Finite groups with a self-centralizing subgroup of

FINITE OF

GROUPS

ORDER V.

1.

WITH

6 AND V.

ONE

A

SUBGROUP

SELF-CENTRALIZING

CLASS

OF

INVOLUTIONS UDC 519.44

Kabanov

Introduction

Finite groups with a s e r f - c e n t r a l i z i n g subgroup of o r d e r 3 were d e s c r i b e d by Felt and Thompson in [7]. A t h e o r e m which they proved allows us to d e t e r m i n e the s t r u c t u r e of finite simple groups with one class of involutions and a s e r f - c e n t r a l i z i n g cyclic subgroup of o r d e r 6. THEOREM. Let a finite simple group ~ with one c l a s s of involutions contain an element $ of o r d e r 6 such that C~(~) = < ~ • Then ~ is i s o m o r p h i c to one of the following groups: ~-2('/'f)j Zz (15), L 3 ( 5 ) , the Mathieu group 2.

Some

/v/v~ , o r the Janko group

Necessary

Notation

and

J'4 of o r d e r 175560. Results

A n is the alternating group of d e g r e e re ; t5 r~ is the s y m m e t r i c group of d e g r e e ~ ; L,7 ( ~ ) is the projective s p e c i a l linear group of dimension rz over a field of c~ e l e m e n t s ; Vp" is an elemental T Abelian group of o r d e r /o n ; 7. rz is the cyclic group of o r d e r rt ; 4~, ~,... > is the group generated by the elements % ~, . .. ; DZ~r=

is the dihedral group, where

ctn---~2 = ~

and go,,{= c t - ' ;

@z~ is the generalized quaternion group of o r d e r 2 a ; A ~ B is the c e n t r a l product of the groups A and B . A. ( F e l t - T h o m p s o n [7]) If a finite group ~ contains an element ~ of o r d e r 3 such that Ca(4 ) = < ~ >, then G has a nilpotent n o r m a l subgroup / 4 , and one of the following a s s e r t i o n s is t r u e : (1)

G//b/ is the cyclic group of o r d e r 3;

(2)

G//-/

(3)

G//-/~-As

and /4 is a 2 - g r o u p ;

(4)

G =Z2(r )

and H is the identity group.

is the dihedral group of o r d e r 6 ;

B. (Higman [13]) Let a finite group g have a n o r m a l 2 - s u b g r o u p (~ such that ~ / 6 2 is the dihedral group of o r d e r 6. Assume that S is an element of o r d e r 3 in ~ that acts f i x e d - p o i n t - f r e e on ~ , and let P be a Sylow 2 - s u b g r o u p of G . Then the following a s s e r t i o n s a r e true: (1) if A is an Abelian subgroup of ~ , then the group

< A,As> is also Abelian;

(2) if I Q I > q , then the c l a s s of ~ is l e s s than the c l a s s of any other subgroup of index 2 in / 9 T r a n s l a t e d f r o m Algebra i Logika, Vol. 11, No. 5, pp. 516-534, S e p t e m b e r - O c t o b e r , 1972. Original article submitted May 11, 1972.

© 197-1 Consultants Bureau, a division of Plenum Publishing Corporation, 227 I['est 17th Street, New York, N, }'. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by an)" means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. .I cop), of this article is available from the publisher for $l.5.00.

284

]~ is well known that in the conditions of T h e o r e m B the c l a s s of the subgroup Q is not g r e a t e r than 2. C. (Higman [13]) Let ~ be a f i n i t e group with a n o r m a l 2 - s u b g r o u p Q s u c h t h a t G / ( ~ ~-L2(2 ~) , 7~ • ~ , and a s s u m e that 5 is an e l e m e n t of o r d e r 3 in G that acts f i x e d - p o i n t - f r e e on Q . Then ~ is . . . . 26 an e l e m e n t a r y Abelian group and is the d i r e c t product of minimal n o r m a l subgroups of o r d e r 2 ° A Sylow 2 - s u b g r o u p P of G has class 2, and if 1~ I > 2 2 h then (~ is the unique Abelian subgroup of i n dex 2 ~ in P . ]1 follows f r o m T h e o r e m C that in case (3) of T h e o r e m A the group /-/ is e l e m e n t a r y Abelian. Henceforth ~ is a finite simple group that satisfies the conditions of the t h e o r e m and does not s a t isfy the conclusion. Set ~--4 3 and s= ¢2, C is the c e n t r a l i z e r of the involution ~ in ~ . Since all involutions in ~ are conjugate, C contains a Sylow 2 - s u b g r o u p P of G . The f a c t o r group C = C / < ~ has a s e l f - c e n t r a l i z i n g subgroup < ~ of o r d e r 3. T h e r e f o r e , C satisfies the conditions of T h e o r e m A. In the sequel we shall call ~ a group of type A(2), where I -~ ; -~ z/ . 3.

C Is

a Group

of Type

A (1)

In this case C is a 2 - c l o s e d group, and since all involutions in G are conjugate, the c e n t r a l i z e r of any involution in G is 2 - c l o s e d . Such groups were studied by M.Suzuki [16], and among them there are no groups with a cyclic s e r f - c e n t r a l i z i n g subgroup of o r d e r 6. 4.

C Is

a Group

of Type

A (2),

and

Z(P)

Is Noncyclic

T h e r e exists in C a nilpotent n o r m a l subgroup ~ such that

C/~7

is the dihedral group of o r d e r 6.

If /4 is the complete preimage of /4 in C , then set T = 0z (/4).

Let Z ( P ) ¢ Z ( T ) . Then there exists a n e l e m e n t ~ E ~ ( P ) and , P ~ T . If T ~ < ' v > , t h e n Cc (7-) = 4 ~) >" Z(TJ'O~(/-/) "~ ~ . Hence T-Cc (7") =POz,(H) ~ C , wtdch contradicts the s t r u c t u r e of g . Thus 7-~ ~ ~> and P is i s o m o r p h i c to an e l e m e n t a r y Abelian group of o r d e r 4. By a t h e o r e m of G o r e n stein and W a l t e r [17], ~ is i s o m o r p h i c to L 2C~) for ~ = ¢ 4 and ~ = ] 5 . In L z(Y¢) and L 2(~3) aIlinvolutions are conjugate, and so they satisfy the conditions of the t h e o r e m .

Let Z(P) c Z (T) . Let us show that Z ( P ) contains an involution u such that 6~z (Cc~ (~)) ¢ T . We shall divide the proof of this fact into two c a s e s : z ~ qO ( r ) and z ¢ cp(T) . Consider the group N=Ng(H).

Then N = N ~ ( T ) ,

s i n c e / 4 = T'CcT(T)

and T is a c h a r a c t e r i s t i c subgroup of H .

If the involution z belongs to the Frattini subgroup qb(T) of 7" , then the f a c t o r group /~/=N/c-P(T) has a s e l f - c e n t r a l i z i n g subgroup < ~ > of o r d e r 3. Indeed, ff ~ e IV and C:~, ~]= ~" and -~ does not b e longto ~> , t h e n N contains a n e l e m e n t ~c s u c h t h a t ~:,s]=4 and ~c does not belong to < s ~ . But in this case [x'~=c~s] = 4 , where x - / ~ x ~ T . Since CT:(s) = < ~ > , we have ~=-'~x= ~ . Thus ~v e Q ( ~ ) = < t ~ , and ~ belongs to < s'> . Contradiction. Since ~ c A/ and the f a c t o r group N / q O (T)

has a s e r f - c e n t r a l i z i n g subgroup of o r d e r 3, /V is a

group of type A(2). Let ~ be a nilpotent n o r m a l subgroup of / ~ , and let F be its complete p r e i m a g e in N • It is c l e a r that F contains H • By a t h e o r e m of Burnside ~ h e o r e m 5.1.4 in [9]), F is a nilpotent group. Thus F=T. Cc~(T) = / 4 , and so N = C . Now since N=/Ve~(T), and ~ ( C ) = < z > , we have

T=/=0 z (C~ ( ~ ) )

for any involution c~ in Z ( P ) .

If r does not belong to gO(T) , then the group can be distinguished as a d i r e c t f a c t o r of 7 - . Now let 02 (~'= (c~)) = 7" for each involution ,~ in Z ( P ) . Then, since all the involutions in Z ( P ) a r e conjugate in ~ , the group 7-=-C~(Z(P)) ~ 7-¢ where ?'t is a subgroup of T . Since X ( P ) c X ( ' 7 - ) , the subgroup ff~(Z('~)) generated by the involutions in Z_ ( P ) coincides with Z ( P ) . Thus 7-=Z(P) × T~. Since the F r a t t i n i subgroup c - ~ ( 7 ) = ~ ( L ) , C_/O(~) is n o r m a l in P , and so c)O(T,)= $ . Hence 7- is an e l e m e n t a r y Abelian group. We conclude f r o m the s t r u c t u r e of C that 7- is the d i r e c t product of < ~ > and minimal n o r m a l subgroups of o r d e r 4 in C . Since Z ( P ) is a noncyclic group, a minimal n o r m a l subgroup of o r d e r 4 in C contains an involution ~ e ~ ( P ) • The involution z is conjugate to ~ in ~ ,

285

a n d s o C ~ ( ~ ) / ~ > ~ - - C ~ (cO/,~cz> and P / < ~ " - p / ~ O . O a t h e other hand, P / < ' ~ > cannot be i s o morphic to P/~: ~ > , since P/ has no d i r e c t f a c t o r s , and P / < Lz> has the d i r e c t f a c t o r < % ~ O / < ~ > . Con£radiction. This concludes the proof that Z ( P ) contains an involution LL with 02 (C~ ( ~ ) ) ~ 7-. Set R=Oz (C~ (~)) . Since ~ is conjugate to 7" in G , T ~ R and a Sylow 2 - s u b g r o u p P of contains two i s o m o r p h i c subgroups of index 2. We now wish to use T h e o r e m B f o r an appropriate f a c t o r group of G in o r d e r to r e s t r i c t the o r d e r of P . F i r s t consider the f a c t o r group C . If the subgroups ~ and R have the same c l a s s , then, by T h e o r e m B(2), I~l *-- 4 . Since Z(P) c Z (T), we have /Yl = 4

and

C/02,(C)~--$ ~. Since Z ( P ) is noncyclic, P%Z)~, × Z 2 and ~ has m o r e than one c l a s s of involutions. This contradicts the conditions of the t h e o r e m . Thus c~(77)< c~(~). Since c~ ( ~ ) ~ Z, we have cC(7)~- Z. Let W be the s m a l l e s t n o r m a l subgroup of C that contains the involutions ~: and ~ . I~ is c l e a r that W = < ~ , ~ , a : s ; * , where s = ~ 2 and W = V 8 . Now let =

c/w

If ~=ZWIV¢" is an Abelian group, then, by T h e o r e m B(2), c~(~) -- cC(R) and CIoz,~Cj ~ g z l . Thus [Tr = 2 j- and Z(T)>~%~I@7 "'. The c o m m u t a t o r subgroup of T cannot be equal to W , since then it would have index 4 in T , T would be a group of maximal c l a s s by T h e o r e m 5.4.5 in [9], and Z ( T ) w o u l d b e a cyclic group. Thus T ' E or T ' = < ~ > × < L L s > . If T ~ E , ~ t ~ , then in this case T = 7-/< z >

is an Abelian h o m o c y c l i c group, since I Tt = 2 9 and ~ acts f i x e d - p o i n t - f r e e on T . Hence, T is i s o m o r p h i c either to V/6 or Z~ x Z 4 - If T'---¢ , t h e n , by T h e o r e m 5 ~ . 3 in [9], 7-=Cr(~)x[r,] and so it is i s o m o r p h i c either to V~2 or Z ~ x Z z z × Z , z. But then ~: cannot be conjugate to zg in ~ , since p / . ~ "~> does not have d i r e c t f a c t o r s of o r d e r 2, and P/< ~g> has the d i r e c t f a c t o r < % e z : > / < ~ > .

Consider the case T ' = < ~ > a n d T = W < x > < ~ / > , w h e r e ~=r,q] = Z and ~ u = ~ ~" . I t i s e a s y t o see that there are two possibilities for and c/a(T)=yV'. Let ~P(T) = < "g>. Then, by T h e o r e m B(1), the group is admissible with r e s p e c t to s and has o r d e r 8. Since a dihedral group has no a u t o m o r p h i s m s of o r d e r 3 and C r ( 5 ) = ,~ ~:>, we have ~ :r~ ~>--- 4)a. Thus 7" = Q#× V4, and /9/~P/< ~ > . Hence the involution "z is not conjugate to ~ in ~ , which c o n t r a d i c t s the conditions of the t h e o r e m . Now let CP(T) = W

. Then 7:-(x~r>)A , zz=x~---~/¢= 4 , and [ar, ~t'] = Z - t h e M i l l e r - M o r e n o

group of o r d e r 2 a" with cyclic c e n t e r . We conclude f r o m the s t r u c t u r e of the group ,[/~z~(Z~t× Z4,) that

,~ZZ x Z e and ,O/,Q'Z'2#_~ph/ . This contradicts the fact that ?" a n d ~ z are conjugate in ~ ' , since ~,(ff.)~Z(',,O..). We have proved that [T"~ < s ' ~ .

Tt#,~'z'> . Thus T ~ 4 u ~ , * j R ~ / r = ~ u , ~ V

C~(~z) . Then, since < ~ ' , u > ~-Z(P)

~ is generated by involutions, it suffices

~'>, and let 8y be an element of o r d e r 3 in

and there e x i s t s in W a subgroup of index 2 in Z ( P )

(without

loss of generality we can a s s u m e that this subgroup equals < ~ , ~ > ) , the group ,~'~ ~ , M ~ ~2 "/2 s > belongs to Z ( T ) .

Thus K contains a subgroup of index 4 that belongs to Z (7") and is generated by involutions.

Hence K can be written as follows:

[(K).

K=(we obtain < ~ ' z / $,' ~d ~'$> ~ V8 , and so A" is an e l e m e n t a r y AbeHan group. Now let K ~-- ~/2s . Then the Hall { 2, 3} - s u b g r o u p ]-/ in ~ can be written as follows: ]4=K , where / - / / K " 5 ' ~ . Moreover, K=xxx x< ~ > ~ < ~ s >

Z, then we again obtain a contradiction to the f a c t that c e n t r a l i z e s Z(7") . Hence IZ(T)I = Z . Since LEMMA 2. The F r a t t i n i subgroup

Cr (s) -- < z > since then ~ d < 7> ~ Z ( T )

(Z (r)) is a 2 - g r o u p , and so 5

the l e m m a is proved.

c~O('7") of T is an e l e m e n t a r y Abelian group and is contained

in Z z ( T ) . P r o o f . F i r s t l e t u s prove that

T ~ is an e l e m e n t a r y Abelian group. If T is an Abelian group, then,

b y W h e o r e m 5.2.3 in [9], T'=~'e>x[,.T~ ,~.$>] . Thus c]~(T~)=~

or else

¢P(T') I"1 Z(T)¢~

and Z ( T )

287

is noncyclic, which contradicts I_emma 1. Now let

7"1 be non-Abelima. Since the element

< 5 > acts

f i x e d - p o i n t - f r e e on ~ , t h e c l a s s of ~ does u o t e x c e e d 2. Thus Tt'ZZz(T) and [ 7 ~ T ' J = < z > . By a l e m m a of Sheriev [4], 7"m~mz~...~/Vl~*Z(T~), where /v/i ( ¢ ~ i ~ k ) is the M i l l e r - M o r e n o group. I ~ t / v / = M i for some ~ - - ¢ , . . . ) k . Since < z > e / v ] and T.~Zz (TJ, ivI is a n o r m a l subgroup of T . Now, in view of a r e s u l t of Bechtell [11], M cannot have a cyclic c e n t e r . Thus Z ( M ) is a noncyclic group. F r o m the s t r u c t u r e of the M i l l e r - M o r e n o groups [15], we know that then M has t h r e e m a x i m a l subgroups ,43 . j - - ~ 2 , ~ , w h e r e ~('Aj)~/Vl'=~'~> . Since / ~ f ~ T , w e

c~p(A]) ,~ T

have

Aj "J 7" for s o m e ] . But then,

and cT)(A])~ x ~ ) > in C , in view of r e s u l t B(1), is i s o m o r p h i c to ~/~ , i.e., ~ V = K Y > x < ~ > × K ~ ) s > . We now wish to apply the w o r k of Held [10-12] to our situation. A d i r e c t application of these r e s u l t s is i m p o s s i b l e , since in our c a s e 02: ( C J can be a nonidentity group. However, the l e m m a s in t h e s e p a p e r s to which we r e f e r a r e t r u e e v e n if ~ (C)$ ¢ . By L e m m a s 1.1 and 1.2 in [12], we conclude that T x ~ V contains involutions cz and ~ such t h a t a J = ~ . d ~ = ~ and, in p a r t i c u l a r , < ~ > < s > ~ A z] . In addition, it follows f r o m [1] that the e l e m e n t d of P which i n v e r t s $ is an involution. If w~ choose ~z and $ just as in [12], we obtain the following two p o s s i b i l i t i e s :

(2)ad=

Z - ~ ) ~ , Sa~--2~fl.V.2)~. ~ . g

Consider c a s e (1). The g r o u p

where Se G/-(Z).

X = < ~ , ~fl~$ >< d >

is dihedral of o r d e r 8 and

V/N X = "/ . By a

r e s u l t of Gaschfitz [8], C splits o v e r ~ / . By a r e m a r k in [11], C/0zF ( f ) is a uniquely defined group that is i s o m o r p h i c to the c e n t r a l i z e r of a c e n t r a l involution in Ag - the a l t e r n a t i n g group of d e g r e e 8. By L e m m a 2.5 in [10], the group G h a s m o r e than one c l a s s of conjugate involutions. In case (2), by L e m m a 1.5 in [12], a Sylow 2 - s u b g r o u p of C ( ' a ) has o r d e r than one c l a s s of involutions in ~ . T h i s p r o v e s the l e m m a .

~

, and so t h e r e is m o r e

LEMMA 7. Let [HI= Z "~ , and let /4 a d m i t a r e g u l a r a u t o m o r p h i s m of o r d e r 3. Then /-/ is an Abelian h o m o c y c l i c group. The proof is i m m e d i a t e . V~ , then T ' ~ ~)(7").

LEMMA 8. If 7"~

T ~-~ V# and T~c/)(T). Then, by T h e o r e m B(1), t h e r e e x i s t s in T a subgroup /4 such that H'=/-//~.'r> = < ~ > x < ~ > )~ , ~ , w h e r e the o r d e r s of Q and 2~ a r e equal to 4 and ~ ( H ) ~ 7" . By L e m m a 2, the exponent of 0 2 (/-]) is equal to 4. If Oz (/-/) is an Abelian group, then, by T h e o r e m 5.2.3 in [9], o,~c/./,)-Qz(Hj($)x[bz(l./),~c$>] . since q~([ 0z Cl"l) ,,~s>J)#'f and 0z(/-/)4 T, we have Proof.

I~t

z('r)n T h i s c o n t r a d i c t s the fact that

Z(T)

is cyclic, since

Z (7") = < Z > = Co~(~p (,~). Let 02(/'/) _benon-Abelian. Then Oz(/-/) =( x < ~ >

,

/v]z = ~ ' C > x < ~ × < ~ > , and 2¢/3=xKO~Z>x , w h i c h a r e Abelian groups of type (1.1.2). Since oJz')

T , for s o m e

Z

the subgroup

is norma

in

Thus

T.

289

¢P(W7, ) ¢ < z>

for any [ = ¢, Z,5 and so < ~> x cP(/v/z) ~ Z ( T ) .

Contradiction. This p r o v e s the l e m m a .

Now let us prove thatonc of the following a s s e r t i o n s holds for group of C , 7"-- 0 z(C)

C = C a < "~>, P

a Sylow 2 - s u b -

,and V=4"~>ו C T ( P ) .

a) there exists in fl an involution

q.o such that

IC,,,(w)t

z

, nd

b) the involution "0 does not belong to the Frattini subgroup

cjD(T) of T ;

c) I T I ~ Z ~ ; d) 1Zz(P)J ~ Z 3 , and ff IZz (P)J = 2 z , then t h e r e exists an involution in P \ 7" that does not c e n -

Z 2 (P).

tralize

By L e m m a 4 , g ) ' ( P ) ¢ ~ ) .

Let V e U ( P )

and V - < z > × < ' 0 . . x . Set t-/=Cp(V)

. Since Z ( P )

is cyclic, V is the subgroup generated by the involutions in Z(/-/). By L e m m a 5, there exists an i n volution ~ in / O k / - / . Let 2) denote Cp(¢o) . Note that V . < ' z o > i s the dihedral group of o r d e r 8, and so ~)~z~z~ -- 2~ w . In addition, q) c e n t r a l i z e s 17 f 7 / 4 . Since t h e r e is one c l a s s of involutions in ~ , • o i s conjugate to "~. Let ~ - "~X, where : c ~ G . I t i s c l e a r t h a t ~ e C G ( ~ o ). Now choose a ~ E C suchthatif

~/=~ar,then

w) - - ' c ~ and

is a S y l o w 2 - s u b g r o u p o f ~ . thins /-/~/.

Since

~ ---P~' contains ~" . Then

IP :T]=Z

and T=Oz(C ) , w e h a v e

C~ (¢,0 contains ~ = P Y , w h i c h 1 5 9 : Q O f l l ~ 2 , and Q con-

A s s u m e that z is not contained in /-/Y. Then z=v) ~/is a c e n t r a l element of H~/. Since [ z , z ] is a central element of V y that commutes with % , we have Ez, Z ] = g o . B follows that k/Y ) , r e s p e c t i v e l y .

In addition, if k+ ~ >5 , then 7

C- 5.

V~ C t ~ "

But Igvi(Vz)~-ICwj(Oz-cP(T))l=2

, and so

7< ~ 5 and

Cr (q)z) would be of index g r e a t e r than 8, which is i m p o s s i b l e .

Hence

/ , we have IT'I = 2 o r T ~'" V8 • By I_emma 6, f T ' t cannot be equal to 2. IP:P'I

If T '~ V8 , then it follows f r o m L e m m a 8 that T / ¢ .qP ( T ) . But t h e n in view of L e m m a 2 a n d t h e f a c t that N c ( < S > ) T / / T ' a c t s o n T / T ' , w e h a v e T/T/~---ZzxZzxZ/~Z@ and c/)(T) ~- V2~. By L e m m a 2 , qb(T)~-Z z {T) • Let qb[T) - Z z ( T ) . T h e n , if ¢z) ¢ 7" by T h e o r e m 4(2) in [16] t h e r e e x i s t s an e l e m e n t S~. , of o r d e r 3 that is i n v e r t e d by 7x~. Since s~ f i x e s only ~ in qb(T) X < ~ > , we have Cp(T)= = < z > x V~x...x Vk, w h e r e the Vz (4---£ 4 k ) a r e a d m i s s i b l e s u b g r o u p s with r e s p e c t to N c (< s>) and VzzVc (< s~>)~.~ 4 (the s y m m e t r i c g r o u p of d e g r e e 4) f o r any i=4,...,k . Since z o e /Vc (< s , > ) , it f o l -

291

lows f r o m the s t r u c t u r e of ~ 4

t h a t ~) c e n t r a l i z e s Z z ( P )

c o n t r a d i c t s t h e c h o i c e of ~ 9 , s i n c e ~.O ~ C p ( V ) . Hence g r o u p of i n d e x 2, s i n c e Z ( T ) [ -~ 2 .

Let C p ( T ) a Z z ( T ) .

~

, s i n c e Z z ( P ) =Czz(r)(~v)r~oc/Z(]-) . T h i s e 7-, and s o it c e n t r a l i z e s in

Z 2 ( 7 ~)

a sub-

But t h e n I Cp (~O) t > 16, w h i c h i s i m p o s s i b l e .

In t h i s c a s e

Zz(7-) i s o f i n d e x 4 in 7 " . T h i s i s a l s o i m p o s s i b l e , s i n c e t h e n t h e c o m m u t a t o r s u b g r o u p of t h e f a c t o r g r o u p T/Z(7-)has o r d e r 2, and < 5 > a c t s r e g u l a r l y on 7-/Z (7"). Now l e t JP : P ' I = 8 . C o n s i d e r the f a c t o r g r o u p C~ Y ' . If Y / F" i s a g r o u p of r a n k g r e a t e r t h a n 4, t h e n , s i n c e /Vc()T'/T' a c t s on 7-/T' , t h e i n d e x of P~ i n P i s g r e a t e r than 8. Hence T/qO(T) ~ Vt~ o r V4~ - F u r t h e r m o r e ,

by v i r t u e o f t h e f a c t t h a t

~ ~" cp(7-j, ( s i n c e ~

~' ] 4 ), w e h a v e [ Cop(T) (20) [ ~ 8 .

If zo e 7", t h e n ~O c e n t r a l i z e s a s u b g r o u p of i n d e x 2 in cp(T), s i n c e cp(7-) c Z z (7-) . [CP(T)I ~ Z ~.

S i n c e s a c t s on

gb(F),wehave ]cP(T)I¢ 2 4.

Thus again

ITI-Z- 2 z.

~O 6 # \ T • Let us show t h a t in t h i s c a s e Z z (T) c a n n o t be an A b e l i a n g r o u p .

Let

Hence

Indeed, let

Z z (7-) be an A b e l i a n g r o u p . T h e n i t i s an e l e m e n t a r y A b e l i a n g r o u p s i n c e Zz(Tj--uzz(T; (s) ~ [ Z 2 ( T ) , < 5 > ] = < z ~ [ Z z(T), < s > ] , a n d i f oz i s a n e l e m e n t of o r d e r 4 i n [Z 2 ( T ) . < s > ] , t h e n t h e s u b g r o u p 4 ~ > × < ~'> i s n o r m a l in T ; t h e r e f o r e , l z 6 Z ( T ) , w h i c h c o n t r a d i c t s L e m m a 1. Now, in v i e w of T h e o r e m 4(2) in [16] and the a c t i o n of N¢() on Z z ( 7-), t h e i n v o l u t i o n ~ c e n t r a l i z e s Z 2 ( P ) , w h i c h c o n t r a d i c t s t h e c h o i c e of ~.o, s i n c e ~ ) ~e C,o ( V ) . Now let

Z z (T)

be a n o n - A b e l i a n g r o u p .

z z (7-) . IT: Cp(T)I=/6 o r 4, a n d we h a v e e a s y to s e e t h a t

T i s an e x t r a - s p e c i a l

T h e n , by L e m m a 2,

c]b ( T )

is strictly contained in

T = Z 2 ( T ) o r tT:Zz(T)[ = 4 • If T = Z z ( Z ) , t h e n i t i s

g r o u p , and t h i s c a s e w a s c o n s i d e r e d in L e m m a 6.

Thus,

[T:Zz(T)I = ~ . But t h e n the c e n t e r of T i s of i n d e x 4 T , and s o T h a s a c o m m u t a t o r s u b g r o u p of o r d e r 2. S i n c e

g a c t s r e g u l a r l y on T , t h i s i s i m p o s s i b l e .

T h l s c o n c l u d e s c a s e a).

Nex£ we c o n s i d e r c a s e c) and d) s i n c e , in o r d e r to c o n s i d e r c a s e b), it i s n e c e s s a r y t h a t c a s e b) hold f o r a n y i n v o l u t i o n -o in Zz(P) \ Z ( T ) . C a s e c). If I T l = 2 3 , t h e n

T ' ~ (R e and

If 17"1= z S , t h e n , by L e m m a 7 , w h i c h c o n t r a d i c t s L e m m a 6.

If

I TI = 2 7 , t h e n Z z ( T )

Y/

U ( P ) = (Z5 • w h i c h c o n t r a d i c t s L e m m a 4. i s a n A b e l i a n g r o u p , and s o

contains a subgroup

~/=

7" i s an e x t r a - s p e c i a l

V8 such that W-~ d • Then

T/W

group,

has order

2 4 a n d , by L e m m a 7, T / ~ A / i s an A b e l i a n g r o u p . T h u s , T ' ~ W , and so !7-'1 = Z o r T ~ - - V ~ / . T h e f i r s t p o s s i b i l i t y i s e l i m i n a t e d by L e m m a 6. But if T " W ' , t h e n by L e m m a 8, 7- ' ~ q~ (7-) . T h e n f r o m L e m m a 7

T / T ' "" Z 4 ~ Z 4 . T h u s , cp(.TJ~ Vzs and i s of i n d e x 4 in 7 - . S i n c e , b y L e m m a 2, cP(T) Z z (7-) , T / Z ( F ) h a s a c e n t e r of i n d e x 4. But t h e n the c o m m u t a t o r s u b g r o u p of T / Z ( T ) h a s o r d e r 2,

we o b t a i n

which is impossible. C a s e d). If Z z ( P ) h a s o r d e r 4, t h e n IZz(?-)l = 8 . W e have: T / Z z (T) is an A b e l i a n g r o u p , and s o f r o m L e m m a s 2, 6, and 8 we o b t a i n a c o n t r a d i c t i o n . L e t I Z z ( P ) I = 8 . T h e n , a p p l y i n g T h e o r e m B(1) to

T/4~>

,weobtain

IZ2(T)I =2 5 . Assumethat

Zz(7- )

isanon-Abeliangroup.

i s e i t h e r a n e x t r a - s p e c i a l g r o u p o r Z 2 ( 7 - ) " Q 8 x Va . T h e n , b y L e m m a

T = Q ~e Cr((2 ) , w h e r e

(~ -~'g2e . L e m m a 6 e l i m i n a t e s

g2 ~ Cr ( ~ ) ~ s i n c e in t h i s c a s e t h a t in t h i s c a s e

cP(T)~ V8 . T h u s

Then Z2(T)

5.4.6 in [9], 7- = Z z (7-)

or

7-=Zz(T) and L e m m a s 2 and 8 e l i m i n a t e T =

~z ( T ) i s an A b e l i a n g r o u p . We have a l r e a d y n o t e d P\T centralizes Z z ( P ) , which

Z z (7-) i s e l e m e n t a r y A b e l i a n , and any i n v o l u t i o n i n

c o n t r a d i c t s c o n d i t i o n d).

Zz (P ) \ Z ( T) does not b e i n g to q0 ( T ) . T h u s Zz(P) i s a n o r m a l s u b g r o u p of P / < ~ > , by L e m m a 2 t h i s i s p o s s i b l e o n l y

W e c a n now a s s u m e t h a t any i n v o l u t i o n i n

f)CP(T)=. S i n c e c ] D ( T ) / , ~ ~ > i f C ~ ( T ) = < ~ > . But t h e n 7" is an e x t r a - s p e c i a l g r o u p , w h i c h is i m p o s s i b l e by I . e m m a 6.

292

Let us s u m m a r i z e the r e s u l t s of this section. If C is a group of type A(2) and Z ( P ~ is a cyclic group, then ~ is i s o m o r p h i c to /-sC3) or /V/~v - t h e Mathieu group of d e g r e e 11; t h e r e f o r e , G does not c o n t r a d i c t the t h e o r e m . 6.

C

Is

a Group

of Type

A(3)

If ~ is a group of type A(3), then, r e p e a t i n g v e r b a t i m the a r g u m e n t s in the second p a r t of T h e o r e m 12 of G. Hig-man's l e c t u r e s [13], we conclude that ~ is i s o m o r p h i c to the Janko group of o r d e r 175560; t h e r e f o r e , ~ does not contradict the t h e o r e m . m

7.

C

Is a Group

of Type

A(4)

If ~ is a group of type A(4), then it is e a s y to see that e i t h e r t h e r e is m o r e than one c l a s s of involutions in ~ , or ~ is a nonsimple group by T h e o r e m V.25.7 in [14] and a r e s u l t in [6]. T h u s , in view of the r e s u l t s of Secs. 3, 4, 5, 6, and 7, we conclude that t h e r e do not e x i s t s i m p l e groups which s a t i s f y the conditions of the t h e o r e m and do not s a t i s f y its conclusion. T h i s p r o v e s the t h e o rem. The author is deeply g r a t e f u l to P r o f . A. I. S t a r o s t i n for posing the p r o b l e m and f o r his scientific guidance; to A. D. Ustyuzhaninov for consultations on the t h e o r y of)o-groups; and to V. M. Sitnikov for his help while d i s c u s s i n g this a r t i c l e . LITERATURE 1. 2. 3.

4. 5. 6. 7. 8.

9. 10. 11. 12. 13.

14. 15.

16o 17.

CITED

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293