroaqJ

'(t+ru) rapro Io dno"r3qns lewrou ou sureluoc C 1"q1 lcBJ aql pue (*) uorllpuoc ruo.ry d11aa.rrp aoIIoJ p1.ro^ op Jo acuels-rxa aq1 ,1 rua.roaq;, urorJ pa^ouar aq plnoc (a) s1saq1od,{q

g

.srrr,o11oJ

Z r.ueroaqJ Jo uolsnlcuoc

aql uaql 'ue11aqe s1 (1sgxa ol u1t\oqs aq uec qclr{A\) ur rapro ;o dno.r8qns .6 ,{q dpuapuadapur peao;d aql JI l"ql sn€quass€Z .H pu€ 1>lnzns

uaaq

cyclic. Hence Theorem 4 immediateiy yields

Corollary. Let

G be a nq4:qyclic simple group. Then G satisfies

the assumptions of Theorem 4

if

and only

if

G

is isomorphic to SL(2,24)

forsomea >1. Theorem

5. Let G be a non-cyclic simple group

of even order whose

Sylow 2-group is abelian. Assume that the centralizer of every element

of order two is nilpolg4!.,T-eq G is isomorphic to SL(2,24) for some

a >1..10 The assumption in Theorem 5 that the Sylow 2 -group of G

is abelian

is essential, since the simple group PSL(2, 7) of order 168 is not isomorphic to SL(2, 2a) for any a, yet the centralizer of every element of

order two in PSL(2, ?) is nilpotent. Another easy consequence of Theorem 4 is the following. Theorem

6. Let G be a simple

group of order 49', where g' is odd.

Assume that the intersection of the Svlow 2-group S, with any other Sylow 2-group of G has order one. Then G

is isomorphic to SL(2, 4),

the

simple group of order 60. There is some reason to hope that

it might

be possible to classify

all simple groups of order 49', with odd g', by these methods. However there are still some serious difficulties to be overcome before this can be accomplished.

Theorems 3,4,5,and 6 all depend on the fact that groups of even or-

der are in some ways easier to handle than groups of odd o"dur.11 We now state a result of a slightly different nature which does not use this fact. We

will consider groups which satisfy

the following condition.

(**) The groupG has order g = m1 . . . mk. k > 1. For each i = 1, . . ,, k, G contains a subgroup M, of order m, which satisfies conditions (i) and (ii)' 26

LZ

>i

sa^€sr luor'ara

-lsuerl

z

,{1dn1-4

d111uap1 aq1 r{1uo r{c_rq^\

s1

ur

uf;#'":iiJlit""::1j;

dno;3 uorlelntu.rad alrlrsu"rl ,{1dn1-4 dpaexe uy '8

pe rrr s sq c :t3l3H3:t"1ltit1";ti: il i;13%f*t* -qns ltrrrrou B u!"luoc pue (*) uorlrpuoc r{ys11zs qalql\ C sdno.r8 aq; ^;

aqr a curr s ur ror aas'

't

.L

aloulooJ ur pauolluau laded 1s.r1; ar{1 aas '9

,:'(eZ 'Z)1S sdno.rg aldrnrg aql Jo uorltzrralJ€req3 y.. ,pep11ua .rad -"d;urruocqrroJ ur palord 9q II!.r g 'g 't suraroaqJ tqtimlo'r.tu aql u1 "readd€ " r{Jlr{/r\ ,,sdno;g uorlelnurrad aarlrsuri.rj flqnoq yo -sgoo.r6 -JIr./b ss"IC ug,, .raded E ur poqsllqnd aq 111rrr Z pu€ I sue,roaqj ;{nzns .W .9t-8I4'dd (eSOf) Z ,.qlElt }o .f .IU.,,spleg allutd ralo sdno.rg al1lta[o.r6 r"Inpoulu,n leuorsuaulq'auo aql .g .C pu? ,I{nzns .141 ,.raneig .U .}C '8 Jo uoll"zlrelcEJ"qC V,, 'II?1K

"qlerd Io

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ppo rrv

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sdno.rg elrulJ uO,, I{nzns 'IAI eruelsur "oy roJ aas esda s1q1-u! yod.rd e .rog 'z -BN E s€A\ roqln" arn arq^r

"""0 "'fr"tJ3$"H"JfiiH"'f";;Hi?"t::gl

'I

seloulooJ

dpnlg pacue^py roJ alnlllsul aqJ puE

Alrsra^lun IIouroJ 'I eq 01 era^\

(l)

s1saq1od.{q

tnaroaqJ, uroJJ po^oruaJ

p paleurrutla aq ppo^r ppo sl 3 1eq1 asec

aq1

'ry ;o 1uaun5r" u? Jo uoll"crJlpow e Sursn dg .1 ua.roeqJ, tuorl zIl{nzns palotuar aq plnoc (a) slsaqloddq JI g uraroaqJ rrrorJ plno^\ slql

^\o[o] rapro uala yo sdno.rS tog '(**) uorlrpuoJ ,{Jslles qclqn sdno.r8 aldtuls dluo aq1 al? '1 < E r{11/r\ ("2 ,Z)tS sdno.rS aql lErI1 .{1a>g1 srnaas 11

" snluaqord E st

g

'dno-r5 snr.ueqo.rg e st

'Z = >1Jf-ffi6-!t3-J.1-ffio-"rF

C.ro6o"rFa1ffi1s e C

faf

s1

C.reqfta

.Z tua"roaqf,

9. See the

first paper mentioned in footnote

3.

10.

In the second paper mentioned in footnote 3, M. Suzuki proves that itie g"oup" Sltzi Za; with a > 1 are the only simple groups in which

11.

For a systematic treatment of gro]ps of even.order, s-eg-R'. Brauer ind K. A. Fowler, "On Groups of nven Order," Ann. of Math., 62 (1955), pp. 565-583.

t2. Cf. M. Suzuki, "The Non-existence of a Certqin-Typg of SiTf le-. Groups of Odti Order," Proc. of the A.M.A., 8 (195?)' pp' 686-695.

28

6Z

parralar sI rap?ar 3ql '996l

01 dn

salpqs Jo fa^rns luallecxa uB roJ 'pe-rllssBIc ,{1rsea 1ou sc1do1 lerceds

auros uo spro.n

lral e d1pu1y puy 'dnor8 snruaqord aql uo sluarndolaaap

fldrlpru ssncslp

Surlrcxa luacor aql pu" sdno.rF alrlrsu€rl

I

11"qs

osIV

'slcnpo,rd alqelnruled pue suolsualxa aarlrsu?.r1 uodn qcnol l1eqs I 'no.{ qlrrn ra8u11

sll leql os areJ l"crl"waql"ru luasa.rd ar{l Jo a1s31 luarc!}Jns B {ro^\ slql Jo asdlu113 e air13 ol pealsur adoq 1 'sdnor8 uorlelnur

Ilrrr\ ro^BIJ aa13 o1 pue

-rad uo salpqs II€

Jo

falrns

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I

sllal d1a.re.r o.r1 rapro Jo luauala dno.r8 e s-r uollnlo^ul uB lerll des o;, 'lsaralul .roleur Jo ualJo '^\om{ 01 r{sl^1' a^\ IIB sn

sI asaql Jo uorl€Iarralul aql poapq 'saull Jo uorlelntu.rad B sB pue slurod Jo

-Ioc

uorltlnrtrrad B

"

sE alqlssaJdxa i{1snoaue11ntu1s sr aueld € ur uorleaurl

puv 'sluawala slt Jo suo-rlelnurrad e.re dno.rE e yo srusyqdrorrroln€

aql snq;, 's1ca[qo sno-rre^ Jo suoll€lnw.rad ;o sdnor3 sr

d11r.rn1eu lsoru

aslrB sdnorE fuew 'a.roru.raq1.rng 'droaql dno.r8 1ae.r1sqe ul r.uJol aIqB -z1u8oca,r d11sea ou al€q 'd1rrrr1rsue.r1 a1dl1ptu .re1ncr1.rtd ur 'd1rrrr1rsue.r1

Io sallradord pl.roleulquroa aqJ 'raqlo aq1 suaqlSuarls uorleluasa.rda"r xlJlBru pu" uorl"lnurrad ;o salJoar{l aql }o qc"a pr.re suorl€luasa,rda.r x-r.r1

-"ru

Jo dlrrue; aql Jo raqrnaw lercads € s€

uorlelnu.rad

y

-osl aralsn€

'rrrroJ r"Incrlred

uy dno.r8

"

u! dnorS

ac"Id sll sa{q uorleluase"rda.r

"

Jo uorl"luasa.rd e sr

lI

'uollel

lc€rlsq" u" u€ql arotu sl dno.r8 uo11e1nu.rad y uollcnporlul

'rf 'II"H leqs.req

dq

sdnouc NorJ,v.Lnwusd No ssrqnls JNsuunc

'I

to Wielandt's notes [3] at Tubingen.

2.

Permutable Products. Traasitive Extensions of Permutation Groups' A and We say that a group G is the permutable product of subgroups

BifG=ABandAnB=l.CIearIyAB=BA.IfeitherAorBisnormal inG,thenGisthefamiliarsemi-directproductofthesegroups.Butin generalneithersubgroupneedbenormal.IfwerepresentGonthecosets may of B, then A is given its right regular representation and the Ietters of G, be regarded as the elements of A. If B contains no normal subgroup as a permuthen the representation of G is faithful and B is represented

tationontheelementsofAdifferentfromtheidentity.ConverselyifG isapermutationgroupwitharegularsubgroupA,thenBbeingthesubgroup fixing a letter, it follows that G is the permutable product of A and B.Wenotethatapermutationgroupoiprimedegreeisnecessarilyof this tyPe. In the study of groups with regular subgroups wielandt [z] iras intro-

ducedwhathecallsanS-ring(SafterSchur).LetAbetheregularsubgroupofG. Forafixeda eAletab,=b5(i)ux(i),i - 1, ' Nasbt ranges over the elements of B. since 1a = a1 it follorvs that a is in the = Sta) is determined by any element in it; set of aOlr;. The set

{tOtrl}

namely, S(a) = S(aO1r)), if ut

(i) '

S(a). In fact, S(a) consists of the letters

ofatransitiveconstituentofB,regardedasapermutationgrouponthe

elementsofA.InthegroupringA*ofAovertheintegers,letusput s(a)

= t, e g

g. Then the s(a) form a basis for a subring of A* and this is

S(a)

tne S-fing. U

e is Abelian,

(s (a) )P =

:

then

ae S(a)

it is easily

seen that

for

any prime p

mod P' s 1a,P) r

Using the S-ring, Wielandt has shown that

if G is primitive

and

if A is

AbelianofcompositeorderandhasatleastonecyclicSylowsubgroup'

thenGmustbedoublytransitive.IfAisofprimeorder,theneithera 30

I8 'srallal o1(1 saxrJ ,(1r1uepr aq1 dluo qclq.r ur g dno.r8 alrlrsuerl dlqnop e sl H ]o uorsualxa enrllsu"Jl E uaql,dnor8 alllrsueJJ.reln8a.r € sl H II

'II ul J

Jo

Jr{J slasoc alqnop Jo sa^tl

-e1uasa.rda.r Suru.ro; s,q Jo las e ro;.{1uo palJ.lJa^ aq poau

(g) uorllpuoc puE

'(t)

II

'H jo uorsualxa alrlrsu€rl E sr

'(g) suorlrpuoc

II

.H

Jo sluawala

{"." r_J }

,ploq = C uaql

(r)

;o s8urdderu Jo surral ur palel

-nr.uroJ aq uea (p) pue '(g) ,(Z) suorllpuoC 'Zqg .raq1o due ,{q pacelda.r aq .{ew

(l) uaqt ,sploq

1"ro, aql

u.l H Jo luatuala

lnq paurrrrralap dlanbrun tou st Zq (l) ut

'H'Zq'q qtt,n Zqxlq = x{x,J I q,H ) q rod (t) 'usrqd.rouolne Jauur rrB sr arenbs asoql\

g

uo rusrqd.rourolne uB socnpur x pu€ .{

I_

(r)

ft)

'''' erall 'd r

-xJ

lua^a due u1 .z Japro

(r,o)

=*, {",*;'=t;

aq ol

x a{q uaae ,{eru

(r)

ain sdno.rS J0 zx elrur; roJ pu€ 'I pue 6 Sul8ueqeralur x lueurala rre u1€luoc 11llr a^ll.lsu"rl ,tlqnop 8ulaq g ueq,L .I rellal Bu1x1y 11 yo dno"rSqns aql aq d la"I

"

:de.tr SurtrolioJ eql ur ualqord uorsuelxe .srallal aql alElnr.uro; ,telu aA\ Surur?ura.r aql uo alrlltsu"rl puE rollal 3

Surxr;

11

dno.rEqns

alrlrsu€Jl

alrlrsu€rl

Jo uorsualxa

"

^{1qnop .d.ra,ra 'es.rnoa

;o ,pw

"

sE

papr"Sar eq deru p dno.r5

,aar1rsue.r1 dlqnop sr

g uaql

,g uo

a^lllsrrerl sl H II 'g dnolSqns € Io slasoc aql uo paluasa.rda.r dno.r8 due dew

II"c (l

p ueql 'H uo pesodrur sr uorlrpuoc

ou JI

.H

Jo uorsualxa

alrlrsu"rl

a^t uaql 's.ra11a1 Sururerua"r aql uo H

"

aq C

sl 0 Bugxr; dno.r5qns eql q?1q.4\ ul = ,s las aql uo dno.r3 uorlelnrn.rad E sl c JI pue s 01 0 raual raqlrnJ

10'si r ulofpe aA\ II

'S srallol Jo las € uo dno.r8 uorlelnru.rad € aq H laT

'g

sdno.rS

alrlrsu"rl dlqnop aq1;o

aEpallrourl

JaqlrnJ osl? pu" g luarualduoa alllrsrr"rlul u€ Jo arnlBu aql uo uorletu

-rolur arour ol sureurar

aABq 01

r1cnl^I

alq€rrsap aq pFo^\ 1I .slFsor asaql pualxa

01 auop aq

'allllsue.r1 dlqnop st C ro Ierurou sr dno.rEqns_d ,troldg

Finite groups of this type correspond to nearfields, and Zassenhaus [1] has given a complete determination of these groups. Zassenhaus [Z] iras also considered the problem of finding transitive extensions

of

ertain

collineation groups. He considers collineation groups H of projective or affine spaces over finite fields requiring that if the space is of dimension n that H should be transitive on sets of n independent points. He finds that

only one.such group possesses a transitive extension. This is the

little

projective (i.e. unimodular) collineation group of the plane with 21 points, and its transitive extensions lead to the Mathieu groups of degrees 23 and 24 which

will be discussed later.

Holyoke

If]

nas given the conditions on transitive extensions in detail

and has shown in particular that

if a simply transitive group H has at most

k extensions to doubly transitive groups which are permutation isomorphic then it has an extension with further letters to a group at most (k-2)-ply

transitive, naturally excluding alternating and symmetric groups.

3. Multipiy Transitive

GrouPs

The problem of the existence of groups which are more than

triply

transitive remains highly mysterious. Apart from alternating and symmetric groups, the only known quadruply transitive groups are the Mathieu groups on 11, 12,23, and 24 letters, those on 12 and 24 letters being quintuply transitive. There is an infinite family of triply transitive groups' the

family of semi-linear fractional groups over a finite field F:

**axl+b,ad_bc*o cxo+d

where

x +- xs are automorphisms of F. Certain subgroups of these

groups

are also triPly transitive. In discussing multiply transitive groups, we shall exclude the alternating and symmetric groups as

trivial for

these purposes.

It was loown

to Jordan that a group of degree n could not be mole than (n/s + 1)-ply 32

88

'alr11sue.r1 ,{lr.ressaceu 1ou q3noq1 'H Jo (d)S dno,r8qns-d no1.{g e ,a:oru.raq1

-rnd 'saop tn aculs H Jo rapro aql sapl^Ip d uaql'ru = l-u 3u1p1,rrp aur.rd e sl d JI 'sJallal 1-u Surureruor or{1 uo alrllsuerl pu" srallel 1 Surxr; g dno-ri -qns € s"q C uaql 'aar1rsue.r1.{1d-(1+1) sl C JI 'sllnser asaql Jo uorssncsrp "ro; aler.rdo.rdde a.re sdno"r8 anllrsuerl ,{1d-1 uo s{rBurar le"reua8 aruog

'a^rJ.suEE-TiA-l wq1 a"roru aq louuec u aaTEap yo p

'ZI< r'27'(1-.r;"r t rl'Zd >I'r+d{-uyy 'sas€a .,{ueru ,{lalrulJul

q

.ra8uo.rls lnq s,rallrtr

moi5T

ueql

'(-ra4.re6.g) ruaroeqJ

l 01 relrurrs sr Ja{r€d lsauJg

luocal V .suorlcrrlsa.r,ra11aq sear8 rrreroaql s,JaIIrIt qclql\ JoJ u Jo sanl€A fueru dlelrurJur ar€ aJarn acurs

ol anp'peqsllqndrm

's,-re111141

opnlcur

1ad se'rna.roaql

'.raaa.4r,oq '1ou seop

llnsar s,lpueIariA .Zt00g.

= g Sof

f

+

tt"tf = c{ltin c/u3ol; ldllecrloldrudsepuruEol t < l saalF stqJ 'tll * I < u ler$ seao.rd lpuelalia dno.r8 e,rrlrsu"rl ploJ-l e ;o ae.r5ap aq1 s-I u JI uoqJ, '[g/]t] = tu allrrl{ sn lal I .ra3alur ue roJ '[t] tRuetatm ot anp sr u1r\ou{ sl qclr{t\ d1r,rr1rsue.r1 uo uollclJlsar urJoJrun 1sa8uo,r1s aq;,

'Z

- yy't <

1 roJ

a^Illsu?rl ploJ-l oq louuEe u ae.r8ap ;o dno.r8 e - x > d > z/x lE^ralur aq1 u] d ernlrdB slslxa

zI < u ro] 13q1 s/r\oIIoJ ll'u a,raq1 '2, < x raqunu 1ea.r due roJ l€r{l 'a'1 lalelnlsod

s,pue.r1,rag Fursn

'dnorJ crJJauurAs Jo;url"uralle aql sI 1l ssalun a^rlrsuerJ pIoJ-(I + r) sE tlanul sE aq louu"c u a5.r5-aa yoffioTllE 'Z = t'T = 1.roy ldarxg '>l < r '>I

< d putarur.rd

€ st d oraq,t .r +

'lrrurl rallaq qcnrrr

d>1

=u

B sear8 9161 ur

laT'(.rantni'y'p) [1]

ra11r1n1

'V'C

rna"roaql

Jo uraroaql V

'dno.r8 cl.rlarnruds .ro 8u-r1eu"ra1le eql sl C acuaq^\

pur; d.ressacau Jr ssoaord

p

dno.r8 aq1

srellal t/u6 u"ql ssal Surceldsrp luawala uB r{111r\ 1.re1s ,fern a^\ I + g/u < I JI 'srallal ur alcdc oerql

Z+\Z "

-ar

uaql

E

sZ ueql arour ou Surceldslp prmoJ 's.ra11a1

pue dsea

aq1 3u11eade.r pue

aq,{eu;yr_1,I_V roltlnturuoa

s ,{11cexa saceldslp V luatuala u€ Jr leql Surrrroqs uo sarl

sr;oo;d eq;, 'ZI = u roJ

paur€118 Suraq 1rwr1 srql'ear1rsue.r1

displaces all the m letters which H displaces since a subgroup of H

fix-

ing a letter has index in H a multiple of m and so its order is not divis-

ible by the highest power of p dividing the order of H. The elements of

t letters fixed by H into themselves form a subgroup K which is intransitive and has S, the symmetric group on t letters as the constituent on the t letters fixed by H. Here H < K and [X:ff] = tt' G which take the

Theorem. Let G be a t-ply transitive group on n letters. Let H be a subgroup fixing t letteqs a4! !q! Q be a Sylow q-subgroup of H, where Q fixes

w > t letters. Thenlllg,nq4qaliZel-in

on tJre w

G oi Q

is t-plv transitive

letters fixed by Q.

Proof: Let

a1,

, &t and

b1, . , bt be two ordered

sets of

t let-

ters both sets being from the w letters fixed by Q. Then since G is t-ply transitive, there is an element x of G taking a, into br, for i = 1,..', t. -1 -1 bt and Q and x ^Qx are Sylow subgroups of Then x-'& fixes br, ,

the group fixing b1, . . . , bt and so conjugate in this group. Hence for some y fixing

br,

,bt

-1 -1 we have y-^(x-^Qx)y = Q. But then z = xy is

an element in normalizer of Q taking

?1,

, a, into b1, . . . , b, in this

order. Z must, of course, take the w letters fixed by Q into themselves. From this it follows that the normalizer of Q is t-ply transitive on the w

letters which it fixes, proving our theorem. Let F be an intransitive group on n letters and let n = r + s where one set of transitive constituents consists of r letters, the other of s letters. Then F may be regarded as the subdirect product of a group A on r letters and a group B on the remaining s letters. If At is the subgroup of A which occurs with the identity of B, and Bt the subgroup of B which

it is not difficult to show* that At 4 A, Br 4 B and that A/ Al = A/Bt. In particular if A and B have only the iden-

occurs with the identity of A, then

*Marshall HaIl, Jr. [2] p.

6S.

34

9t

-t t/ft->l)

{y

,re qcttg 'saceldslp

l! srallel 4 aq1 uo dno.rE ly yo dno.rEqns ? 'e'l '{y auo fllcexa Eu11eu.re1p aW prre srallal r1-1 3u1x1y suleluoc g dno.r8qns arn g,/(I + lf) < { paxtJ E roJ l"r{l reqlrnJ asoddns a6 'g dnorEqns ? Io slasoc ,ro ly ;o uorleluasa.rdor ar{l sl g uaql ily oi clqd (f

-q)

-roruosl sl u pu" I"rurou

"

eABq

su1?luoc

>r

I

= N ler{l 1s,r1; asoddnS

lsntu

g '1ca.rrp lou sl €J pue lv

'lg/tll=

a1dw1s sI 111aculg

'lV

= N/S leql qcns N dno.rtqns yo lcnpo.rd laarrpqns aql pu"

trlN < II t"qt serrnbar slr{l t"r{l Eullroqs u1 slslsuoc;oo.rd s,lpuelel1t'srallal W eql Jo erour.ro;1eq Fur

-xlI g

Jo sluawale

1.rer{cog aq1 ,{ys11es

Iu qtr^\

tll

=

qll/{ A\ ur pa.rled 1ou ly;o alcdc-g e 1eq1 'uo-rlrpuoc "1 ol repro ur 'asoddns r(srn an pue ,€I pu" lV ;o lcnpo.rd

laarrp aql lou sl slr{l

6 yo flrpurulru

eql dg 'g pue

ly

;o lanpo"d

1ce.r1p

srellel I + ru aql uo pu" srallal ru des uo g luanlrlsuoc a^Il -Isrr€Jl rer{lrn} s"q /r\ usqJ, 'I"rululru 16 esoddns deu ait\ srallal 1 lsrll

-qns aql sl

aql uo

-rell"

\

71l

"

a.re qatq/r\ C yo sdno.r8qns

aq1 aq

-a.r r{.reae Io

I

II" JO 'dno.rE cr.r1auu,{s

.ro Eurleu

lsnul C lreqcog Io uaroaql aql fq uoql luenlllsuoc Eururern

srallal aql

lsr-rJ aql uo

ly

Jo arortr

q"1q^

"1 a^-rlrsu"rl dtd-t aq C

ro l1eq

11 dno.r8qns

ieT 'lpuelalt\

"

sa*1y

ly

uo aladc-g B II

'srallsl

'srallal u uo , ? I aql 01 urnl sn laT

seq C uaqJ

Jo uroJoer{l

'srallaI ne; e dluo 3u1ce1dslp sluauala ureluoc ,{eru sdno.rS aal11rn1.rdrul 'aslnoc

Io 'puv 'a1cdc o^11 B pue elcdc ae.rq1 e dlal-rlcadsar u-reluoc qclqm sdnolE cr.rlewwfs pue Eulleuralp aql ol lserluoc 3u14rr1s ur sdno.r8 alrllsuBrl .{ldn.rpenb yo $.rado.rd E sllqlqxa lror1co€J Jo waroaql aql leql alou

ar16

'sratlal I - f?- tsBaIJr sa5"dfl p lueluale AJeAa Alrluept arn roJ lclacxa 'clnoJb errlauur^s Jo SurleuJall€ or{}

u

'(l.ra{cog) ruaroar{f

'[t]

traVcoA Io uaroaql arfl Io asn a{Bru osIB lpuelalA,\ put ra{J€d

1ou-

'1cnpo.rd

-qns rlaql pue E =

lcarlp rraql paapur

Ig 'V = IV u"ql aEerur clqd.roruoruoq

s1

lcnpotd lcorrp

uouuroc e se

,{1r1

cycles. From this it fotlows that a given 3-cycle is contained in

MFF+itB

conjugates of

I{. This is the number of letters fixed in

the

representation of the 3-cycle in the representation B of A1, being the representation of A, on cosets of H. For Least M/2 contrary to our choice of the

k

> (4t +1)/5 this number is at

transitive constituent B of W to

satisfy the Bochert condition. with a number of intermediate results, Wielandt shows that essentially this situation must hold for any constituent B on

M < N(t) Ietters

even when B

is homomorphic to A, and not

necessarily isomorphic.

Miller's result is more easily obtained. With n = k p + r, r > k, p > k, r > 3 a group G of degree n which is (r + l)-fold transitive contains a subgroup H fixing r letters and transitive on the remaining kp

letters. A Sylow p-subgroup P of H must displace all kp letters and as k < p there are less than p2 lutt""" and so P is the subdirect product of k cycles of length p. U, the normalizer of P is the symmetric group

first r letters. U has a subgroup V of index 2 which is A, on the first r-letters. Because P is simply a subdirect product of k < r pcycles it is not difficult to show that B, the part of V on the kp letters displaced by V contains no factor isomorphic to A, for r > 5. Hence V S" on the

is the direct product of A, and B. But as A, contains a 3-cycle, G contains a 3-cycle and so must be the alternating or symmetric group. For

r

= 3 and 4

similar considerations give the same result.

Parker's result is an elaboration of the Milter technique but depends ultimately on the Bochert theorem. Suppose G is of degree n = k p + r'

k < r(r-1)/2, r 212, and that G is (r + l)-ply transitive. Then G contains a subgroup H fixing r letters and its Sylow p-subgroup P has

k.

p2,

constituents of Iength p or pz displacing kp letters in all. Since a p-group of degree

p',

can be moderately complicated, the nature of P in this case

is far more complicated than in the case considered by Miller. G contains 36

,,8

'(4 s1 raqrunu aql tZy{ dno.r8 nagqle4 oql roJ) srailal

ro I

(g (p dlleexe sexl; 'g H;o dno.r8-6 noldg

'Jopro uaAa Jo f,11.ressecau sl srellal .rno;

e'Zd leql s^\oIIoJ Surxr; g dno.rSqns e g

II ua^a lI sdno.r8

ly olrlrsuerl fldn.rpenb (a]lqI) reqlo il" rog 'TI^ dno.r8 nalqleyq aq1 "o rlg ;o auo flr.ressacau s! rapro ppo alIuIJ Jo sl srallal .rno; 8ur '9V '9S -xgy dnorEqns 1eq1 8u1,noqs

"

qcrq/r\ u1 'e11u-tyu-t ,{1q1ssod 'dno.r8

llnser

"

e

[1] roqln" aq;, 'II^ dno.r8 na1q1ery dluo uea d1g1uap1 aql sl sralla1 .rno; Surxr; dno.r8

s-Iql papuaga seq

aql ro 9V '9S '?S -qns

e^rlrsu"rl dldn.rpenb

"q

qclq^t ul dno.r8 aa111sue.4 dldn.rpenb

a11u1y

B lern pa^\oqs

[6] uzp.rog

'eIqeIIB^" sI uolleruro;ul Jo lunoure pallwll e dpo '.rey os '1nq eq lsnru sdno.rt al1llstrerl dldn.rpenb uo suorlcrrlsar

(8uo.r1s

alllelllenb 'uorlBJal

-1e lnoqll^\ elqeclldde s1 droaql slr{ Jo

sI r{c1rltr

'6

Eurzrlerurou aJoJaq se

n

lsar ar{J 'srallal r lsrlJ

aq1 uo

dno.r8qns B u-rtluoc lsnur C

ry

&Irrlll

-srrBrl aldn.rpunb ;o sgsaqlodfq arn Japrm asrd\raqlo rod 'paleloll aq lsnrrr

?,/G- r) r > {'rU t nsuolllpuocarnr+d:I=uquaqlsrellalrsaxrJ pue H Io dno.r8qns-d ,troldg e sI dldn.rpenb aql Io

srallal

.rno;

d II pue u aa.rSep Io C dno"r8 anrlrsue"rl 3u1x1; g dno"r8qns Jo rapro aq1 saplllp d

Jl arag 'a^Illsuerl .{ldn.rpenb sl t{c!qr\

tZ <

"

u t{1l,vr (uS .ro uy 1ou; u aar6

-ap tso/trol ;o dno.r8 e .ro; Surqc.r€as ul uollcnpu-I ;o sasod.rnd roJ aq deru slql pu€

ppon

'VZ'tZ'Zl'lI f r teqt dluo ueau dern s1q1 1eq1 dldrug

a8pe1r\ou{ luasard .rng

,{ldn,rpenb B

l"rll

par.unssB

qcns roqunu €

'rg

sl srallal "o "y

sl r IBI{I asoddng

-penb o1 elqeclldde s1 d.roaql sIrI Jo lsow

.r uo dno.r8

allllsuerl

'sdno"r8 al11lsur"q [1dn.r

l"ql pa^rasqo sBrI ra{r€d

'rxra.roeql lraqcog aq1 o1 drz.rluoc

srellal

r lsrlJ eql uaql re{r"d '"v ol

u eql JI"q uer{l ssel 3u1ce1ds1p slueuele q€1uot }snw sJal}el

s1 dno.rSqns s,t,or{s = s 'sV sI qclq/$ A ;o " l"ql "o t5] crqd.roruosr dno.rEqns B u1€luoc lolru"c d Jo luanlnsuoc E Io srusrqd.roru -olne eq;, 'srellel r lsrll aql uo "y q"1q^ 6 Surzrleu.rou n dno.r8qns e "1

J. Tits If ] iras studied in considerable generality groups which are transitive on "independent" sets of k elements. His motivation is geometric and the properties of independence he postulates are those satisfied by k points in a k-dimensional space not lying in a proper subspace.

4.

Frobenius Groups

In 1901 Frobenius [1] proved that a transitive permutation group

G

of degree n, in which only the identity fixes as many as two letters, has a normal subgroup consisting of the identity and the

n-

1 elements

dis-

placing all letters. His original proof has been simplified, but aII known proofs depend on the theory of characters of matrix representations. The normal subgroup N may be regarded as given by its regular repre-

sentation, and the subgroup H fixing the identity of N as a group of automorphisms of N. The group H must be of a very special kind. A subgroup

of H of order

, p'or

pq is necessarily cyclic. Thus the Sylow subgroup is

cyclic or generalized quaternion. Zassenhaus It] tras determined all such groups explicitly.

is doubly transitive, then N is an Abelian group, and G may

When G

be regarded as the group of mappings

x +- xm

+ b, m

#

O

in a nearfield

K, N being the additive group of K and H the multiplicative group. The

distributive law (x + y) m = xm + ym holds but not the other. As we have already remarked, all finite nearfields have been determined by Zassenhaus. The author [2, p. fAZ] has shown that an infinite doubly transitive

Frobenius group is also the group of linear mappings in a nearfield provided that we also assume a Euclidean property; namely, that there is

at most one permutation displacing all letters and taking a given letter i into a given letter j. The structure of the group N in a Frobenius group has only recently been settled. able then N

It was

is

shown by W. Feit [f ] tnat

if we assume N to be solv-

indeed nilpotent. By developing some very deep techniques 38

6t uaql 'U

<

1t\

r{ll1r\ a^rlrsuErl d1d-lr s1 C

}I 'a^llrsur.r1

qc-rq.r ur sas?c aql 3ur1e311saaur uaaq s?q rornn?

.{1d_(1+ru_u) 1ou sI C

aqJ

.a^-rl-rsue.r1

,{1qnop

ul sl C (11) pue alrlrsutrl ,{1d-(1+ru-u) sl C a^rlrrul.rd st g lt (l) leql pa.roqs It] ueprol'saaeldslp H qJ-rq^\ srallal w aql uo a^rlrsu"rl pue 1uaaa,{ue

srallal tu-u Surxr; dno.r8qns

€ H pu€ u aa.rSap ;o dno.r5 B aq C lorl ',{1tado.rd srql

r{ll,n

,8

t

xapur }o raqloue pu€ Z xapul ;o dno"r8qns e pue ^e.rado.rd

.g.

s1q1 q11in

'u^\ou4m dlsnorzla.rd ara.r qclqa Ig lZ rapro ;o dno.r8 e sr araqJ

aa.r8ap ;o sdno.r5 rar.nrnJ pue 6Z9 ,lZT ,6V ,gT,,6 = u r{llrrr (I_u)u srapro ;o sdno.rS plallrEau aql apnlcul asoql .sdno.r8 a^Ilrsu"rl ,{1qnop aIqB^Ios

Jo raqurnu ollurJ ? dluo are a.raq1 'aseq1;o sdnor8qns pue sdno.rS asaql

l"ql u.roqs sBrI II] l"raddng .alq€^Ios pue alrlrsue.rl ,{1qnop s1 g plall allull ? rolo q + -<- x dno.r8 uoll"urroJsuerl Jeaurltruas aqJ sx"

wo-ry 1.rede

'suolleluasa.rda.r yo f.ro

-aql

aql Jo ap?w sr asn a^lsuelxa ,11nsa.r

qclq,l\

0I

I€uorldacxa

aa.r3ap ;o dno"r8 aarlrurr.rd

sr{J 'I +

zru

"

slql rod .alrlrsuerl

,{1qnop 1ou

s1

paapq s.l oroql pu" aslr€ u?c as€c

ruroJ aql Jo sl d7 uaqn,{1q1ssod ldacxa alrlrsuerl

,{1qnop sr dg aa.r5ap yo dno.r8 earlrwl.rd B l"q1 udoqs

s"q [Z] lpu?Ieli6

sllnsag lelcadg atuos 'g

fetu suorlrpuoc raqlrnJ uaAa lnq ,g yo .rap.ro aq1 Surplarp arurrd d.raaa .roy (d)l ,iq paprmoq sl N Jo ssBlc aql 'lap.ro alrsodruoc ;o 'p1oq

sI H qJlr1^t ul as€e aql a{Bl a,n ;r ,.raq1.rr\!J .alq€rlsap sI (dX }o uor}Burru -ralep rasolC 'sdnor8 snruoqord uo auop eq 01 ilIls {ro^t sl aror{I

's=(t)r{'r=(z){ a.lag 'd uo dluo Eulpuadap .ra8alur ue 3u1aq slql ,(d){ lsoru lB ssela Jo sl N uaq; 'fi11uap1 aq1 dluo 8u1x1y d rapro Jo usrqd.roruoln€ uE sllwpe qclq^\ N dno.r8 luelodllu '1ua1od11u aq

"

Jo ss?Ic aql pe1e811saaur seq

II]

uern8lg ureq?rg

lsnw N ler{l Suuroqs ur papaacrns /tlou s"q

[t]

uosdruoqJ uqof

a subgroup G, of G fixing w-2 letters

will

be doubly but not

triply transi-

tive and will contain the subgroup H. Hence our original G is at most a

transitive extenBion of a group G of degree n which is doubly but not triply transitive and contains a subgroup H fixing n-m letters and transitive on the m

letters which it displaces. We now replace H, if necessary, with

a maximal group fixing

n-k

=m

k > 3 letters and transitive

on the remaining

letters. We now find that the sets of letters fixed by H and the

conjugates of H from the blocks of an incomplete block design D and that G may be regarded as an automorphism group of D. We 1+

r(k-l)

where

will

r is the number of times any given letter

have n

=

appears in dif -

ferent blocks. If b is the number of blocks (in G, b is the number of conjugates of H) we also have the relation bk = nr =

"2k

- ,2 *

".

Any pair

of distinct letters occurs together in precisely one block. Conversely, suppose we are given a block design D with b blocks of

k letters each, a total of n distinct letters, each letter occurring in r dis-

tinct blocks and any pair of distinct letters occurring together in exactly one block. The parameters must satisfy the relations

bk=nr r(k-1) = n

1.

will satisfy our requirements if transitive on the letters of D and if the subgroup of G fixing

Then a group G of automorphisms of D G is doubly

all the letters of a block of D is transitive on the remaining letters.

If k = 3 the design D is a Steiner triple system such a group of collineations only

geometry over GF(2) or

and D

will

possess

if D is the set of lines in a projective

if D is the set of lines in an affine geometry

over GF(3).

California Institute of Technology* Pasadena, California -_---T_This paper was written at the Ohio State University. 40

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le.lnV

REVIEW OF SOME RESULTS IN COLLINEATION GROUPS by D. R. Hughesl

1.

Introduction

ln this paper we will attempt to review a part of the subject of collineation groups of finite projective planes. We largely disregard the more or less classical aspects of Desarguesian geometry, and we also skip the infinite planes. An example due to Marshall HaIl shows that

there are infinite projective planes with trivial collineation groups (and

it is also probably possible to construct

an infinite plane with any pre-

scribed countably infinite group as exactly its full collineation group). But in the finite case, every known plane has a rather "Iarge" collinea-

tion group. Whether this is an accident of our limited knowledge, or

a

theorem yet to be proved, is one of the most important problems in the study of finite planes. Section 2 is devoted to basic material, including nonexistence theo-

rems and configuration theorems. In Section 3 we list the finite nonDesarguesian planes known at the present time, and in Section 4 their

collineation groups, so far as they have been determined, are given. Sec-

tion 5 gives some special results. Statements of results

il

the paper are either followed by a reference

to a specific source, or else they can be found in Pickert's book;23 the two papersl6'12 by HaII are also very good references.

1. Supported in part by the United States No. AF 18(600)-1383. 42

Air Force under Contract

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tions with the same axis clearly form a group; the subset of elations in

this group is a normal subgroup. We call d a (V, K) central collineation,

if d is central, with center

V, axis K.

Let V and K be point and line, respectively, in a projective plane

..

P, are points of z such that neither is on K, neither is equal to V, but W'PZ is a line; if for all such choices of Pt and Pr, there is a (V, K) central collineation such that P1 d = P2, then we say that ' Suppose

Pt

and

is (V, K) transitive. It is not hard to show that there is at most one (V, K) central collineation sending P, onto P2r so (V, K) transitivity amounts to demanding that the group of (V, K) central collineation be as large as pos-

sible: (simpty) transitive and regular on the "non-fixed" points of a "non-

fixed" line through V. Without saying what Desargues' Theorem is,

we

remark that o is Desarguesian if and only if it is (V, K) transitive for

all

choices of V and K. One of the most interesting and

fruitful studies

of

non-Desarguesian geometry has resulted from considering projective planes with only a limited amount of (V' K) transitivity.

For any projective plane, we can construct an algebraic system called a "planar ternary ring"; this is an analogue of the use of a field

to co-ordinatize Desarguesian geometries. We will not need to be concerned with the most general of these planar ternary rings, but

will

con-

tent ourselves with certain special ones which are the most important in

terms of present knowledge.

Let (R, +, .), or merely R, be a system with at least two distinct elements 0, 1 and with two binary operations, addition (+), and multiplication (written a . b or ab),'all satisfYing:

(i) (ii)

under addition, R is a group with "identity" 0; under multiplication, the non-zero elements of R form a loop

with identity

(iii)

1;

x(y + z) = xy + xz, all x, Y, 44

z

e R;

s?

cnm l<-

+ (o) (x) -+ (x)

'I

(c0)

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(A)

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Then

{ is a collineation of r,

?.rd indeed

is an elation with axis

L-,

center (m) where ma +b = 0, if a + 0, or center (co) if a = 0. In fact, r is (P, L-) transitive for every point P on L-. This last property is characteristic of (left) V-W planes:

If o is (V,K) transitive for two points V on K, then z is a (left) V-W plane with K as L-, and thus o is (V' K) transitive Theorem 2.1.

for all points V on K. Also we have: Theorem 2.2.

If

is a (left) V-W plane with K as

ordinate rings with K as

L-

L-,

then

all co-

are (left) V-W systems, but not necessarily

isomorphic.

Let us restrict attention to the finite case' R is a proper (left or right) V-W system if it is not a field, and r is a proper V-W plane if it

is co-ordinatized by a proper v-w system. If one of the v-w systems of a v-w plane is a field, then all the co-ordinate rings for the plane are fields and the plane is Desarguesian. We have: Theorem 2.3. If, for two different choices of the Iine Lco (the point (co)

), the plane

r is co-ordinatized

r is Desarguesian.

So a

by a left (right) V-W system, then

proper left (right) V-W plane has the property

that none of its collineations move the line

L-

(the point (co) ).

finite non-associative division ring plane, then no collineation the line

ff r is a of o moves

L- or the Point (co).

This theorem reduces the study of collineation groups of non-associative division ring planes to a simply stated problem. Let R be a division ring, and a, b non-zero elements of R, T an additive one-to-one mapping of R upon R. Then the triple (T, a' b) is an autotopism of R

if

(xa)T(bv)T=(xY) T, for all x, j e R. The autotopisms of R form a group under the operation

LN

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II.

The Hughes planus.13 These cannot be co-ordinatized by V-W

systems, and are the only known finite planes with this property. Orders p2t, for aII odd primes p, positive integers t. Probably self-dual, but this

is only established 1o. o2t = 9; uniqueness still open. rIL The Hall V-W pl"n"r.10 Let F be the Galois field GF(q), q,+

2,

and let R be a two dimensional vector space over F, with basis elements

1, tr. Let f.(z) = z2 -

az

-

b be an irreducible quadratic over F, and then

define multiplication in R by:

(i) x(u+ Iv) =xu+ tr'xv _1 (ii) if y1O,(x+ \y)(u+ rv)=)m- y ^vf(x) + I(yu-xv+av). Roughly, this corresponde to demanding that every element of R, except-

ing elements in F, shall satisfy f(z) = O. The system R is always aproper

left V-W system and the multiplication is associative only in the case that F = GF(3) andl(z) = 12 *

1.

The right distributive law never holds'

Al-

though there are several Hall V-W systems of any fixed order q2, th"y

all co-ordinatize the same plane, and the plane is never self-dual; dual plane

the

is a right V-W plane co-ordinatized by the system anti-iso-

morphic to R.

W.

The Andr6 V-W planes.4 Lut R be a Galois field and

.l

a non-

identity automorphism of R; Iet K be the subfield of elements fixed by o.

xot1,*h"r"ti" ForeachXe R,define ,(x) =x xo'*o2'..' the order of a; then also R has degree t over K. Furthermore r (x) e K, aII x e R. Let p be any mapping of K into the integers modulo t, with only the proviso that

p

(1) = 0

= p (0). We define a system R,

addition is the same as R, but with multiplication given by

xoY=1Y wnefe dx=

x

c' su(x) 48

whose

6t

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a/Yr

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ro

tl < l pue eurlrd

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ppo u" sr d araqlr'rd.rapro.{.ra,ta;o

dg rualsds

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,6-n

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twalsds 16-n.rado.rd"

.rado.rd €

-11d111nru eq1 yo

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aql uaq& r{lasrce.rd aarle

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s! d u uaqJ

Then R

a non-associative division ring.

is

VIII.

Dickson's commutative division ring planes.l When c = -1r the

division rings of type VI are commutative. But another class is known.

Let F be a Galois field with

r

a non-identity automorphism, and let R be

a two-dimensional vector space over F with basis elements

1' tr ' Multi-

plication is:

(x+ ry) (u+ rv)=(xu+ Eyo v")+ r(xv+yu) where 3 is an element of F. If F has characteristic not two, then R is a division ring if and only if E is a non-square in F; R is always non-asso-

ciative(if

IX.

S

+0).

We add here a remark that other division rings exist, whose na-

ture is not completely understood. Knowledge of them is fragmentary,

it

and

seems best to disregard them here.

4.

Known collineation groups We

wili list here, with references, some of what is known about the

collineation groups of the planes of Section 3. The Desarguesian planes (Class I) are of course classical, and we only remark that they all have non- solvable collineation groups.

II.

The Hughes planes have been analyzed by Zappa and Rosati.24'27

A typical Hughes plane n has order p2n, p an odd prime, and contains a

ro of order pn. Th" projective group of ro P€rto all of r I the fuII group of r is the product of this

Desarguesian sub-plane

mits an extension

projective group with the automorphism group of a near-field of order t^ p'tt. Thus the group is non-solvable. It has the additional interesting prop-

r into iust two transitive one of which is the set of points of ro.

erty of breaking the points (and lines) of stituents,

III.

con-

The HaIl V-W planes have also been completely analyzed.l? Th"

description of the group is somewhat complicated, but it is always non50

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5.

Other theorems about collineation groups Besides the study of the groups of specific planes, there consists a

large body of more general theorems about the implications of the existence of collineations. Two of the more basic of these are:

Theorem 5.1.

fi

,l' is a collineation of a finite projective plane, then

the number of points fixed

by ,f equals the number of Iines fixed by {

'

Theorem 5.2.8'15'22 If G is a collineation group of a finite projec-

. , N1 (N2) the number of transitive constituents of points (lines) of r with respect to G, then Nf = NZ. Many of the theorems concern circumstances under which r must

tive plane

be Desarguesian. Gleason and Andr6 have proved:

If for every pair (P, L) of the finite projective plane o , where P is a point on the line L (not on the Iine L), there is Theorem 5.3.5'9

non-identity (P, L) translation (homology), then

a

r is Desarguesian.

Planes with transitive collineation groups have been studied

for

some

time. singer discovered that a finite Desarguesian plane of order n has a

cyclic collineation group of o"der n2 + n + 1 which is transitive on points and lines.25 Subsequent *o"k11 has indicated that probably only Desar-

guesian planes have this property. (It is interesting that

for infinite planes

the correct conjecture appears to be that all planes with this property are non-Desarguesian!) Indeed,

if r

has a transitive collineation group, then

o is probably Desarguesian. Failing to prove this theorem, wagner have shown the following, which is

still

a high-point in the field:

Theorem 5,4.20,21 V the finite projective plane group G which is doubly transiiive on points, then G contains all the elations

of r

Ostrom and

z

has a collineation

. is Desarguesian,

and

.

Also, the following is a step in the direction of analyzing transitive planes: 52

t9

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=

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II"',

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1o

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sl r

uaql 'uo11e

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a

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point P and one line L, where P is on L. The above result has some interest in

its bearing

on

order 10, since

that is the smallest order for which the existence of the plane is in doubt.

Bibliography

1. A. A. Albert,

On non-associative division algebras, Trans. Amer.

Soc., vol. 72 (1952),296-309.

2.

, Finite noncommutative division algebras, Proc. Amer.

3.

, On the collineation groups associated with twisted fields,

Math. Soc., vol. 9 (1958), 928-932.

Bull. CalEutfa-[Iath. Soc. (to appear).

4. Johannes Andr6, Uber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. 2., vol. 60 (1954), 156-186.

5. , Uber Perspektivitd.ten in endlichen projektiven Ebenen, Arch. MdTh., voi 6 (1954), 29-32. 6.

(1955), 15F16-0-.

, Projektive Ebenen iiber Fastkijrpern, Math.

2., vol.

62

?. R. H. Bruck and H. J. Ryser, The nonexistence of certainfinite projective planes, Canadian J. Mith.,'vol 1 (1949), 88-93. 8. H. P. Dembowski, Verallgemeinerungen von Transitivititsklassen endlicher projektiver Ebenen, Math.2., vol. 69 (1958) , 58-89. 9. A. M. Gleason, Finite Fano

planes, Amer. J. Math.,

vol

78 (1956),

797- 807.

10. Marshall Hall, Jr., Projective planes, Trans. Amer. Math. Soc.t vol. 54 (1943), 229-277.

11.

, Cyclic projective planes, Duke Math. J., vol. 14 (1947),

t2.

, Projective planes and related topics, CalU. Inst. Tech.,

1079- 1090. 19

54.

13. D. R. Hughes, A class of non-Desarguesian projective planes, Canadian J. Math., vol. 9 (1957), 3?8-388.

14.

, Regular collineation groups, Proc. Amer. Math. Soc.,

vol. 8 ( 195?)--T59- 164.

15. . Collineations and generalized incidence matrices, Trans. Amer. Math---Soa., vol 86 (1957), 284-296. 16.

. Generalized incidence matrices over group algebras,

IIIinois J:-NIt[h-.', vol. 1 (195?),545-551. 54

9S

'o8€clqC ]o l{llsra^Iun aqJ, 'OZZ-Lgl'(ggOf ) II 'Io^ '8.rnqueg 'Arun -uras 'ql"W 'qqy'"rad;g{ls"J aqallpua rac1l 'snequasse2 suEI{ '8U

r"ul

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pue i{.rlaruoaF aNlca[o.rd al]ulJ

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'ZZ

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alqnoq 'wol1sg 'D

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SOME FINITE GROUPS WITH GEOMETRICAL PROPERTIES by W. Magnus

1. Introduction This is an expository paper. The term geometrical refers to the classical meaning of the word, namely to those properties of a group of transformations which permit the definition of congruence in the corresponding geometry.

first attempt at characterizing a group of motions in terms of properties of the group itself seems to be a paper by M. DEHN1, in which The

he studies the abstract properties of the group of motions (including re-

flections) of the hyperbolic Non-Euclidean plane and describes an abstract characterization of other continuous groups. Although we shall not discuss projective groups,

it

seems appropri-

ate to quote here a paper by N. S. MENDELSOHN,2 the methods of which

are closely related to ours.* Mendelsohn succeeds in characterizing completely the plane projective groups by two sets of conjugate subgroups,

their normalizers and their intersection. (EIis methods can be generalized for higher dimensions.) The proofs of the theorems stated here are to appear soon in the extended version of the Ph.D. thesis of G. Bachman.3 We shall present his results with some modifications.

2.

Axioms

Definition 1. A group M is called a group of n-dimensional motions

if it satisfies the following

(i)

conditions:

There exists a set of proper subgroups Pt of M which con-

*I am indebted to Professor G. de B. Robinson for calling my attention to this paper at the Symposium on Finite Groups. 56

L9

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observations lead to the following

Definition conjugate

2. Let M be a group which contains a complete set of subgroups Pi (i = 1, . m) satisfying the postulation of

Definition 1. Then the P, shall be called the points of a geometry f . The group M acts as a group of transformations on I' according to the rule that, for any element

t of M, the map t(Pt) of Pt is

defined by

t(Pi) = tPit-1. The d-dimensional zubgroups of M are called the d-dimensional subspaces

of t. ; in particular, we shall call the one-dimensional

subspaces

'lines'

and the two-dimensional subspaces 'planes'' A

point P, is called incident on a subspace L6

if

LO

is a subgroup

of

Pr. The number n is called the dimension of the geometry' The action of t on LO is defined.bY t(Ld) = Two sets

if

tldt-1.

:1 and t,

of points or subspaces are called p-Sqglgg$

there exists an element t of M such that

t(>1)=:2' Now elements

t of M are called the motions of the geometry'

From the axioms (i) to (iv) for M, we can derive the following Statement of properties

of t

:

(i) AnY two Points are congruent. (ii) Any two d-dimensional subspaces are congruent' (iii) Let P be a point and let Lo and ti be two d-dimensional zubspaces incident on P where d > 1. Then there exists a motionwhichleavesPfixedandmapsLoontoLj.Thecorresponding fact is true for two (d + k)-dimensional subspaces incident on a d-dimensional subsPace' 58

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tively and without fix points on the points of the corresponding geometrY.

obviously, Frobenius' Theorem cannot be extended to infinite groups since the motions of the Non-Euclidean (hyperbolic) plane provide a coun-

ter example. In the two-dtmensional case, a finite group M, of motions determines a geometry uniquely. We shall state this as

Bachman's Theorem:

If a finite

group M contains a complete set of

conjugate subgroups Pt which are their own normalizers in M and anv two of which intersect in the identitv, then the P, are unjggelX

determined by M. The proof follows from the result stated in Bachman,3 Theorem 3.2, and an application of J. G. Thompson's Theorem. Bachman showed:

It

M

finite group containing two subgroups P and Q, each of which is its own normalizer in M such that both P and Q have the property that each

is

a

of them intersects with any one oI its conjugates in the identity only, then

either P contains one of the conjugates of Q or Q contains one of the conjugates of P. Obviously, we may assume that P contains Q. We wish to show that P = Q. Let Nn anci Nn, resPectively, be the

normal divisors consisting of the identity together with the elements not in P or its conjugates and not in Q or its conjugates' If P contains Q, then Nn contains Nn. According to Burnside,4 the orders of P and Nn are coprime and the same is true for the orders of Q and Nn. Therefore' N^ must contain at least one sylow subgroup of P. According to Thompr.J

son, the group Nn

is nilpotent

and

its sylow subgroups are characteris-

tic in Nn. since Nn is normal in M, its sylow subgroups are also normal in M. Therefore, P must contain a normal divisor of M which is incompatible with the assumption that the coniugates of P intersect in the iden-

tity only. Therefore, P = Q. 60

I9

'l+3=lu o^€q a^\ tasec s1q1 uI 'paxrJ

sr J ) L

aJaq&

(/ -x)Zn = /. -.x suorlnlllsqns Jo sdnoJ3 aql arE IC

'(07")

pu" J ,'d 'o

".{1

aJal{/t\

d+*zo =-*

suorlnlrlsqns ;o dno.r8 aql aq ol uasoqc aq deru 6 dno.r8

aqJ 'tW dno.r3 e s1 3 .rapro Jo J plalJ slol€C

ur sluarclJJaoa qu^\

"

Z

aa.rSep yo (3'7)AT dno.r3 leuorlc€rJ r€aurl ,t.reaa aculs aldrurs aq deru try tlce; ur 18W uI suoll"Isu"rl;o dnol8qns € lslxa lou IIr^\ a.raq1 'ye.reua3 u1

'({*!)

l={oulq l"q1 pu€ 6 u1 ale8ntuot aJ?

lu'"'tI=!'Id

U

d=Iq

sdno.rE aq1

l"ql

pu"

6 slenba tw ut d Jo (d)N razll"urrou aql l"ql qcns 116 = 4 ku'''''I Ia sale8nfuoa sll pue 6 dno.r8qns e a^Bq a,n ttry dno.r8 e u1 = l) suorlo6 l?uolsuaunq-g ;o sdno;g allul.{

't

'rapro arul.rd

sl ;, ssalun uotuuloc

yo sdno.r8

u€ql aroru taurl t111,n tleraua8 3 uo aq s.{urn1e 111n slutod on1 saurl o^\1 lnq u1 arreq c11cdc Jo urns

lcarlp

aq1

u1 1u1od auo

i{uy'as.raaul sl1 oluo V Jo luaurale ,{:aaa sdeu qc1qo. , tuslqdJotuoln" aql dq rapro ppo ;o y dnorS crlcdc-uou tuellaqe u" Jo uolsualxa aql IUoJJ saslre aldruexa ue lsaurl aql uo d1ea111sue.r1 lce lou lllfi, ZW tle.raua8 u1';, yo

1 dnor8qns

c11cdc

l"tulx"tu "

Jo

sluauala aq1 qSnorql sunr Y aJaq^r

'-lsdrru Jo auo

sI qclq&) 6

1u1od

t

-uolsuaunp-odu Jo Zliq dno.r3

dq paullap aq B

u"r

ul suorl"lsu€rl

s11

au11

Jo

I

Y

dY

pue (!d sdnor8qns

t

aq1

'u,tou1 sI suo!1otu I€

dno.r8qns aql acuo

Since

LF(2,5)

=

LF(2,4i,

we also have an example of a group MU which has two different sets of subgroups P, satisfying

all the postulates for a group of three-dimen-

sional motions. BaehmanS has found necessary and sufficient conditions

for

an

M,

to have a normal divisor T of translations in the case where the D, are

their own normalizers in P. Geometrically, t}tis means that, if a one-dimentional subspace (a line) is mapped onto itseU in such a manner that one point remains fixed, the mapping

is the identity.

Let M, be a finite group of three-dimensional rnotions and let P,Pir D = DZ and D, be defined as above. Assume that the normalizer Nn(D) of

D in P satisfies Np(D) = D. Then

M, has a normal divisor T which acts transitively

points on the Pi

(i)

if

and only

and without fixed

if

The normalizer N(D) of D in

M, is the direct product of D and

, of N(D). (il) f,et ? run through the elements of P outside

a subgroup

jugates. Assume that the smallest subgroup T of aLL

D and its con-

M, containing d

and

n 0n-1 h"" no element in common with P. Bachman has shown that T

is always a normal divisir of Mr. If T has

no element in common with P, then

wttlt =.v. We shall prove now that conditions (i) and

(ii) are necessary. (That they

are also sufficient has already been shown by Bachman who also exhibited examples of groups MU with a normal divisor 62

T.)

Thus, let us assume

89

'suoll"cllqnd ra^og' gg61 palug.rd

eplsurng '1t\

-aU 'uorllpg pu3 'rapro allulJ Io sdno.rF ;o d.roaql 'OL

'Z'q1epsdnorS

a11u1y

'696I '6tt-99? u!€lrac u1 d.rlatuoeg '88

'0t 'Io 'sa1.rag prg 'lU uorl?as 'aog 'dog 'su€r; 'dno.r8

'gt6l

1e.raua8 arn Jo

-gg

u"uqr€g '0 '8

-/,t

'"pEu"C uorl"aurlloc anlcafo.rd

dery

uorl"zlralc"reqc crlaroaql dno.r8 g

'It6I g '1g.rr1ssp1;, 'l€W 'alrlawoaC

't

uqoslapuatrl's'N 'z

't8 .rap uaEel

uqac'w 'I

-punr5 upp ul ua8unqcs.rog aranau aElura raqan sacuaJaJou

'sdno.r8 3u11eu.ra11e aql roJ pue tB "rap.ro Jo pIaI]

sloI"D e u1 (3'ur)91 sdno.r8 l€uorla"rJ r"aull aql roJ anrl s! slql

'tr1nc11red uI 'uIN addl yo sdno.r8 se pa1o.rd.ralul aq u"c sdno.rS a1du1s

requnu

J,frJ

a3.re1

e 'Z

< u.ro; t1eq1 {J?uIaJ

eql

qlll\

apnlcuoc deru

yo

a7y1

{sllelap ;o .{1a1.rea e rod '*I ;, lerll = g'u€tuqc"g aas 'I ruoJ} slrtolloJ 1!'d p I78W aculg = d u*J'a.rola.raq;'s1ulod

xlJ a^Eq1ou op

| * L Jo sluatuala aql aauls I

l_u0u aq1 -Il"rus 'a.ro1a.raql 'pu€ l_,t 0,'

= dU;'d1sno1,rqg

aql pu€ B f,q pale.rauaS*I dno.rS lsa sdno.r8 aql Jo IIB ululuoo lsnru

I

a\oN

'd.ressacau s1 (1) (a.royaraqJ '(q)N =

eq1 slenba

q Suruleluoc

ex 0 al"q a^\'(C)H ut C Jo xapul !6 ;o aql aauls 'g x d lcnpo.rd lcarlp aq1 "aq*.tu

suleluoc (q)g'e.royaraq;, 'paxlJ d sal€al q searaq^\ paxlJ !d due aaeal

IIIII\

d lo T f

sluawala aq1 aculs

(q)N ut I€urou

I = qud

1e*"ou sI I acurs replsuoc pu" slsrxa I1"q1

pu€

sr d '^Ir"aIO '(A)NUI = d

84

1ou

u1

SYMMETRICAL DEFINITIONS FOR THE BINARY POLYHEDRAL GROUPS by H. S. M. Coxeter

1. Introduction. The binary

polyhedral groups arise in at least

four ways:

(1) abstractly. as the groups determined by the relations

Rl =sm=1r=RST, where

l,

m, n are small enough to make the order finite;

(2) algebraically, as finite groups of quaternionsl (3) geometrically. as.finite groups of Clifford translations in spherical 3-space; and hence

(4) topologically, as the fundamental groups of certain 3-dimensional manifolds discovered by Seifert and Threlfall.

After considering each of these aspects in turn, we shall derive some new abstract definitions ("presentations") which are symmetrical

in the sense that they admit an automorphism which permutes the gen-

erators. Some of these presentations were communicated to me privately by Dr. R. G. de Buda and Dr. B. H. Neumann, to both of whom I would ex-

press my gratitude.

2.

Rotations and quaternions. The finite groups of rotations in

three dimensions are the polyhedral groups ( l, m, n)

2.t

R/ =sm=tx =RST=1,

where

k=mn+n! + lm- lmn >0 64

def ined by

'uzO :arlcdc

=

't)

(u 'u

s1 dno.r8,,c11c,tc d.reu1q,, aq1 ,.re1ncr1.red u1

.[69 .d,g]

ISU=uI=*S= UE ,{q pauryap

rapro Jo (u -.ra1enb

Jo' (u'* 'tr)

dnot? 1e.rpeq,{1odXr€ulq eq1 sp1ar,{

t'Z

'\/wn

\/uwV

tr

V

Z rapJo

'ru '7 ) dno.r8 le.rpaqdlod aq1 :1ea.r8 s€ ocr^\l sr rapro asoq& suolu .suoru.ralenb 1o dno.r8 e sp1a1d suorlslor yo dno.r3 elrulJ r{cta snrl;,

lrrm pue suollelor uaal\laq acuapuodsa.rroc o1y\1-auo B sr araql ,uo11eru -roJsrr€rl aq1 Eu1ra11e lnoqll/rr e- dq paceldor aq uec e aculg '[Ogf .d 't]

gf

ursl{8a + gzd+

rlq + + f

soc = z

z'z

oJaqlr 'gxe olut x sruJoJsu€Jl 8d 'Zd sau.rsoc uollceJlp

qll/{ aull aql lnoq" / q3noql uorlelor

'Id

ar{l leql putJ aA\

'>1tx+tZx*rI*=" uolu.ralenb e"rnd eq1 dq

(6x 'Zx

'Ix)

saleulprooc u"!salr€C qlll\ lurod

aq1 luasa.rda.r am

y1

'1enba

lsnw olt\l Jal{lo o{1 '.,, s1 ,,413ue111,, lucl.raqds e yo a13ue auo yr '12q1 lD€J aql qll^\ pal"lcossB sl I = 7 uaq/tl u = ru uoll3lr1sa.r pa11drul aq;

aq

.>t/ulntr €InurroJ

'l '?,'t'V 'fu rapro Jo

qcea ur JapJo aq1

'0g 'v?, 'zl 'uZ 'u

Jo rapro Jo rapro Jo rapro Jo rapro

'u7 ate

{

=

(g 'g

dq ual13 sg asrc

}o sanl€A aallcedsa.r aql aculs

'z) (l 'g'z) = 'uU (g ', 'z) = 'uO = 1u'r'r', 'tg (u 'u 'I) = '?S

or{1

z

dno.r8 lz.rpaqesocr aql dno.r8 le.rpaqelco aql dno.r8 1e.rpaq"rlal aql dno.r8 le.rpaqlp aql dno.r8 cl1cfc aq1

:dlaweu '[gs 'gt-t

t

'66

,e1

group (2, 2, n) is dicyclic [4, p. Z]. fhe dicyclic groups include, as special cases, the quaternion group (2, Z, Z)

and the "binary dihedral"

(Hamilton's t2 = i2 = k2 = ijk) and the "generalized quaternion group"

<2,2, zq> [ro, p. rzo]. It is true,

2.4

though

tricky to prove [2, pp. 369-?0], that the relations

Rl =Sm=T"=RST=z

fmpty 22 = 1 (provided,

2, m,n, k are all positive; but if any of .{,, m,

n are negative the period of Z is greater than 2, and

if k = 0, Z is free

[2, p. sza]).

3. Permutations. The fact that the relations 2,4 do not imply Z = 1 is obvious for (1, n, n) , and well known for (2, 2, n) .In the more complicated cases

it

may be deduced either from the representation by

quaternions [2, pp. 3?0-?1] or from the following permutations.

for (2, S, S)

(degree

8):

R, =

(ac'a'c)(bd'b'd),

S= (bdab'd'a')(cc'), a = (abca'b'c')(dd').

ror (2, 3, 4)

(degree 16):

3 = (ae'a'e)(bc'b' g = (cahc'a'

c)(dhd'h' )(fg'f' g),

h' )(dged' g' e' )(bb'

)(ff '),

1' = (abcda'b' c'd' )(efghe'f' g'h' ).

For (2, S, s) 11

(degree 24):

= (aga' g' ) (bd'b'

d) (cf ' c' f

)(ek' e'k) (hj' h' j) (i .(,, i, .!,

1,

5 = (kdak'd'a')(cfbc'f'b')(jgej'g'e')(i Xhi',L'h'),

1

= (abcdea' b' c' d' e' ) (ghijkg' h' i' j'k' ) (ff' ) ( .t, 2' ).

In each case Z =

(aa')(bb') ., and the central quotient group is

derived by identifying the primed letters with the corresponding un-

primed letters. When this has been done, we have the ordinary tetrahedral, octahedral and icosahedral groups, representecl by permutations of the faces of the tetrahedron, octahedron and dodecahedron, which we

t9 -Enb

('e'r) suoruJal?nb llun

Jo 1s!suoJ

lsnur suoluJalenb Jo dno.r8

a11ur;

,{ue 's-ro1ee; aql Jo sruJou eql Jo lcnpo.rd aq1 o1 lenba sr suoruJal?nb o.rrl

;o lcnpo.rd aql Jo rtrrou aql acurs

'

1 e.rnSrg

q V o i'gg aal8ap Jo uorlsluosa"rda.r ou seq {l 'e '21 dno.rE T?;rpaqslco d.reulq aq.t! 'I arn8tg ur srue.rSerp 1a8e1qcg riq a1a.qsn111

ternions of norm 1). Since any unit quaternion may be expressed in the

*

form 2.2 (witn pf

* ni = t), any finite group of quaternions must be either polyhedral or binary polyhedral. The former possibility will arise

if

p!,,

-1 does not belong to the group of quaternions, in which case

the corresponding binary group is the direct product of the group of

quaternions and the group of order 2 generated by -1.

if possible, that ( l, m, n) is the direct product of (!,,m, n)andthegroupof order2generatedbyZ. Then (l ,^, n)must Suppose,

contain elements Ro =

RZr,

To = Tzt

So = SZs,

satisfying

*j=ti=4=Rosoro=1. This requires

| + rl

= 1 +sm = 1

which can happen only

if

+tn = I

.!., m, n (and

+

r +s +t

(mod 2), =0

r, s, t) are all

odd. Hence the

only case of a direct product is

(t,n,n) and the only

=

Gzn

3

tsn

x

G,

(n odd),

finite group of quaternions that is not a binarv polyhedral

group is the cyclic group generated by a single quaternion of odd period [0, p. ses].

It is, perhaps, worth while to point

out that

all the groups discussed

here are finite subgroups of the classical group A1, which is the continuous group of all unit quaternions [p, p. eZ6].

5.

The representation by points on a hypersphere. Representing

the quaternion

*=*o+xri+x2j+xrk 68

69

z'11 uorlaunrl at"tpotrrralut aql p1ar,{ s, {t'u't} { s 'l 'e } leco.rdraa.r on1 aIil pue'lvzt'aTt {"2} " {rt} lcnpo.rd .re1nsue1ca.r .ro usr.rd Ieuors - uarutp- t aq1 plard ur o.rt aql :[lZe 'd ,i] ru.rogrm r{po 1nq .reln5ar ", { }

lou sr lr q3noq1 'ado1d1od B prmoq aq1 '(a1dtu1s oo1

't

{e 'e '?}

sl

qctq/r'(r'r'I)

alnc.raudr{ aql prmoq

''3'a) adoldlod

1111s s1u1od

aql 1€ saueld.rad,{q lua8ur}

8u1pn1axa) sasec Eulureua.r aq1 u1

'f+ = Ix

I+ -lx'"

.re1n8a.r lecordrca.r aql prmoq a.raqds-g aql 01 saueldladfq

1ueSuq Surpuodsa.r.roc aql g aql ar€ daql qcrqn loJ

'{{ f ' ( Z,

'S ,Z

'g}

,?,)

,,adoilIod sso.rc,, aql }o sarlpa^ dno.rS,uoru.relenb aq1 Jo asec aql

ur "8'a) ado/1od teln8ar e18urs B Jo sacllra^ aql arB

uaqrh

saueld.red.{q g aql

s1u1od >I/urJJT V

a\l

'

'slasor rraql pu"

{rnwr*t} '{r} sdno.r:qns aq1 ' (r't 'Z) ul 'luasarda.r suo8elao 8I stH ',g + g s11ec [g] .d il] uos ,l ,e} teco.rdtcar ot\l aqJ, -ulqou qcrqir uorlsrnSr;uoc aql urroJ ", {e Irac-009 eqr

'( g't'z)roJ

pu" ia,raqds.rad.{q arrr's

r;lt-;:""*

,l "rr ,s} n""-lz 'l'e} teco.iarcar o^r1 ' (v't 'z) .roy: ", {e {e "ur

' (t 't '27 tol 'saueld 1zuo5oq1.ro dlalaldruoc uf u7} o^f ' (u,Z,Z) ",{ rl .rol : .re1n3ar .ro; :sadoltlod .re1n8a.r on1 {uZ}'uo3-u6 "' (r'u'1) ro auo Jo sacrlral aql are s1u1od aq1 as?c r{cea uI .[t ZZ .d ,i ltt t .d -Z] suorlcalJal dq pale;aua8 dno.r8 allur] B Jo ,,sJorJIur I?nlJ-rA,, pu€ sJoJ

-rrru aql ar?

r{crq/n 'saue1d.rad,{{ >{/uru Z Z u1 suol}ca1ya.r dq pa8ueqa.ralul

are slurod ppodrlue ;o stred \/wnV Z aql snqJ, 'I_x - = x - ,q x uorural -enb 11rm qcea saceldar I + saEurqc.ralul qclqa uorlcallar aql leql aaras -qo ol aaeq dla.reu an '.{1.rado.rd on1 due sa8ueqc.ralur

s1q1 qs11qelsa

lsql uollcalJar aql dq

oJ 'sJaqruaur ppodllue

nas11

olq

paurJoJsr.rerl sr

las aloq^! aql l€ql dlrado.rd aW qll/t\'I = xx a.raqds-g aql uo s1u1od;o 1as e dq paluasa.rda.r s1 dno.r8 1e.rpaq,{1od.d"reurq qcea l"ql put} am'aceds-7 uespllcng ur 1ty 'Z*

'I* 'o*; sal€urprooc

uersalre3 q1lin lugod aql dq

[f , p. aSf ], which may be described as foilows"

Consider (in Euciidean 4-space) two 24-cells in reciprocal positions. one so large that

it entirety

encloses the other. Let the Iarge one shrink

(or ttre small one grow) untit the corners of the small one are cut ofi by the bounding hyperplanes of the large one. The part of space that lies inside both polytopes is called a truncation of either. Eventually the large polytope becomes small, and the small one large, so that their roles are interchanged. At certain stages (when the polygonal faces are regular)

the truncation is a uniform polytope. In the notation of "Wythoff's con-

struction" It], ttre complete series of such trunctions of the 24-cell is follows:

as

IL

sE.{rt ,, ,uz} qrnocdauoq oloq,,r arn roJ loq "t ,"2} Erpaqlp arB qcn{.,r\ -ru,{s r1;epcg aler.rdo.rdd€ oql snqJ .[f 'o ,Ol] {Z r) sIIac uZ puE uZ olnlrlsuoc sasual pue sataqdsrwaq 1ea.r3 tuZi ,,saceJ,, 'I

aIqEJ ur

aq,1 'sa8pa uZ olur ,c esodruocap sacrlJaA asaqJ .,a Buole paceds dluaaa 'sac11.raa u6 1r Surar8 dq qruoc,fauoq I"np-FIas E

olul lr a{Bru uE; alA

'[gg '0 'g-] suelpg.rau alaldruoc u dq sarml uZ olul a.raqds ,!.reurp.ro ur ;o ,r} uol1r1red aql sl qcrq/{'{u, uollellassal lecl.raqds aq1 ;o an3opue I"uolsuaurp-g E se pap.re8at aq.{eru a.rnEly

a\I'u/L

el8ue ue 1e Eur

-laaur sa.raqds alllncasuoc '(,c sluasa.rda,r qcrqrrr) olcrrc auo q3no.rq1

Eur

-ssed 1p sa.raqds d.reulp.ro u dq.raqloue auo urorJ pale.redas ar" sasual aql Iapou slr{1 uI 'aceds-g utaprlcng B oluo uorlcafo.rd crqde"r8oa.rals

I"uorsuaurp-t € dq palanrlsuoc oq ppoc s€ qcns Iapou leru.ro;uoc e dq pazrlenst^ dlrsea lsour sr 'eceds-g lecr.raqds aq1 3ur11r1,sasue1 qens uZ ;o qwoc,{auoq aqJ, 'c o1 (,,.re1od,,.ro) puo8oq1.ro,{1e1a1druoc alc.rrc

1ea.r3

aql :,c,,.ro1€nba,, uotuuroc e 8uo1r laaw qclql\ sa.raqdsrruar{ }ea.r8 o4.{q paprmoq suel e sr uorSat lalqclrle aqi pue 'c elc.rqc 1ea.r8 e Buole paards .41ua,re a.rz slurod uA aq1 asec

' ( r 'u 'I)

srql

q

'papnl?xa dlsnorrra.rd alrr qc1r{ilr

dnor8 cr1c,{a ar{l roJ osIE punoJ aq uea uor5a.r B qcns

'[VZ'A

tT]

uolEa-r letuarueprm] € sr IIac aq1 :dno.r8

p.rpaqdlod dteulq aql ]o luaurala r{cea .ro; uor8a.r qcns auo sl araql snql

'sraqlo aq1;o due aq1

;o 1:ed

aq1 yo

01

u"ql 1ulod

s1q1 01

rareau sl leql eceds leer.raqds

Sullsgsuoa'[6I-gIZ'dd'6]

,,uo13o.r laFIclJ!61,, B Jo Jarru?u eq1 ur slurod

-.rns qwoadauoq slql Jo IIer qc"S '[ggI

-q"lqo snql

\/u,ay

louoro1l, ro

? or{1 Jo auo spunor

qruoc.{euoq Iecrraqds e

8u-r

'a.raqds-g Dlrluacuoc B oluo raluac slt urorJ ado1,{1od qcea

Eu11caford dq aceds-g lecr-raqds o1

'", r 't 'ti)--lEnba

1t

e1pplru aql

'd'l]

,,uo.rpaqd1od

acrds-p u€ap-rlang rno"r; ssed all

o/$1 Jo

q sl 'sn su.raruoc dlyelqc

lualuoJ uolnluoc aql :sarras aql

Jo

qclq,r\'{e ,l ,e} z'11 ado6lod aq;

produ"t" {zrr} * {zt , which arises from the } dicyclic group (2, 2, n) , seems at first to be unlike all the other honeycombs. But in fact it can be derived from two reciprocal {Zn,Z, Zn}'s

The "rectangular

in the same manner that we derived,r,,

rl

13, 4,

3

J's.

{t.

+,

l} irom two reciprocal

The appropriate graPh is

which shows that, instead of calling

it {Zn} . {r"}

well have called tt ,1,2 \2n,2,2n1

,

(l

The quaternion

, we could just as

group (2,2,2) has been listed separately (after

the general dicyclic group) because the rectangular product of two squares happens to be the hypercube sociated with the fact

{+,

S,

that <2,2,2> , unlike

S

} . tnt" regularity is as-

the other dicyclic groups,

has an outer automorphism of period 3.

?.

The representation by Clifford translations. When a binary poly-

hedral group is represented by a spherical honeycomb, the cell that represents a quaternion q (belonging to the group) is derived from the cell that represents 1 by a Clifford translation [9, p. t+Z;

!!, p. 138] which

transforms the general point x into xq. This is a left-handed screwdisplacement: the product of an ordinary translation along a line (i.e., a great circle) and a rotation about the same line, with the special condition that the angle of the rotation is equal to the angular measure of the

translation. In the case o{ the cyclic group generated by a single quaternion, the generating clifford translation transforms each lens into the next, with an accompanying rotation. when the rectangular product

{rt} 72

>r

{Zt} t" regarded as a pair

tt

I\9

/\,.\/\,^. Nt w i\9

\9

F

C.,

C.t

C,

t9

19

I

ctra(rt\9lF \,/ \./

v'

v

v

o o t r5

7a i:o .iF

3!0 d'

o

O) t9 O@t\9HAl.9 O@t}O)i'I

Ft

t\,

o

tn

l{

19 Ctr O-fcgqt@19 oo)o)tgF:J

-J

19 O 13 o 5

fi Y d

HF

A(g AI9 or1 o:. f;s

EE to -a

o.

sq

t\,

(o OJ -3. 6

E o a

Nt t5 o '.o tr

o a

t\9 4,5 5I Ne =\e t\3o) p I FA

H a df; Jo aa

Eo

Fi

o a

N9l5t\9@At\9 o@A^lF

o.grotrr5o o-lodE3 otstloF.r 6-lgthgQ

SUdBf +*EsrAj ox: 6EP

t0tr o

a

o o (n

Fl

ri 14

of interlocked rings of stacked prisms

(2,2, n)

lg p. tZ+], the generator

T of

permutes the prisms in each ring, rotating them at the same

time. The case when n = 3 is illustrated in Figure 2, where the edges of a hexagonal prism are marked so as to indicate the manner in which T transforms the bottom hexagon into the top one. The other kind of generator, s (or R), has its axis "horizontal" instead of "vertical," transforming one of the squares (such as 1524) into the opposite one'

I -\\

I

i2 P-- ----\ 4

Figure

Similarly, the generators 24 octahedral cells

or {f,

+,

S and

S}

T

2

of (2, 3, 3)

each permute the

in 4 rings of 6. We can obtain some

idea of the appearance of such a ring by stacking six octahedra (so that

neighbors share a face). With this model before us, we immediately see

certain sets of six oblique edges winding up the column in the manner of a helix. when the top and bottom triangles of the column are brought to-

gether (by rotations through 60o about the planes of the remaining horizontal triangles), these six edges close up to form a skew hexagon which

is shifted along itself by the clifford translation. Figure 3 shows how each face of an octahedron

is transformed into the opposite face by such

a "screw -displacement, " 74

st -tnetr Eurlsorelul u? leql prmoJ [OL-tS '00 TT]

lra]las pue IIBJIarq,l, ,sepod

-11ue Euldyrluepl dq aceds lecg.raqds ruo"ry pa^-lrap aq u?c aceds c11d1ga se

lsnf

'pIoJIuBru l"uo-rsuaurp-aerql E Io clnorb lquarrEpr.mJ aqJ, .g

(',f,

ant

a^Eqeq s.role.reua8 aq1

I1e qclqAt u1 'dno.r8 lBJpaqesocl r{.reu1q aql purJ IIBqs a^\

IIs q

lo suorl"luasa.rd Jeqlo eruos '1! qll/r\ uor.uluoc ur xaua^ B ua^a 1ou Surleq'a1ele

-des d1a;11ua sI 1"ql raqlou€

olq nec qc?a srrrrolsuBrl

'[grz 'a 'zt lcg'g '31,{ 'Ig 'd '7] t-rpaqe-r1a1

S "role.raua8 aq;,)

-re1n5ar paqlrcsul g Suorue

uollnqlrlstp rraql ol Surp.rocce sacllJa^ aq1 5u14retu dq paleclpur

s1

ul {S ,t ,S}

lo

IIac € Io sace; alrsoddo;o 8u1qc1eru oql pue'g1 ;o s8ur.r Erpaqeaapop

:^rr"urr' Jo r rolerauaS aq1

0zI aqt salnruead (s't,'z)

'sa8pa

oql Jo p"alsul sacllral

aq1

Eulqteu dq

allsoddo uaa/r\laq acuepuodsa.rroc aql

pa1ec1pu1 dlalenbape aq 01 saa€J

to; paleclldruoc flluarcryps

sg (p

am8rg) aqnc palecrmrl aqJ 'saceJrolrn puo8elco qll/t\ g ;o s8urt ul tuaql selnurad (g pouad Jo) r roleraua8 eq1 pue 'sace;.re1ur .relnSuerrl r{lllt\ g

;o

p saqnc pal"crmrl gt ar{l salnurrad (g por.rad dno.r8 lerpatlelco d.reurq aql rod Jo) S rolereuaS aq1 '(''t'Z)

s8u1.r

u1

{g'U't}

Z'11

g a.rn81g

z

(, L,

t

,i,B

B

^A-. F Figure

4

ily of 3 -dimensional manifolds can be derived by identifying ali the points related to a given point by any finite group of Clifford translations. The simplest way to do this is to take a fundamental region for the group, as

in Figure 2, 3, or 4, and identify pairs of opposite faces as indicated. In this way the honeycomb is reduced to a "polytope" having only one cell, as in Table

II.

(The numbers of vertices and other elements in Table I

simply have to be divided by the number of cells.) Elliptic space itself arises in this manner from the group (1, 1, honeycomb

rl

72,2,2 |

1)

and the very simple

.

Following Tietze [t+, p. 111], threUall and Seffert describe not only

trom { 2n,2, znl but the more gen< q < p/2 eral lens space (p, q) $rat comes tro* {n, 2,p/q}, wnere L

the simple lens space (2n, 1) derived

and (p, q) = 1. (When g > 1, the gener"ator of the fundamental group is no

longer a Clifford translation.) They give necessary and sufficient conditions for two such spaces (with the same p) to be homeomorphic ?6

[E p. SSt].

LL

\9 q,

/\

t\9

4.. t\9

a\ 19

\9

F

Cr9

q,

N,

\9

F

v

\/

\./

.v,

AA(l,T\9i'I

v

\,,

g,FlOordl' (nFJ

tsl U)o tr Eed EE Eg 8' 's *g[ 3s 3& oo.'9

3.I F!o

I

5F

f3 iE

o H o

o A D')

€

e

o.

o !

o th

Fl F

o

O) r5 6 i' i.f

t\9

t9

Cr,A .)f. llin gE O al'

0c

o a

A ai 3. br E

t' U) F C p

FiJ t\ro 5F I d ocp

; 9B E dY ,dE o a :io o (h

d

rt p.

H t9

I

t

oc

9

t0

o o (n

"'a

HHPPFF

o.footto. of.lcF= nts+*J O;tF:6oo Ao5trO. iiD(iPE :yRo-o

6X!!' 6E9

lo) OF o

C)

o

l.

trl

Along with the spherical dodecahedron space derived

tto* {S, t,t}.

they consider also a hyperbolic dodecahedron space that comes from the

s, l} ltz, p.218; 6, 157]. {0, t_i

hyperbolic honeycomb

9.

Svmmetrical presentations

for

<2,

2,2> and (2, 3, 3) .

We

have so far presented the binary polyhedral groups in terms of three

generators. But

it is quite easy to Eliminate

any one of the generators.

This is particularly desirable when n = 2, so that we can substitute ST

for R, obtaining

9.1

sm = Tn = (sr)z

t)r

9.2

TST =

s*-1,

srs

=

C-1.

r-or instance, the quaternion group <2, 2,2> is simply

9.3

jkj

kjk = 3,

and the binary tetrahedral

9.4

TST =

= k,

group ( 2, 3, 3> is

s2,

srs

= T2.

The substitution S = A, T = B-1 changes the

.r.i

gazg = e,

latter into

aa2A = n.

ri'hence ._; -eA=R_1 _i

^A ^=BAB_1^

_r-

= C. say. We thus obtain e new presentation in terms oi 3 generators:

t.6 i-.etinrng

iB

= AC ..

n = g-iC-1

llA.

ABC =

1.

= C-1e-1 = A-18-1,'we obtain a presentation in

terms ol 4 generators:

!l"t

DCB = CDA = BAD = ABC =

ii:i*: rlo Seiieri and Thretfail [1?, p. 218].

78

1,

flnally. the substitution A = S,

6L

'z=e(JS) =pI=eS uearB t,{les,raluoC

' (V't'Z)

dno.r8 lerparielro.{.reurq aql aur}ap 01 u.,rlou{

ar"

qcrq^\

'r(,r,s)=rJ=ts suorl"Ior

aq1

dldtur (f_X = J pue

z'0r

fI = S r{lln) I.0I suo1lelar .J = J .If .I .II =

aql snql

'VJ = ZL.fJ,I

.fl = ,(rs) = ,(rn) = If 'IJ eS puB Jf = IJ = S acnpap a,n,r_>I = J,fI = g Eururyag ,I=fi>I 'X=IXf 'f =Xfl I'OI suorlelar aq1 .{;sr1es ol uaes dlrsea

U ,2./ 'ET=r ^ FT=x

a.re

,U 'Fi=r suolu

-ralenb aartg aq;, '

(V't'Z)

l"clrlaIIIIII^s .0I

roJ suorlElueserd

'[ol 'd'e]

,-{ ' ut O x (u 'tu '?,)

=

(u ,*- ,Z) ur€}qo at\

u(su;=*S=zu ,{q l'O 8ulce1da"r ',{1.re11rn1g

'9a dno.r8 c11cdc aql oulJap ,{qenlce 'g'6 alqwese.l

'p=cq

'c=e€

,{1p1cgy.radns qc1q,n

,s=pc

'q=ep

suorl"Ier aql

13q1

s{r"rrrar uu?wneN

- }"6 Jo IIv)

('dno.r8 1erpar{"4a1 ,{.reu1q aql Jo suoll€1uasa"rd are g'6

'A=nS 'n=JA 'I=Sn ruroJ sruu?urnsN olur s1q1 slnd

I_A

'S= =

q 'n

J =C

8'6

'I-I

=E

we can define

J=sr-l

I=T-1s,

K=T-1, and deduce

u = t-1s2r-1 - T-1s-1T-12 KU = T-1s = I,

=

s,

K-1 = T,

uK= sK- sr-l

= J,

JKr = sr-3s = srsZ-1 = r-1 = x. Hence 10.1 and 10.2 are equivalent,

i.e., 10.1 is a valid presentation for

the binary octahedral group. Given 10.1, we can equate four expressions

for I to obtain

= 6_1r_1* = J_lKI = KJK-1. 'r_1,1_1 These three equations are not independent but can all be derived from

the two relations

10.3

JIcI = IUK,

.l-ltcl-l

JKI = KJK,

]f,jJ2K = J2.

=

x.l-lx

or

10.4

Hence 10.3 and 10.4 each suffice to define the binary octahedral group. Since the relation

Kl2K

= .12

ir, 1o.e can be replaced by JK2J = K2,

we might be tempted to expect the two relations

r;.2K = J2,

.lx2.l = x2

to suffice. However, the group so defined is obviously infinite, since it has the infinite dihedral group

f=:r<2=t [1 p. 21], where a finite factor x2lx2 = J, J2KJ2 = K is described.)

as a factor group. (Cf.

finite group

The infinite group defined by the single relation

group of the in-

I8

'Sr-r =,-SJS =A

'1_,lS =SJI_S =O

,I_S de pur g .{q I }o srtrro}su"rl aql raplsuoc

'z=zt's)=gr=gs

I'II

dno"rE le.rpaq

-Bsoc1 dreulq aql q . (e ,e ,Z;

'[pSl '0 $] suolu.ralenb

Z/6+ !+t+I+)

1e.r3a1ur

,>f+ .[+

;o walsds s,zg^unH Io

,!+

,I+ sllrm

VZaql 1sn[ e"re dno.r8 1e.rperi"rlal ,{.reurq aql }o

'l*fr--t=c 'a-f,-=a

sluatua;

,1ce; VZ aq1

u1

'n-t',.*r=v puu

olr 'suolu.ra1enb sy

.L.6 dno.r8qns lBrpor{?rla1 d.reulq e aletaue8 snql pu€

'I = I_Xf .I_fI.r_DI = CAV 'I = I-XI_f .ft_I .I}I = (IVA 'I = .,-IX' t-Xt_f .fI = yCC 'I = xr_f .fI . I_II->I = scc d;sr1es qclq.n

,If=xI=tr=t_a 'tr-"=I-Xf = fI=J 'tr-t= I-fI= IX=A

'tr_t= I_IX=)If=V

sluaurala rnoJ aql suleluoc 1.91 dno.r8 l€rpeqelco ,!.rru1q eq;

'[eg 'a

5'i]

toua

llolarl

aq1

;o dno.r8 lelue.o€prml eql se

!

azlell

os1e aas

iggl 'd

dq peuorluaru

)II)I = fXf

s1

g'0I

We find that

uv-1u-lv = s-1ts.s-1r.TS-1.

STs-1 = S-1T4S-1 = T,

vt-1v-lt

=

STS-1.r-1.

s-lt.T

ru-1r-lu

=

T.TS-1 . t-1

.s-lrs

=

St4z-1

=

sr-1=

U,

= T4zS = r-ls = V.

Thus

uv-lu-lv = t,

t:.z

vr-lv-lT = u,

TU-1T-1u = v

and obviously also

VU=TV=UT=S. De Buda has observed that the relations 11.2 suffice to define the

binary icosahedral group. One step in his proof is the deduction of

Vu = TV = UT. This is achieved as follows. The relations 11.2 (beginning with the last) are equivalent to

LL.zL T=uv-1Tu-l, u=vr-luv-l, v=ru-lvr-l or, in terms of three new elements

11.3 A=vr-l, B=TU-l, c=uv-l, 11.31 T=CB, U=AC, V=BA. Thus

11.4

T=BU=BAC, U=CV=CBA, V=AT=ACB; W

and

= ACB.AC = AC .BAC = UT,

similarly these are equal to TV. We have now proved that

11.5

VU = TV = UT,

as desired. In terms of S = VU = TV = UT, we have

82

t8 Zq

= BqI_"qB

'Z? = qEf_q?q

6'I

I

uollrulJap polelor ,{1aso1c aqg,

'VAV = A.VE

'EVE = VaEV

:3 pa^Irap

'I

s1 dno.r8 l"Jpaqesoc1 d.reulq

3u11eu1rug1a

8'I

I

dq 1'11 tuo.ry

eql rol uorleluasa.rd qlrnoJ V

l"ql paao.rd aAEq ad\ ,d11e1uap-rcu1 'aleldruoa sI uolleclJlluapl ar{l snq;,'Z'II salldtur ,u.rn1 u1 ,qclqin IZ.II sa11dru1 IS'II uaq&'l{1.re11ru1s sltiolloJ t.II Jo lsor aql it_IA = v a^sq alrl = CEV d1dru1 2,.11 suorl"Iar aql

'IV=gOV=VA=A

q 't'II pue g'11 d1dru1 'Ig'II rn1,n Suop 'daq1 1eq1 paao.rd aq ol su!"wal fpo U ,(t.II r1llr\ IS'II Sur.redrsoc ttq) g'ff pue Z'II ruorJ suorlBlar esaql acurg ^roIIoJ 'acv = va 'vgc = cY 'cvg = gc ,.'II aculs '1cey

s1

dnor8 Ierpaq?soJl f.reulq or{l ro} uorleluase.rd

prlq

V

'uoI1"lue -sa.rd

plpl dlpnba uB s" Z'II

paqs-rlqqsa

palcnrlsuocar a^sq o^\ snr{J'_(,tS) =

'r_surr_s =

f-sztz(g_srgs)I-s I_SIS' I_S,l'

=

puB'I'II suorlelar Eurul;ap aq1

^S'g'II dq,os1y.gI = tS a?ueqa

,-srrr-srrrs = =I = Z-S.I,ZS r_Sr-IrS

'_SIZS'

pIeId

ol A,-n,_AO = J uI_pelnlllsqns eq u?a srerllo aqJ, 'A pue n q11,n d1le11 t- t-tnls prre 'J qll,n salnruuro? gS l€r{l st\oqs (r-SJ,rS = I) uorl?Ier lsBI eq;,

't-SIgS = r_Sr-JrS = r_AS =,L

'r-sr_rrs=

,-SIS =,-OS =A

.S 'r_sJrs = ,_SIS

=

o ,_,LS = 'rS

=

JSI

9'II

(a = A, b = B-1) has been cited by Neumann as one of the "shortest" pos-

sible presentations of this group. In terms of the "golden section" number

,

=!2{u'U + 1), we have the

quaternions

s

=

| tr-i-:-r.),

t=it, -i-,-1k), u=*t,-k-'-1i), u=it,-i-'-li), o = i t , -i+'-1i), u = a = it"-k+'-1i). The presentation 11.2 shows that the binary icosahedral group

'(,-1+'-1k),

is its

ItS, p. 140]. It seems to be the smallest nonsimple group of this kind. Neumann has conjectured that it may be the

own commutator group

only such group which can be defined by as few as 2 relations (and the

It is natural to ask what

same number of generators).

would happen

if

we

replaced the relations 11.2 by

u-1v-luv

=

T, v-lT-lvr = u, t-lu-lru

=

v.

The rather surprising answer is that the group so defined collapses com-

pletely: its order is 1.'

It is interesting also to observe that the relations

11.5 are equiva-

lent to UVU = VUV (cf. 10.5).

12. A direct product. In the binary icosahedral group

R2=s3=1.5=RST=2, consider the transform of T by R:

w = R-1rR

= Rz

-1:rR

= RS-1 =

r-1s.

We lind that

wt2w = T-lsrs

rw2r

=

r-lRs = T-22 = T3,

=

r.R-1r2R .T

=

z(s-lt)z

=

z-1(rnr)z

= zrtt-2 = w3. 84

98

I_IX'LZZI_XJ,X = ,Zr-IXr_IaIr_XrX zz

z\ _ Lx) z(, _xrr), _,rx

= rz r(r_ rx) zr

_

rx

=

=, zr(, _,rx) pue

z

=

,(r-xrt)

a^sq e& 'X'I_LXZ = XgW = nX =

,XrX

r_re(rx)zx z&zx

=

=

,(xJ,)rxrl.rx

=

I_re(rx)trcXcf

=

,I

aculs ,f,es ,7

=

r_rrJrr = nr acuor{A\

ruorJ srr\ollo}

'r(,r,x)r,rrx = ,I le$ uorl"Iar lsBI stql .Z(JX)rf,rX osIB eculs 1I 'JXTJX = = tJ, tI fIr"IIrEts pu?

xrx' rexzJ = xJX. zxr = XJrXr = rx .X ' ,xrrrx = x erxzxr = XrJrXr = XZI' ,I,'XI'I = X'JX' ZXZL = rJ-rXrJ. = rJrXrJ

,(x,r,)rxr.r,

=

3ugd1du1

'rx sawoceq

I'ZI l"rtl

os 'r_X dq

11

=

J,rxr

=

xrrx

z.zl

eaeldar ol lualualuoc 1l spug aH

'(n = Z 8u1pas

sl

'rr

,{q paallap) dno.r8 lrrpaq"soc-r

Io dno.r8.ro1ceJ I?r4lrl-uou dluo aq1 seareqrr ,g.rap.ro ;o dno.r8 rolce; E s€r{ I'ZI dq paulyap dno.r5 aq1 1eq1 Eulnoqs ,1 = ol

aql


'22

rI

an;r '.rog .dno.r8 leJpaq"socr ^d"reu1q aql aur; lou op I'zI suollBlar aql l"ql pa^rasqo sBr.I acl

ecnpar do,D'r_I = d\las -ap

01 ac-rlJns

"png

'S = lt\J

osle dlsnor,rqo pu€

,At = IZA\J

'rJ

= ,tArJ1tl

t'zt snqr

=

xT6x-1Tx6T-1 = (xr3x-1)2(rfr-1)2

= $2x-2rzrx2r-212 =

1.

Interchanging T and X, we have similarly (rx-1)323 = 1. Thus

z Defining

S=

=

(xr-\-3 = 11;-1;3 = z-3.

TX-1, so that (TS)2 = (r21-1rZ = Z, we have

15=(sr)z=g,

53 = 23,

The further substitution T =

26=L.

t

T Z' yields T!,5

= z-9

-3. , (st')z = z

Thus

another presentation for the same group is s3 =

r'5 = lsy'12 = g3,

sz =

zs,

TZ =

zr.

1st')2 = z' impry z'2 = 1, it is no longer necessary to mention Z6 = L.) This is the group that was denoted elsewhere by <2, g, 5> [9, pp. ?0-?1]. It is clearly the direct product of (since the relations s3 =

1'5

=

t-}

cvclic s"oun {22 Thus 12.1 and 12.2 are symmetrical presentations for the group

the binary icosahedral gto.tp

(2, 3, 5) x

{s,

ana the

}

.

G3,

of order 360.

References 0.

S. A.

Amitsur, Finite subgroups of division rings, Trans. Amer.

1.

$;i;

Y;f,l;:',ftk

2.

Math.soc.So(lffi

'

ralizations

- The binarv polvhedr

ot the quaternlon g!o!rp, ljuKe

, Quaternions and reflections, Amer. Math. Monthly, 53(1946),

!..

Fp.T36-j+e

.

t

Regular PolytoPes. London, 1948.

q.

Extreme forms, Canad. J. Math. 3(1951), pp. 391-441.

6.

-

rboli

-, 86

Proc. Internat.

,.8

'rua.o..'oc

tl,:8"'.T".9 '9I

.Rrr-r

.aa ,(sonr)ET .sdqa .qren .r{Sreugnr ,Edrrd-{FrJle75!ffi8fr JaIBUO!suaIIIrprqau ualu"-rr"Arll uaqcsrboloctol arp raqn .azlerl .H

'rl

'dd lg

saurn?l{ uaqcs cllpue aqc!aJ

rap

IbOI

'7961 '31zdya1

,

'1.ray1ag

'ffi'1e;1a.rq;

uaur

rnurluo{s!q

'H pue II"JIarqJ '111

pue

lra]las

a

rap uollcnpau alp

uaqcs!

.H

'zI

'qlell'f

'j

larp llw uarrrJoJ

,, '1a1qcr.rrq'1

'g

'0r

'6

'(' J - N)'q1e1a1'qa8.r g'Fil[-o-.r-5-5]5"raffi 's 'H '8 'rasolN 'r 'o '/I\ puB ralaxoc

'n

JoJ suollBlar p

'dd'(SSOf)f

'tr

,Jlx{,;18?;0"{$ .II

'gI6I 'uopyo.I '(tggl 'SIzdlaT ' ua8unselro1 Io uorlelsrre.r;) uo.rpaqesoJr ar{l uo Sarntcaf 'ulaDl 'ina3ue

'71t

'peueC

ffi

,___El:gl

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APPLICATIONS OF GROUP CHARACTERS by Michio Suzuki

We shall describe a method of applying the theory of group charac-

ters to a partlcular type of problem in the theory of finite groups. The particular type of problem which we are interested in is the following one: To what extent and how does the knowledge ol a subgroup H deter-

mine the structure of a finite group G which contains

H ? The

theory of

group characters provides a method to attack this line of investigation. Suppose that we are in the most favorable situation and know everything

concerning the subgroup H. Then we can determine

all the irreducible

characters of H. From the knowledge of these characters of H we can get information on irreducible characters and finally on the structure of G. Our main concern is how to get this information. Essentially, this is done by applying the Frobenius

reciprocity law. We shall however give

a

modified form tn the second section, which may be more convenient ln

particular cases. The first section is devoted to the discussion of basic material necessary for the later development, and in the last two sections we shall give a few applications of our method, including a proof of the following theorem recently proved by R. Brauer and myself: a 2Sylow subgroup of a simple group of even order cannot be a generalized quaternion group.

1.

We conslder a

finite group G of order g. Let X!, .

.. ,

X

n

be the

totality of distinct irreducible characters. We shall restrict our consideration to the classical case of Frobenius characters so that the number of distinct irreducible charactere is equal to the number of distfnct conjugate classes of G. A (generalized) character 0 is a linear combination 88

68

ra5u"p ou sr araql Jl (C)W dq,{1dru1s .ro'(C)DW,{q palouap sr O uo qsrut^ qclqa DIAI Jo sluaurala yo,{111e1o1 aqJ'D u} C Jo luerualduoc aq1 3 dq alou Jo auo yo u1oI r s1 -ap ar! tC las pasolc qc"a JoJ 'sass"Ic q?ns IEJaAas

pall"c st C Jo C lasqns V 'sassrlr acualzalnba olur sluauala dno.r8 ;o uor111.red sacnpu! aloqt paurJap uorlelar acualennba aqtr,

C JI pasola

"

pue (ur poLtr)

t

.I = (u,)t)

=

{

leql qcns

{

la8a1u1 u€ sr araql uaql

'u/ru pu€ I

=

ar€ u'rtt'T JI l€ql slrass? qclr{a\ €ur -rual lBcllaurwlr" aq? pasn a^"q a,t;oo"rd aq1 ;o lted rallBl aql uI'H Jo rol -e.rauaE e o1 a1e8n[uoc sr , lsr{l suBaur qJlr{ltt''0 #(t) *0 al€q a& acuag (r;u'tr)

'*0

y1

pue s.ra8a1ur I€uorlBJ aarr{l

Jo uollrurJap eq1 o1

d;e;1uoc 0 = *, lEql su"au

slql 'I

= (3,{) qll,tr )t

)*B aruaH'0 = (d)*d uolllurJap,tquoql .0 =(o) *g ;r pu" d o1 luapllnba src II 'H Jo .role.raue8 auos 01 e1e8n[uoc 1ou sl r Jl 0 = (") *0 raaad\oH '0 f *0 pue DW Jo luawela u" sr d urorJ paonpu! *A relc"reqc eq;,'H Jo / rolerauaS-uou due .ro; 0 = (r) d pue 1 fi1c11dlllnu qlra H Jo relc"r"qc l"I^lrl aql sur€luoc d 1"r{l qcns HW ul d luaurala u€ sr araql uaqJ' d,{q pale.raua3 g i, } = dno.r3 -qns aql Japlsuoc eA! asJaluoc aq1 aao.rd ol rapro uI 'd 01 lualennbe sr o uaql'I = (3')t) qlle {d o1 ale8nluoc s! , ]l 1"q1 snopqo d1.r1ey s1 11 IIB JoJ0 = (oa

'd -uoc

ol ruarearnba s1

,

Jr

sI o p fluo pu" JI d

8uytro11o; aql alrel{

acuaH'I o1

=

(s';ril:t::t;:"::::;'"t;r:r:::

lualealnba

su

luawala

uy

an dlaueu lasec aql ralat\oq sI sIqI 'uoll"Iar acual

-earnba u" saurlap,{11rn1ce uorl"Iar aloq" aql

0 = (d

)/

sagldrul

y,g

r

p

l"r0

snorAqo lou

roJ 0 =

sluaurala o^\1 IIBo

atl'u {uer yo (dnor8 u€Ilaq€

l"ql rtalc s! u 'O Jo sralc"rtqc I=I 'u'3

=

d

yo d111t1o1

(,

sl

1I

)p

tuatei\in65 C

Io

aa.ry) blnporu E

sI

lf

pu",

'1 rua.roaqg,

aql alouap Oy{ =

d IAI

w 1"T 'l x I"

:slualclJlaoc 1rr5a1u1 l?uoll"r qlrd\ sralc€r€qc alqrcnporJ!

Jo

of confusion. Clearly the set M(C) is a submodule of M which is in fact a direct summand of M.

Lemma

1.

There

exists , e M(C) such that d takes a constant

value c on C, where c is a rational integer dividing the group order g.

This follows from the fact

that laeC x (a) is rational for any ir-

reducible character X of G and

foi

any closed set C.

As a corollary we get Lemma

2. If C, and Crare disjoint

closed sets of G, then the index

tM(c1 U Cr):M(Cr) u M(Cz)l is finite. Let r(M(C)) be the rank of M(C). Then from Lemma 2 we get

r(M(C1u . . . . . ucs)) = x i 3 1r(rvr(Ci)) if c1,,

cs are mutually

disjoint closed sets of G. Theorem

2. Let C be a closed set. If C consists

classes, then r(M(C) ) =

of

t conjugate

1.

Let 0y. .. , 0r be linearly independent elements of M(C). Then each 0 i is a class function taking values in the field F of the g-th roots of unity. Suppose we have a relation 2a, 0 i = 0 with a, e F. Taking a

base

rationals we can express di as linear combina-

{.0} ot F over tions :k aik ,k with rational

Let X be an irreducible character of G. If aO is the multiplicity of X in t , I t aro d rr then loi di= f ( Xa,O 0 i),ycontatns Xwiththemultiplicity I"k,k. 0. Since allthe Since t i1 r "i di= 0, we conclude that xk coefficients atO.

"k'k=

ak are rational numbers, we must have aO - 0. SinCe X was any irreduc-

ible characterr rve conclude that

!i I t *it di = 0. From the definition,

the d, are linearly independent over rationals, whence aik = 0 for all i 0rare andk.Thereforeweget oi=0 (i= 1r2,...,r)'Thus d1', linearly independent over F andhence r < t. If C1, . . , C" are the totality of closed sets of G which cannot be expressed as a join of two dis90

I6

*/' "'rlp

uolleulquoc r€aull

1o

sluawala luepuadapul

" "l

f1.reaug1

ld'' " "'l

ul

I are oraql 'g ;o sasselc

slslsuor (I II 'H ]o o las posolc -.r1

Id qces'(q)Hw

raprsuoc 'H

a1e3n[uoc 1 ;o

sralcsr"qc alqlcnpar

]o " ' rlf tat'gyodno.rsqns"aqHlaT

pd111e1olaWaq*4 ''

0

'(9)w

'Z

= (c)Ur tew aas o1 fsza

lr'((O)W)r = ((O)IS)r snqJ,'l - u = ((C)w).r - u F ((C)rS)r aauaH '0 = (C)W U (C)r,JpuBqraqloaqruO't - u =((g)r,r).r < ((C)rJI)ruaqJ, 'sass"Ic aqeEnfuoc I Jo lslsuoc C laT '(C)ry;o 1.red E sl (0)W d1rea13 &ou sr

'(O)w = 131t9 '8 tuaroaqr

'(C)w' f ue roJ0= C

<

,1.

' f, >

1"q1 qans W Jo

4

slua{uala;o

d111t1o1

oql's! trrn'(C)W

Jo

an'3 1as pasolc fue.rog .!ql'I=!3=C<,l,rf >

luarualdruoa 1euo6oq1.ro aql (C)Id dq alouap

p1ald suoglela.r d11puo3oqlro aql uaql

!x !q 3 =4, putlx !" g

=+

}I '0 = + ]l fluo pu" }I sploq o

<

(

f

)Crn

aql pu"

( /)D,n

uaql 'C < -ap sl

d111enba

#'

+>

s"

paulJ

u 'Cyt ra^o l?uolleun; rBeuIM z sg lcnpo.rd rauul slql uaql

'(r-.)i (')f c"3 @/tl = c dq lcnpo.rd Jeuul uB aulJap an Ctrq

'z

"ruuarl

u1

urorl srl\ollo} as"c le.rauaE eqJ, 'slas pasola 1untu!

-ulru aql JoJ Z r.ualoaq;, salo.rd s1q; 'u51s

d111enba

aql

aABI{

lsnru

arrr acual{

.u=Il 3 > ((!c)w)r< 1aB

'((!c)rnr)r!< =(("cn

a.r tsasselc ale8nfuoc ]1 yo slslsuoc IC

n I3)nf)r

=u

;1

ua{l 's1as pasolc dldua-uou 1u1o[

with integral coefficients: o

L=

2flr "'.

dr..

Let A be the matrix (arO) of those coefficients. The induced characters 0

|

are elements of M6i let the decomposition be

,l= . b. x I >n ts=r rF I The coefficient matrix g =

(blr ) is an integral matrlx. If we know the matrices A and B, then the values of the irreducible characters Xp on the set D can be obtained. In fact we have the following theorem.

4. Let X = (xO u ) be a solution of the matrix equatlon B = AX with integer" *k, Then for any element p of D we have Theorem

xn(n)=:[rxon

ds( r).

In order to prove this result consider the.element

x= xplH- t[r*ku where

d1,

Xp lH indicates the restriction

of Xn to H. Clearly

that f . Mrf(D). By Theorem that ,i . f4H(D). We have

We want to show

the proposition

3, this

,,l,

.

MH.

is equivalent to

<*, oi>H= .*nlH- :[r ttu 6k, oi> H = <x nln, ei >H - : [t *r." < d6, di> g. By the Frobenius reciprocity law we get

.*nlH, di >H= .Xp,

di*t G=birr. 0' since 01,'', Hence <*, 0i > H = bi* :[t "il.*k,, = iinearly independent, the above equations imply that P . ftH(D) Corollary. the

Suppose

values x* (n) for

grare .

that { r r, . , 0 t} i" basis of Mn(D). " p

€

Then

D are completely determinedby the matrices

A and B. In fact we can find two unimodular matrices S and T such that SAT 92

86

-sod Jo raqrrnu allul; 1r.r5a1u1

dluo a.re areql pu? (g'og) ur.roy eql s€q g ,x1t1eru

"

u" sI g aculs 'tr

Jo

xlrl€tu pasodsue.rl aql sal"alpu! Ll araq,r

g.g 'v.v {t = uorlrnba xuletu aql sp1a1d

suoll"Iar dlllruo8oqpo

qlld\ paulquoc

aq1

a^oq€ ar{J,

"urtual ' 4 > 1a3 ain,(q)w,

'uoluass" rno pu" 0 =H < f / rg aculg'(C)Httt ,4 aJaqAt i + 0 = Hl*d tzqt aas ad\ eurrual dg

'H.g'Hl*at 1eq1 sa11dtu1 ,ne1

=C

a*,!r'*0>

firco.rdgcar snluaqord aql lcEJ uI

'H.+r0,

=C<*f ,*d>

uaql'1q;Hry ,

9 '6i II 'tEruwa-I

aao.rd arn

'gt d

.roy

(d

s1q1 Eu1s61

"uuroT

)d = (d )*, uoqt'(C)H4 ,p JI .t "ruua-I a^?q aA\ (z) pue

-s" aql rapun l"ql s^\oqs sralc"reqc 'H uI paur"luoc sI D ul

pacnpur aql Io uollelndruoc aql

d Jo Jazll"rluac aq1 ueql '6 r 6 il

ale8nfuoc a.re

u1 a1e3n[uoc

ar€

6J

Jo

uaql

(Z)

tg

ur

sluauala o,[l

11

pue

daql uaql rg

(I) suollduns

(I) :suoIl

-lpuoc leuolllppu on1

uoll"Ial -uroJ dJaA

aruos pulJ

"

3u1rno11o]

utc

ad\ asec

aql satJsll"s g les eql l€r{1 asoddng

'g pu" v

uaa&laq

ltrcads e ug'.raaa,tog'rauueur

pa1ec11d

ul C Jo arnlcnJls aql uo spuadap g xrrltur

'xy = s uollenba

aq1 le.raua8 u1

ar{l Jo uollnlos € s}

("or)

xlrl"u

l

=

eql uaqJ '1 aa.rSap Jo xrJl"ru llun aql s!

x

I araqar (O'f) =

sibilities for Bo if the matrix A is known. A particular case of Lemma 4 gives the relation wO(

for

any

de Mg(D).

ff d is a linear

A

*) = wH(

combination of two or three

d)

irre-

ducible characters with multiplicities 11, then d* has a similar form.

3. If the order of a finite group is even we have a method to investigate the structure, which is not available otherwise and which has been

proved to be quite powerful in many cases. This method was

first

ap-

plied by L. Red6i in 1950 and shortly afterwards, independently, by the author and by R. Brauer and K. A. Fowler. The method depends on the

If z and r are two involutions of a finite group G, , then the subgroup {, , } Senerated by the elements r and z is a dihedral group. Let p be the product r n of r and r. Then we have a Pr -t '= p -t '. Hence the involutions r and z are contained in the

following remark.

normalizer of the subgro"p

{. n

senerated'by p .

}

Brauer and Fowler have stated the above idea in the language of group rings. Let T be the sum of all the involutions of G in the group

ring of G over a field. The sum K of elements in a conjugate class of

is in the center of the group ring and the elements K1,

.

G

,Kn corre-

sponding to n distinct classes form a basis of the center. In particular

t=Kt+...+K, where Now

Kl, . . . K, are all the conjugate classes consisting of involutions. ,

T'can

be expressed as a llnear combination of K1,

. . , , Kn with

non-negative integral coefficients:

^2 = -n Ki. Tx i=r "i The coefficient c, indicates how often a fixed element p of K, can be expressed as products of involutions

r and o . This number coincides

with or is one less t}ran the number of involutions in G which translorm P into its inverse o -1. Th.r" if we know the structure of the normalizer 94

96

aql qlFr paurquoc a^oq€ paqrJrsap poqlaur pue In7 v'n[v

aqtr uaqJ, 'i{1aa11cadsa

"

zn

Tv

aq N ul " puzl t ' ! Jo srazll€rluac aq1 Jo srepro aql laT 'r{ }o rosr^rp sl r uaqJ, ' T z aq s }o razllErluac aql Jo rapro aql laT " '(Z poru) I = q 'qg aq N Jo rapro ar{l lal pu" t, uollnlonur aql Jo N raztlerl -uac aql raprsuoC'C ur raqlo qcta 01 ale8nfuoc aJ? r pu? Z, ,l r leql Z

luatualels aql ol lualelrnba sg Z xapul ;o dno.rSqns I€rurou

acualslxa

" ]o Z, = luawala

sr ul " ,"} 'Ir t_o = ul",{t,

-uou aql uo uorldtunss" aq;, 'uollnlolu! uE osIE

=t ;I,

-ur raqlou€ pue

sl

r

Zu

r

aqJ

uorlnloa

dq pale;aua8 sg g dno.r8 aqtr, 'S Jo

luawala

ar{1

l€ql qcns'? JapJo Jo

r

uotlnlolul l"rluac aql Juaurala u" sur"luoc S.Z

xapu! Jo dno.rSqns Isturou ou suleluoJ C 1eql pu€ 'g rapro ]o dno.r8 p.rpaq

-Ip

"

st s lBql aulnss€ a/t\ 'g ;o dno.rSqns noldg-z

;o dno.rE

larl 'rapro ua^a allulJ B eq D 1e.I 'asec lercads E raplsuoa ain alduexa ue sy

"

aq s

'pasn aq u?c sralaEreqa Io anl€ll aql alndruoc ol uorlcas snorla.rd at11 ut paqtrcsap poqlaw aql uorlcauuoc slql uI 'E .rap.ro dno,r5 aql roJ EInuroJ saa.rEep

tu,nou4

""e

r

1aB o1 alqrssod

snolrtA Euowe uorlelar

"

sl

1I

uallo 'F pue ( d x )tA

saqsllqBlsa uoglrnba a^oqt aql uaql

!a pu" sralc?r"qa alqlcnparl

II"

roJ ( d ;

/

1

sanlea aq1

'r

JO

11

Jazll"Jluac

arfi Jo rapro aq1 Eulaq ( r )u pu" suorlnlolu! ;o 3u11s1suoc sass"Ic a1e8n[uoc

uro"rl

|r,

{ }

saAll"luosaJda.r yo las E

ralo uaqel 3u1aq araq uorJer,uuns

,(')u/(,;/x'3

aq}

=1d1;Ca

pur's.ra1ce.r€qc alqrcnparJl tC dno.rE eql ]o rapro aq1 s1 3 araqa aql II€ raao saEus.r uoll"ruruns aql

(rx)Bq/(d)dxr{dx)Dr al€q ad\

dK g=Ic

(*)

'!r

uorssa.rdxa ue p1a1d sralcerBqc dno.r8 ;o suollelar "o1 d111euoEoq1.ro aql puer{ raqlo aql uo 'lc anl€^ aql &ou{ a& u"qr ' lo

{, }

computation method in the precedihg section enables us to obtain the fol-

lowing result. The order g of G has the form g = o4hu2(ut*,rr)2i{i

+, )/(t -,

)2

f is the degree of an irreductble character X and f = e (mod 8) with e = * 1. Actually we can say a little more. The group G has 5 distinguished irreducible characters Xo,X1, , . , ,X4, of which Xo is the principal (trivial) character and X, is the character X mentioned before. Let the degree of Xt be ft: fo = 1, f1 = f. Among those degrees we have where

two linear relations:

f4=f* e=fZ+ €'f3 where e ' is either 1or -1. As a matter of fact the equations

x4(p) = x( p) + . - x2( P)1 .' X3(r) are true for all 2-regular elements p of G. For the 2-singular elements the values of those characters can be computed. Putting

3-

e

e', we get

X1 = e 'e

x'=

t

-e ' x3 = 6 , -5

x4= o'-2'' first number in each row stands for the value on elements of order divisible by 4 and the second one for the value on

In the above equations the

elements of order divisible by exactly 2. The degrees are connected by another relation: g = 256uh2lt

rfr/l (t+.)(fr-.)(fB-

s ).

If 6 = -1, then it is easy to show that rL2-_+t3-_(f.+,)/2. |

Moreover we have the

equality.l

= u2.

If 6 = 1, then those

equalities

are in general not true. As examples we mention the groups LF(2,q) with

1,6

H q d 'n+> ((^+)Ba/.('+ D

r

. *o ' xu

1a3

-sn'0=H < 0',1,>

an

aABr{adr(C)Ey,f

/r( Q)Ht') '< q-(Hl ^

"*

)((

/i a

(*,r)

,ne1 r{11co.rdrcar snluaqo.rg aq1 Bu1

,B

due.ro1 aruaH.CluosoqsluBA ^

d

)Hl) '.q=

/ 11'*)Bitz1 1;c1;

Q(('

tx)cl) x)Bcr/z(

+ )8o.

d3 3 =

uolssa.rdxa aW

,l

l"ql su"au slq;.

'H Jo sJalc"reqc alqrcnparrl aql II€ JaAo saSue.r uolJeruuns aql alaqa\

'(^f )ao11a1"4 r{'+)H,r, niq=t" 'd

.roJ anIEA aurgs aql seq (1) o1 .re11ru1s H roJ

"InruroJ

aql acuaH

.I_d olut d ruro} -sutJl r{clq& c Jo suo}lnlo^ul }o raqunu aql s! 1 + lc .to lc uaql'lc .ro; .roJ (i.) BlnruroJ aql allrlrr a,t tl1 uI sI g Jo d luelua1e ue Bursoddng 'H ul ar" suorlnlolul r{cns IIE (g) uol11puoc aql dg

sI r uaql JlpuBq rd J!(z) 'H uI

'cJo, uollnlo^uraluosroJl_d =l_tdt fg;o

lasqns pasolc

"

sl C (I)

:suolllpuoc 3urno11o; aq1,{yslqes

qclq,[ H ]o q lasqns E pu" D 1o g dno.rSqns B s! araql lEr.Il aulnssv 'd11nc1yg1p

ran"rg

Jo

str{l Jo p1.r

1a3 01 Z

uollros }o poqlatu aql pu" rald,od pu"

poqlau aql aulquoa II€qs adyalnduoc ol llnclJJrp are sral

'lx uI d luaruala aql uo (a ) d x sanl"A sJalc"r"qc Jo sural ur !c Sulssa.rdxa (*) elnurol aq;, 'V

-c"r€qc Jo sonl€^ esaql dllensn aql sa,rlolur

'aJar{ passncsrp aq lou Jaaall\oq

seq g dno"r8qns

lltfi\ qatq&'sl1nsa.r J"Irurs ur"lqo ar\ pu".rap.ro.ra8.re1 rnoldg-g arg uaqa sasec uI s{JoA! IIIIs poqlaru rng

'tt

zt

1uo dno.r3 3ur1eu.ra11r aq;, 'Zn = In pu" = ZJ sdenle aa€q ar\ as€c stql uI '(gI potu) r F b lnq (g poru) r = b

1tq1 aldruexa

tj

f

u" sapllord

s.ra11a1

forany B.Mr(D). If the subgroup H and the subset D satisfy the assumptions (1) and (2) of the section 2 we can apply the results of that section. In particular

if

MH(D) has a character g with

smallwt( a),

then w6( d*)

is also

small. Hence in the above formula the summations on both sides contain only a small number of terms. We shall therefore get a relation between characters of G and those ol H, which is fairly easy tohandle. As an illustration we consider a special case. Let G be a group of even order and let s be one of

its 2-sylow subgroups. we assume that s

is a generalized quaternion group of order

clic subgroup z

of.

2

16. Then S contains a cy-

index 2. since the order of Z is at least equal lo 8, z

contains a unique subgroup P of order 4. Let H be the normalizer of P

in G. It is not too difficult to show that H contains a normal subgroup M

suchthatM

n

S= eandMS= H. LetUbethe subgroupof H suchthat

V -- MZ. S contains only one involution which we denote group V generated by M and

r is a normal subgroup

by '.

The sub-

of H. Let D be the

set of elements of U not in V. It is easy to verify all the conditions of this and the second sections Since

degree 2.

for H and D.

H/M 3 S, all non-Iinear irreducible characters otH/M are ol

Let

d be one of the non-linear irreducible characters of H,/V.

We have at least one such character since H,/V

induced character from a linear character

outside of U.

Let I o be the principal

trivial character ot H/V.

of"

) 8' { is an

U/Y and hence vanishes

character of H and 'l 1 the non-

Then the character

0 = no+ ,1 1-

is of order

d

defined by

Q

w"( d) = 3, we have wO( ar) = 3. Hence d* is a linear combination of three irreducible characters of G with multiplicities + 1. One of those characters must be the principal character 1 of G and

belongs to MH(D). Since

the multiplicity is 1. Hence we have 98

66

sIoutItI Jo dllsre^Iun 'Z JapJo ]o luauala auo rtluo suleluoc N,/C dno.r8 Jolc"J

pu" rapro ppo Jo g dno.r8qns l€Iurou pazrle.rauaS €

st C dno.r3 a11u1;

aq1

suleluoa C uaql 'dno.r8 uolu.ralenb e;o dno.r8qns rttoldg-g e y1 'tua.roaql

"

'uraroaql

Eu1no11oy aq1 1aB ain

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uaar8 oq lou

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lou s!

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'J=x

Jo

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"

ral]V

t o=(-7+*-**tr z" ^' z s€ spear mou (**)

"InuroJ 3g

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uo saqslu"A

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=

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aqJ,

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aculs

ON MAXIMAL SUBGROUPS by W. E. Deskins

Consideration of the question, What do certain intrinsic properties

of a maximal subgroup H of a finite group G imply about

G

? has led to a

number of interesting results. For example, G is a solvable group

if

(i)

H is abelian (Herstein4), (ii) H is nilpotent with regular (in the sense of

P. Hall) Sylow subgroups (HuppertS), (iii) H is nilpotent with Sylow subgroups of class 2 (Deskinsl),

or (iv) H is nilpotent

and of odd order

lTho*pson6). So one might expect G to have some nice properties when H is supersolvable. Following this line of thought we prove in section 3 that when H and certain other subgroups of G are supersolvable, G is solvable (3.3). In proving this another variety of problem involving maximal subgroups was encountered, the investigation of which should be useful in handling the question mentioned above.

If

.1 is afamily of maximal sub-

groups M of G determined by some "external" relationship between M

is the nature of. S(1), the intersection of all the Mof. 7? These generalized Frattini subgroups will be considered for several different 7's, families selected according to the index of M or variations and G, what

thereon.

family I is through consideration of the indices of the maximal subgroups of G. So, for a fixed

1. Index. A rather

natural way to select a

rational prime p define g = 3p to be the set of all those maximal subgroups M of G with (p,[G:M]) = 1,[G:MJ denoting the index of M in G.

is a metanilpotent normal subgroup of G' 1.1. 6 (7 pp ^) = d '. (A metanilpotent group is an extension of a nilpotent group by a nil100

IOI

'H Jo dnor8qns_D l?rrrlxeru aql aq X lal pu"'TW = g,{1.rado.rd aql qlllrr O }o.I sdno.r8qns I€rurou Jo ss"Ic aq1 yo (laq1o ou ur paur€Juoc auo)

uollou raulJ B roJ paau e s1sa33ns

luatuala l?lururlu € aq H laT .xapur

srql

aql s" anJl lou sI asraluoa aql'aru!.rd e ;o dno.r8qns

l"wlx"tu

Jo

's^\oqs ggI rapJo yo dno.r8 aldruls

"

Jo ranrod e sr dno.r8 aIqEAIos

d.raaa yo xapur aql a1rq11 'xapu1 1Eru.ro1q .g

'(g)o uo uorlcnpur dq paao.rd aq .t€tu sllnsar asaql

'5 aIq"AIos

rrurou

"

sl 'd,

uaql'dnoJ5 alqr^Ios-, !

1o

AnoT5qns

estcu's'r

'(slsrxa W qcns ou JI C = ' ,i,) r ul sarulJd 11e o1 aur.rd.{1an1e1a.r [f,I:D] qll,t\ D Jo Itr sdno.t8qns l"rurxetu aql II€ u Jo uollaasJalul aql alouap Q l"l pue',a u! r Jo saul.rd aql Jo auo lnq IIB ro

II"

q11n

saul,rd Jo slas o,ttl aq i! pue

,

1a1

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a.ro141

'g -qns aIq"AIos l"urrou 3 s! b'd

p uaql-'dnorg

aICIE^Ios-d € sr

yo Tn:o"rll

c lt .l.r 'sdnorB

artros roJ d1111qrl1os troqs 01 alqgssod s! Jr rola.uoH .salsrlsuoruap .g = 6

put

I lsEal l"

=d

qtlrn'gg dno:8 cr.rlaruuds aql sE os 1ou sI slr{l 1nq,a1qe,r1os

aq plno,r,

b'dp 1rql

adoq lqigru auo

'w

asaql Jo uorlcesraru! aql

alouap b'dg l"I pu"'I = ([W:D]'b) = (tW:gl'd) q1,1 C;o ry sdno.r8qns l"ru -lx€u yo dllrury aql alouap b'dg 1"I'b puz d sarur.rd;o rled aql rod

'd'p aroqrr\ 1u-aJralodltu

r,

st C .g.i

1,/C

st

.c@,(c)P

=b+ u d+ bpu"dn'Z't

uaql'(g)o '(C)o'C 3urno11o;

Jo

aq;

rapro aql uo uollcnpur

^{q

paulrlqo aq,{eu salr"Iloroc

'dno.r8qns tur11"rJ aq1 ;o .{cua1od11u aq1 8u1ao.rd ur pasn

leql ol snoSoleue rauu?ru z

u1 paao.rd aq deur

llnsar srq; ('dno.r8

1ua1od

2.1. The number n = o(H/K) is unique for M. Moreover. if [G:M] is a power of a prime. then H and K are also unique.

This may be proved by using induction on o(G) and by investigating the centralizer ol H in G. Hereafter n will be referred to as the normal index of M in G. We

list

some results concerning n.

2.2. G is solvable if

and only

il

every maximal subgroup of G has

prime power normal index.

2.3. The intersection of those maximal subgroups of G whose normal indices are composite is precisely the maximal normal solvable subgroup of G.

2.4. The intersection J of those maximal subgroups of G whose normal indices are prime powers is a subgroup with the property that the socleof

G/Jis@.

2.5. The index is iJ and only

if

a

divisor of the normal iqdq4,4t4! G is solvable

the index and the normal index are equal

for each maximal

subgroup of G.

To prove the sufficiency of the last statement, proceed by induction

If G is simple then the identity element forms the only maximal subgroup of G, so G is abelian. So let L be a minimal normal subgroup

on o(G).

of G; then G/L is solvable by the induction hypothesis' Let p be a prime

divisor of o(f,). Then any maximal subgroup of G which does not contain L has index divisible by p. Therefore L 9 do and hence is solvable. so G

is

an extension of a solvable group by a solvable group and hence is

solvable itself. Proceeding in a direction suggested by one of the characterizations of the hypercenter, we consider the family of all those maximal subgroups for which the previously definedg/r is not in Z(G/K), the center

t02

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-radns sl. qcr.qa g

qlt,n

c

(H)

"

uaql'a1durls sI D II

q73

.1er.rr-r.r1uou

HUC

n ut (q'3) m g clno-rB

@

aq1

.g.g

;1

uollcnpoJl

-u! arn ur pauorluaut llnsal

aq1

;o ;oo.rd B al€crpur pue alels a& 6[oN 'srazIIErluac

snolr€^ Eur.raprsuoc pu€ uorlcnpul 3u1sn dq paao.rd aq deu asaql

'c @a/cqtyrr(q'3)@ (6) p qcgqrn .ro1 1,{

'g'g

'u"11aq€

c/C

qr./r'

(q'C) @

(W)tr

c 'I'g

'}t

1o xaliluloc xapul ,W .roy .r1ecl-tu-r-1afiftuod tas aql pu" '1,11

aql pall"c sr s.ned qcns E

palpc s! (C 'C)

.r1ed

II" Io (W) F oql '3 yo dno.r8qns-C lerurxetu aql

,c

Jo

lal

{w,r} , "qf {w,C}

(q

T dno.riqns-g due .ro; nr =

'C =,IAI pue 3 dqpalz.raua8 1"ql qcns g

aq O

go dno.r8qns E aq C

dno.rSqns

lal

pu€

(e

D yo 1q dno.rSqns l"urlx€ru

B

roJ 'x/H luaralJlp aql Jo sallradold aq1 raplsuoc II"qs ad\ (x/g)o Jaqrnnu aql Jo p"alsur tog'1so1 sl (;4/H)o;o ssauanblun aql pa^otua.r s1 uorlcas Eur -paca.rd aql ul H uo luaurarlnba.r flrleu.rou aql y1 'xa1cffio3r xapul 'g

'( (c)

+

7912 fiaslcalils1-sdnol5 @) a

'zlnqcsBc u

01 anp

llnsar

"

/D

urg'z

ol sp"al slqJ, 'y/D

Jo

needs careful scrutiny

is ttre one arising

when G has a unique minimal

normal subgroupKrvithK n H= 1. So let Mbe amaximal subgroup of G not containing K, Then M

is supersolvable,

and we shall show that M

is conjugate to H. As a supersolvable group, M is an extension of a nilpotent group by an abelian group, as is H. So, if H is not abelia.n, there exists a prime p such that both H and M contain normal p-subgroups. Furthermore, the

following simple lemma shows that a p-Sylow subgroup S of H is a pSylow subgroup of G.

Lemma. If P is a p-group expressible as PtP, with P, normal in P and

P, 0

P2 = 1, then Z(PZ)

n

Z@)

+ l.

Therefore M contains a p-Sylow subgroup St of G, and the centers

Z ardZ, of S and Srr resPectively, are normal subgroups of H and M, respectively. So there exists x e G such that x Z *-L = Zl, and consideration of the normalizers of.Z and Zt leads to the conclusion that xH

x-l= u. Then,

if q is a prime divisor of [G:H], it follows simply that K 9

dO,

which is metanilpotent by 1.1. Therefore G is an extension of a solvable group K by a solvable group H and hence

is itself solvable.

References 1.

Deskins. W. E.. An extension of a theorem of Herstein, Abstract 548-

2.

Gaschiitz, W.,' Ueber die d -Gruppe der endlicher Gruppen, Math. z e-iilT3_(1 e 5 3I-IforI?O.

HalI, M., The Theory of Groups, New York, 1959. 4. Herstein. I. N.. A remark on finite groups, Proc. Amer. Math. Soc.

3.

ffi

5. Huppert,B.,Nor@

6.

Thompson, J., Proof of a

endlicher

conjectut 104

, these Proceedings.

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all

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J ralo

aqJ,

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1 e.rqa81e alT reaull aql urroJ uaql a^\ ,{1pcg1uap1

qslue^ lou saop (1) ru.roy aql qclrl^\ uo V Jo S lasqns € uror]

3ug1re1g

'alqlrnparrl .4lalnlosqe sI C uar{^\ ''a''l 'J.reao aldrurs I€rluac sl H ;r 'aaue1su1 ,ro; 'asec at11 oq IIIA slqJ 'g uo f11rc11uapr qsrrrBl lou saop (qe) .tr = (q'e)

(r)

ac"rl aql lBql aurnssB a1d 'c Jo sluatuala oql Jo J ralo suoll€urquoc r?aurl IIE Jo tos aql'sI leql'C Jo IInq r€ourl aql aq H ruroJ rBaulM

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Ig] roqlne

aq1

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lpl^\ cpslralcereqr arur.rd;o serqa8le Jo

aq1 Suru.raauoc

1o e.rqa8

pu" sdno.rS

all

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dq

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'p

nn automorphism of

L/Lo which we denote Uy 0 g. Then g -+- d, is a

homomorphism of the given linear group G into the automorphism group

Aof.L/Lo,

and the structure of A, at least in principle, is known from

what was said above.(3)

Ho*"u"r, in general it may

happen

that

d

(G) is

"too small," so that the knowledge obtained does not give sufficient insight. Significant results may be expected

if

d

(G) can be proved to be

the restriction to F of a certain algebraic subgroup of A. For this purpose the choice of the set S has to be made in a special way.

To give an example let us assume that G is generated by a set S of

*rt"i"""(4) ("eF)

exp("X)

(2)

where each X satisfies

(3)

XP = 0.

In this event the matrices (2), for any fixed X, form a subgroup U of exponent p of G that is the restriction to F of a one parameter algebraic

e.(5) wtr"r, can we conclude that the group G that is generated by the re-

strictions to F of the algebraic groups gAg-1 (g . G) is the restriction to F of the algebraic group B generated by all gAg-1? This problem in general is not solved. However, let us make the additional assumption that for any matrix

in (2) we have (4)

for Ye

xpYxp-ts

=0

L, t'=1,2,,.., p-1i (4) isfor

instancesatisfied if f S(p +l)/2.

In this case we have the identity exp (o X)Y[exp where X(Y) = XY

(" x)]-1 = exp (" x)(Y)

- YX. It follows that the Lie algebra generated over

by the matrices X is transformed by exp ( " X) into itself, so that this 106

F

t0I

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rapro

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(4) HereexpTdenotes l+T/lt +r2/Z! +... *tn-176-f)t. (5) Indeed, on account of (3) we have exp ("X) exp (BX) = exp[(" +6)X] for any a , B in an extension field of F,

108

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XgCINI

TITS, J., 38 Trace bilinear form, Transfer map, 1?

Quadruply transitive group, 3?

Quaternion, 65 group, ?8

Translations, 43 Trefoil knot, 81 Truncation, ?0 TVisted field planes, 49

Radical of a form, 104 Regular S -group, 18 polytope, 69 subgroup, 22

Unit quaternion,

SCHLEGEL, V., 6? SEIFERT, H., ?5, 78

Semi-nuclear rings, 49 Singer's theorem, 52 Solvable group, 150 Spherical dodecahedron space, Spherical honeycomb, 70

S-ring, 30 Steiner triple system, 40 suzuKI, M., 25 Tetrahedral group, 65 THRELFALL, W., 75,78 TIETZE, H. ?6

104

6?

Veblen-Wedderburn planes, 48

systems,

78

VINCENT, G., VORONOI, G.,

45 87 ?O

WALL, G.8.,24 WIELANDT, H., 33, 39 WYTHOFF, W. A., ?0 ZASSENHAUS, H., 25

Zero dimensional group,

110

56