Factorizations of nonsimple finite groups

FACTORIZATIONS OF NONSIMPLE FINITE GROUPS

L. A. Shemetkov UDC 519,44

Beginning in 1954, S. A. Chunikhin published a series of papers (see the survey [i] and

the book [2]) devoted to extending P. Hall's theorem on Sylow systems in solvable finite

groups to arbitrary finite groups. A solution in principle of this problem was obtained by

Chunikhin in 1958. It turned out that to each collection of nonempty subsets ~1~/~2,...,,M~ of the set ~ of all indices of some principal series of a finite group $ , whose union

is /~, there corresponds at least one factorization of the form $=~,..~, where ~,

~ .... ~ are certain pairwise commuting subgroups of order ~l~z~ ~..., ~FC>, respectively,

and ~(~)c~(raf),~ being the product of all elements of ~-~ f=4z .... ,F'

Having become interested in Chunikhin's factorization results, M. I. Kargapolov carried

out a series of investigations [3-6] on the same theme. He isolated the essence of the

matter in [6], where he proved the following

THEOREM. Suppose J=~CSF.,.C~ =~ is a principal series of the finite group ~ .

Then ~ contains pairwise commuting subgroups ~,Rz,...,~ such that for any ~=~8 .... ,~

the following conditions are satisfied:

a) R I ~i-! = Lri ;

b) the intersection ,~ 13 ~ is nilpotent for ~ ' ;

d) ~ contains a nilpotent normal subgroup ~ such that ~ /~ ~---~ /~-t"

Kargapolov called the collection of subgroups ~,~ .... ~ a principal decomposition of $.

By multiplying together suitable members of a principal decomposition it is possible to ob-

tain a Chunikhin factorization. Moreover, Kargapolov's theorem provides important informa-

tion on the connection be~Teen the factors of the factorization and those of the principal

series.

Kargapolov's theorem was refined in [7], where it was shown, in particular, that the

$~o in condition d) can be replaced by ~ n~_! . It was also shown that Chunikhin's theorem

can be augmented by the solvability of the intersections /~i ~ ~' if ~ ~ .

After 1968 there was no significant progress in this direction, although one could not

assert with confidence that the issue was exhausted. While the problem of finding an analog

of Sylow systems for arbitrary finite groups was completely solved, there remained open many

questions related to the discovery of "generalized" normalizers for Chunikhin and Kargapolov

Translated from Algebra i Logika, Vol. 15, No. 6, pp. 684-715, November-December, 1976. Original article submitted November 27, 1976.

This material is protected by copyright registered in the name o f Plenum Publishing Corporaffon, 227 West 1 7th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retn'eval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy o f this article is available from the publisher for $Z50.

425

factorizations, which in the solvable case coincided with system normalizers and ~-normalizers

[8].

I n t h i s p r e s e n t p a p e r we o f f e r a f o r m a t i o n a l g e n e r a l i z a t i o n o f t h e a b o v e t h e o r e m o f

K a r g a p o l o v . We w i l l p r o v e t h a t t o e a c h p r i n c i p a l s e r i e s o f a f i n i t e g r o u p $ p a s s i n g t h r o u g h

i t s ~ - - c o r a d i c a l ~ ( ~ a homogeneous l o c a l f o r m a t i o n ) t h e r e c o r r e s p o n d s a s o - c a l l e d

S -decomposition ,~,,R z ..... Z~+/, consisting of pairwise commuting subgroups with specific

properties. This ~-decomposition becomes a principal decomposition in Kargapolv's sense

when 3 f is a formation of trivial groups. The last member ~+! of the indicated ~ -decom-

position in the case of a solvable group coincides with the Carter--Hawkes ~-normalizer

[8] and has a number of interesting properties in general; in particular, it covers all f

central principal factors, which leads to the following proposition.

Proposition. Any finite group 0 contains subgroups A and B such that A is

nilpotent, is contained in J , and covers each central principal factor of G, and ~ is

supersolvable and covers each cyclic principal factor of ~ �9

Our plan is as follows. In w we present some necessary information and lemmas. Also

of an auxiliary nature is w In w we introduce the concept of ~ -supplement, which plays

an important role in obtaining the main results. In w we introduce the concept of ~-

decomposition and prove a formational generalization of Kargapolov's theorem. The study

of ~-decompositions is continued in w where we consider ~-normalizers.

The main results of this paper were announced by the author in 1975 at the All-Union

Algebra Symposium in Gomel' [9].

i. NECESSARY INFORMATION AND LEMMAS

We consider only finite groups. Therefore, by group we will always mean finite group.

Recall that, according to [I0, Ii], a screen f is a function which assigns to each group

some (possibly empty) formation ~(H#, and satisfies the following conditions:

i) if % is a homomorphism of /'/ , then ~(H) is contained in fI~O~(~e~), where

~ is the image and ~e~ the kernel of ~ ;

2) ~[lJ~ , where / is the trivial group.

A normal section H/~ of a group ~ is called f-central in ~ if ~/~ (H/K)~ f(~/~)

and f -eccentric otherwise. A series of normal subgroups of ~ is called f-central if

each factor is ~-central in ~ The set ~ of all groups possessing f-central series A

is a nonempty formation; we say that f is a screen~ of the formation 3~--? A screen f A

is called inner if f(~)~f for any group ~ .

In the rest of this section, ~ will denote a formation and f will denote some inner

screen of this formation. Let ~ denote the ~-coradical of ~ which is the intersection

of all normal subgroups of G whose factor groups belong to ~ Also, by an ~-central

(~-eccentric) principal factor we mean any f-central (resp., f-eccentric) principal

factor; this definition does not depend on the choice of ~ , as Lemma 1.4 of [I0] shows.

426

A maximal subgroup of ~ is called f-normal if it contains ~, and Y--abnormal

otherwise. We will freely use the fact that, according to Lemma 1.6 of [10], an ~ -normal

(~--abnormal) maximal subgroup covers all ~-eccentric (resp., all F-central) principal

factors. Recall that a subgroup #f covers (isolates) a section H//( if /~K-~/-/ (resp,,

~o~/-c/(). Let /~G denote the core of the subgroup /~ in ~ , i.e., the intersection of

all subgroups conjugate to #f in ~ . It is not difficult to establish the following lemma.

LEMMA i.I. Suppose a maximal subgroup ~ of a group ~ does not cover some principal

factor L/~ of ~. Then the following assertions are true:

l) if ~ /~ is ~-central, then ~ does not cover the factor L~/K~ ~, which is

-isomorphic to ~/K ;

2) if ~/~ is ~-eccentric, then ~ does not cover the factor ~ F / ~ , which

is ~ -isomorphic to ~/J( ;

3) ~ does not cover the factor ~/~0/~$ , which is ~ -isomorphic to ~/~

LEMMA 1.2. Suppose ~=H~ p, ~ . Then /~F~=~F~. In particular, if ~_~F then

Proof. Under the natural homomorphism of ~ onto ~/~, the subgroup ~ is mapped onto

~/f2~ , and the subgroups ~f and ~ are mapped (see Lemma 1.5 of [i0]) onto the same group F. ,~ F,

H ~/~= ~ ~/~, from which follows the desired equality ~ ~7=~ ~.

The following two results are well known (see Theorems 3.3, 3.5, 4.4, 4.5 of Chap. III

of [12], and also Lemma 1.4 of [16]).

LEMMA 1.3. Suppose F is a nilpotent normal subgroup of ~. If ~Q~I~I-I , then 7 r

is complemented in ~ and is equal to the direct product of certain minimal normal subgroups

of G

LEMMA 1.4. Suppose K ~ G and K//(~qS~O) has a normal ~-subgroup, where qr is some

set of primes. Then ~ also has a normal ~r-subgroup.

An ~r (i.e., a Hall qr -subgroup ) is a q~ -subgroup whose index is not divis-

ible by the numbers in ~7.

Following Chunikhin [2], by rye solvability measure S (G) of a group ~ we mean the

product of the orders of all primary factors of some principal series. If ~ has no primary

principal factors, we put $(~)=/. We also put $I(~)=IGI:5(~). A primary group is one whose

order is a power of a prime.

LEMMAI.5. Suppose ~'= ~,S(K)=/, If all ~ -principal factors of K are ~r-central

in ~, then ~/~G (K) E ~.

Proof. Suppose K~f and 2P is the intersection of the centralizers in $ of all ~-

principal factors of ~ . By hypothesis, $/y_DE ~. The group, A=D/C G (K) can be viewed as a stable group of automorphisms of K . Therefore, ~A)~-fr(~r(K)) (see, e.g., Lemma 2.5

of [10]). Since S(KI=I , we have 77(/()'f, hence ~ffif , and the lemma is proved.

427

Suppose ~ ,K is a ~ -principal factor of SF. If L / K is ~--central LEMMA 1.6.

in ~ , then ~/K ~ Z(~F/~).

Proof. If ~/K is f-central in $, then O/CG (~/K)~ftL/K)~ ~, hence $~

C G (L/~) as required.

LEMMA 1.7. Each non-Frattini S-principal factor of SF is f-eccentric,*

Proof. Suppose ~/K is a principal factor of ~ and ~ ~ Assume that L/K is

non-Frattini, i.e. L /~ is not contained in G~I~/K) . Then it is clear that some maximal

subgroup ~ of ~ does not cover L/K . Therefore, M does not contain ~ , and so is

-~--abnormal. But then, as we observed earlier, ~ /K must be f-eccentric in ~ .

The lemma is proved.

A normal subgroup ,~ of 0 is called ~-ultimate if ~/R~961G~ is an ~-eccentric

principal factor of ~ .

LEMMA 1.8. Suppose ~ ~ and Kn O~is not contained in ~(~) �9 Then 0 has an

~-ultimate normal subgroup which is contained in ~G

Proof. Assume that H=A'n~F is not contained in~($1. Then ~=~nqS{~)#H. Let

R/Z be a minimal normal subgroup of H/~ contained in ~/L. Since ~/L)=q~ the

factor R/~ is non-Frattini and, by Lemma 1.7, ~--eccentric. Thus ~ is Y~-ultimate in

, and the lemma is proved.

Recall that by ~(m) we mean the set of all prime divisors of the natural number ~

LEM>~ 1.9. Suppose R is an F-ultimate normal subgroup of ~, ~=q(R/K), K=c/~

(~)~ R . Then ~ has a normal ~-subgroup ~ , and the following groups are ~-isomorphic

and are therefore ~-eccentric principal factors:

Moreover, any maximal subgroup of ~ which does not cover R/K does not cover any of the indicated principal factors In particular R~(~I, $ and ~n ~E . , , are ~--ultimate normal

subgroups of

Proof. By eemma 1.4, ~=~T where T~q6=o~I$) and ~ is an ~-subgroup of

Obviously, ,~ ,d~/~-~c~ R/A and ~/$~ are $ -isomorphic. Furthermore, we have a

-isomorphism

~ ~. KOF , ~/~ ~.~F_~ ~/~(~j.

The subgroup K(~O~ coincides with either ~ or ~ Suppose Ki~o~= K. Then, ac-

cording to the above $ -isomorphism, ~0~/K$ ~ is an ~-eccentric principal factor of ~ .

*The property of a factor being non-Frattini is here understood with respect to the whole

group G

428

But this contradicts the fact that between G and ~*- the factors of a principal series

are ~-central. rhus,/((~ ~-) R, which implies that ~//~ and ~ O ~ / K ~ ~ = are

- isomorphic.

Suppose a maximal subgroup /~ does not cover ~//~ . Clearly, M does not cover

~/~ and ~/$0~. Assume that ~(/(~ ~) contains ~ . Then, since K~96=/~ the

subgroup /# contains i~f]~]~$=~, which is impossible.


LEMMA i.i0. Suppose ~=~RI=/~fR ~ where /V is maximal in ~ , and R I and ~ are dis-

tinct nilpotent ~-ultimate normal subgroups of ~ contained in G F such that ~ f]Re-~9~=

G~C~). Then R=~f]~7~ is also an ~- -ultimate normal subgroup of ~ , and ~ ---- ~2

and the following groups are ~-isomorphic:

R, R /R, R 2 /

Proof. Since ~=M and s is Abelian, it follows that #,~. Also, ~'ffi

~II~fT~IR~)=~R 2 . Let RnR1=i( . Then /(~, and if ~qS, then /(=kl , since ~i/r is a

minimal normal subgroup of ~/~. But then /~7 c-R and R, 7~ z = ~?~ =R~M, which is impossible,

Therefore, ~@~1=q O. Obviously ,~fn~z=~. From this we immediately obtain a ~-isomorphism:

Consequently, ,~/q~ is ~-eccentrlc and R is ~--ultimate.


We now introduce an important concept. A maximal subgroup /~ of a group ~ is called

~--critical in ~ relative to normal subgroup K of ~ if $ contains an ~-ulti-

mate normal subgroup ~ such that ~ K and ~=~. By an yC-critical subgroup of ~ we

mean any subgroup which is maximal in $ and .~ -critical in ~ relative to 6~ f . Note that

any ~-critical subgroup is an #~-abnormal maximal subgroup. The concept of ~--critical

subgroup comes from [8], where it is defined in other terms for solvable groups.

LEM~IA i.ii. Suppose ,~FI~J = ~, where hf is an ~"-abnormal maximal subgroup of ~ �9

Then ~ is Y~-critical in ~ .

Proof. Consider ~/~=/~,/~" ~/v�9 where GE=~;~ ~ffi~). By Lemma 1.3, ~7/q6 can be

written as a direct product

. . . ~ a~l,

where ~/~6 is a minimal normal subgroup of ~/~. Obviously, ~7 tq~ is not contained in

M/~ for some K �9 Then /~ does not cover ~ /q5 , hence ~ /q~ is ~-eccentric in ~.

Consequently, ~ is ~-ultimate in $ Since ~=/~ it follows that /~ is y~-critical,

as required.

LEM~tA 1.12. Suppose /V~ = ~, where ~ is maximal in ~, and ~ is an ~-ultimate

normal subgroup of ~ contained in ~. Suppose also that ~ is an ~-abnormal maximal

subgroup of ~ containing ~ . Then L O ~ is an #--abnormal maximal subgroup M .

429

Proof. Since ~= ~ and L-DR, it follows that M~=$, L=R(LITM)~ The mapping

~:fn~-~-m(fl/n]~), ms , is an isomorphism of &/]~ onto ~/~nR, Since L=~(L~M), the

function ~ for meLoM sends the maximal subgroup Z/• of P/A? into MnL/Mf]R. There-

fore, L n M is maximal in M Since MR=S, R _c~ s . , we have MFR - G f . Therefore,

since L-DR, we see that ZnM does not contain /~jc, i.e., ZfTM is YC-abnormal in fi4.

LEMMA 1.13. Suppose ~=~i--~;~, where ~7 and ~z are maximal subgroups of G,

and ~ and ~ are distinct nilpotent ~-ultimate normal subgroups of ~F contained in

such that s --c/V/Z, R,n~=-~=r $'~ Then 4riM;is F-critical both in ~/, and Z;.

Proof. Since ~ - ~ A ~ , ~ = ~ and ~ is nilpotent, it follows from Theorem 111 .4 . 3

of [12] t h a t ~ induces on ~ / q b the same group o f automorphisms as ~g . T h e r e f o r e , ~ / q b

i s an ~ - e c c e n t r i c p r i n c i p a l f a c t o r o f M 2, S ince (A~InMe)Rz=Mz, i t i s c l e a r t h a t / ~ n M a

i s an f - a b n o r m a l maximal subgroup of 6 . Since R~ is n i l p o t e n t , we have I ~ / n M ~ / Z / M ~ ) - ~

hence, by Lem~a 1 .11 , 4 , ~ is 7 - c r i t i c a l in ~ . l f & = M , , then, by the same argu-

ment, ~ O M z i s # - - c r i t i c a l in M~ .

Suppose P/ ,~Z=~. By Lemma 1.12, M,~]M z is an ~-abnormal maximal subgroup of ~1.

By hypothesis, ~)~ . By eemma !.!0, R=~nT?~ ~ is an ~-ultimate normal subgroup of

and ~ =~1~z. The subgroup ~ is not contained in ~2 , since, if we assume the contrary,

~z=R~1~Cz,whichis impossible. Thus, ~R=~, (/~I@Mz)R=/~ ! Since R is nilpo-

tent and h~10 ~$ is it_abnormal in A41 , it follows from Lemma i.ii that ,~ M~ is Jr-critical

in ~$ , and the lemma is proved.

If 7~ is a set of primes, then we denote by ~' the complement of ~ in the set of all

primes, and by ~(~) the largest normal ~-subgroup of $ .

LEMEA 1.14 [13]. Suppose a group ~ has a ~ormal p -solvable subgroup ~ (p a prime)

generated by its ~-elements. If a Sylow /~-subgroup P of ~ is Abelian, then P possesses

complements in ~ and any two of them are conjugated.

Proof. By a theorem of Chunikhin, K possesses an ,-subgroup ~ , and any two

subgroups of A / are conjugate in ,~ . Therefore A//(=/V/?=~, where A/=~/$ (~). We will show

that A/O~=I. It is easy to see that ~ (/:))=P(N~]IVK(P)) and ~/~f(~)=~ , where ~c__)D

~ c~ Therefore, using Grun's theorem and the fact that ~ is generated by its ~'-

elements, we obtain that ~ (/7) is generated by its ~-elements. By Theorem 111.13.4 of

[12], ~/=~7Y=/. Thus, ~ is a complement of /!7 in ~.

Suppose ~ is any other complement of P in ~. Then /~K is normal in H and is an

2//-subgroup of /( Clearly, #/=/d& (//n ~). We now see that the conjugacy of complements of

P is a consequence of the conjugacy of 2p,-subgroups of f.


For p>2 the Abelian condition in Lemma 1.14 can be replaced by modularity.

A screen f is called homogeneous [i0] if the following conditions are satisfied:

I) f(,D)=F(~) for any two primary groups P and ~ such that ~(~)-q~(~) ;

430

2) for any group $ and any primary subgroup ~ we have f (~) ~ f(P). A

If f is homogeneous, the formation ~=~ is called homogeneous.

The following lemma is of interest in two respects. First, it is a formational gener-

alization of the Ore -- Chunikhin theorem on the conjugacy of maximal subgroups with the same

core (see [i], Theorem 1.9.3). Second, it can be used to prove the conjugacy of ~'-proJectors

LEMMA 1.15. Suppose ~ is a homogeneous formation, ~ is /~ -solvable for some prime

P , and /~ and ~ are 9~-abnormal maximal subgroups of $ If ~G n ~--HG n ~ ~ and

p divides (I~:MI, l~:~I) then ~ and ~ are conjugate in

Proof. Suppose $ is a group of smallest order for which the lemma is false. Suppose

and ~ satisfy the hypothesis of the lemma, but are not conjugate.

Assume that ~=~$nHG~/, and consider $/D. Since ~fP/~7 is the ~-coradical of

~/~7 , it is easy to see that the hypothesis of the lemma holds for $//_~ and the maximal

subgroups ~/D and HID. By assumption, ~/~7 and ///2 are conjugate, hence ~ and ~ are

conjugate. Contradiction.

Thus, ~$ n ~G = ~" Assume that ~f~ ~f. Suppose ~ ~ M~, where ~! is a minimal normal

subgroup of ~ . Then ~I is not contained HG, hence ~I= ~" Since M G O HQ = / we have

LI~= / , hence ~z is ~r-central in $ . But this contradicts the fact that H is

-abnormal and does not cover L 7 Consequently, ~= f. Similarly, HG =/.

Suppose ~ is a minimal normal subgroup of ~ contained in ~f. Then ~L=~ = ~.

In view of the hypothesis ~ is a ~-group. Put ~=~ IL). Obviously, ~ C =L (MoC),

hence ~@~. Since ~=/, we have ~oC =/ . Thus, ~ coincides with its centralizer in ~.

If O~(~/L)=R/~ is nontrivial, then ~ is generated by its p~-elements, and, by

eemma 1.14, ~ and ~ are conjugate. Since L =~G (L} , we have ~p(~/L)=L/~ Thus,

~p (~/L)=~/L �9 But then ~/L~ ~L} , since ~/L belongs to 9 s and, in view of the

homogeneity of f and Lemma VI.5.4 of [12], is an extension of ~/~I by a group in

f(L) �9 This is a contradiction, since, in view of the ~-eccentricity of ~ , the group

G/L does not belong to ~[L).


In the sequel we will assume that the screen f is homogeneous. Thus, from this point

on y~ will denote a formation having an inner homogeneous screen f By Lemma 3.2 of [i0],

Y~ is a saturated formation, i.e., A/qOIA)E~ always implies ~f. Let ~(F) denote the

union of the sets ~(~) for all OEf. Obviously, ~(~#=~[~N~), where ~ is the class

of all nilpotent groups.

2. s

Suppose ~ is a partition of the set of all primes into pairwise disjoint, nonempty

subsets, which we will call ~-layer~. If we intersect each ~ -layer with ~), where

$~/ , and discard the empty sets obtained in the process, we obtain the O-partition ~,

~,...,~ of the set ~(G). A system of subgroups ~,H2,...,/~ ~ is called a ~ -system of

if the following conditions are satisfied:

431

l) .....

2) the subgroups ~,/~z ..... H~ commute pairwise when ~>/ ;

3) ~(SI(~))~(~) for some /, I~.

If ~ is the trivial group, then we take a V

-system to be a single ~ and a ~ - 6

partition to be a single empty set. V

An arbitrary group ~ possesses a $6 -system for some fixed partition 6 if and only

if 97(S'(G}) is entirely contained in one of the 6-1dyers. This follows from the theorem

of Chunikhin mentioned in the introduction.

If the partition o is unimportant, we will omit the subscript ~ Thus, a C-system

is a [-system for some 6 We now introduce the concept of a minimal C-system. By this 6

v

we mean any C-system ~, ~...,F/ having the following properties:

I) g<n is primary for any ~ ;

2) either ~ is primary or ~IH~)=~(sr(G)). v

As a justification of this definition, note that if we are given a C~ -system, then it

V

can be obtained by multiplying certain terms of some minimal $ -system. Of course, a group

v

can have several minimal ~ -systems. Note also that in the case of a solvable group the con-

v

cepts of minimal S-system and Sylow system coincide. &,

If ~=~T,,,~...,~ 1 is a ~ -system of a subgroup ~ of ~ , then for any ~e~IH)

the system ~= IT ~,T2 ~ .... ,~ ] is also a C -system of H . We say that the systems ~-and

yx are conjugate (by means of ~ ). The subgroup #/G (Jr)={~l~s is called the nor-

realizer of system ~ in $ If /{G (1)= $ , we say that 5 ~ is normal in G �9 A group v

is called ~-decomposable if it possesses a normal ~-system. In other words, a group

is ~-decomposable if it can be represented as a direct product $=~61 x ~z x ... • ~ga ,

~6~ -subgroup of where ~,62 ..... ~n is the 6-partition of the set %(GI , isan ~L

We say that a C-system $ of a group ~ is reduced in a subgroup H of ~, if some

[ -system ~ of /~ can be obtained by intersecting the members of $ with // (trivial

intersections, if any, are discarded). In this case we employ the notation ~=$QFf and say

that $ is an extension of ~ If ~ is the kernel of a homomorphism ~ of ~ , then we

denote by ~ (by ~"/(/X ) the j -system of the group #/x (the group#//(//<) into which

is sent the ~-system ~ of the subgroup I/ of ~ under the homomorphism ~ (under the

natural homomorphism of $ onto ~/K ). Here also we must take into account that if #/~#/

the members of its $ -systems are nontrivial, hence ~ consists of the nontrivial images

of the members of I

We present two ]_emmas on [ -systems. The proof of the first we omit, since the reader

can easily reconstruct it by applying Chunikhin's theorem on ~-solvable groups and using

as a model known facts from the theory of Sylow systems of solvable groups.

432

LEMMA 2.1. Suppose a partition ~ and a group ~ are such that ~($'(~)) is contained

in one of the ~ - l a y e r s . Then G has at l e a s t one go-sys tem ~ and the fo l lowing a s s e r t i o n s

are true:

i)

are conjugate;

2) 4/G (S)

3) if H

4) if

has exactly [~ :J~ (S)] distinct V V

~ -systems, hence any two ~6-systems of

LEMMA 2.2.

is ~ -decomposable;

is a subgroup of ~ , then there exists s ~ such that Sz is reduced in

is a homomorphism of ~ , then ~$ ($)~= N~ ($~).

Suppose ~ ~ is ~ -solvable for some prime Suppose also that a

Sylow ~ -subgroup ~ of ~ is a member of some ~-system $ of SF . Finally, sup-

pose ~ contains an ~-abnormal maximal subgroup /~ whose index in $ is a power of /9,

and ~ r]G ~ is not in ~ and ~ is reduced in ~f. Then ~G(~) ~(~flmF'). Proof. By Lermnal.2, ~Fc---~ F. Let ~=$N/~ F . Assume first that the core ~ of /~-

in ~ is nontrivial, and consider the naturalhomomorphism ~-~=~/~. Obviously, ~=~/L

satisfies the hypothesis of thelemma. Itisclear that the d-system ~ of ~--~/~ is an

extension of the C -system ~ of ~--F= ~F/~, and ~=PL/L~S. By induction, ~(S) ~ ~/~ I~)-

Since, by Lemma 2.1, we have ~${$)~/L=~)If), ~/M (~')~/L=~pT(F), we obtain the desired inclu-

sion ~(S) c- /JM (~" Thus, we may assume in what follows that ~n~ ~ contains no nontrivial

normal subgroups of $ Let S=[P=/~7,H~ ..... Ha]. Since ~Fis ~-solvable, $ has a minimal

normal subgroup K such that K_c~4~K=~)~oK=/, ]Kl=p #. By Lemma 1.2, ~F~ = ~F and

/ ~ ~ is not normal in ~ . But then P~ $~ , i.e., Nm~. Let ~ be a minimal

normal subgroup of #f contained in ~ Since S is an extension of ~ , we have ~

~>/. Clearly, ~ is contained in H~ for some #>{. The subgroup ~/$($) normalizes #/~

and 7 rK since ~-K~$ Consequently ~/$($} normalizes the intersection ~0~=7:, i.e.,

~/~(S)~#/~(F)=~. Thus ~ ($) normalizes ~ and ~ for any & Thus N~(S) normalizes ,~n S=~,

Lemma 2.2 is proved.

eEb~IA 2.3. Suppose ~= ~(Jc), 0~,(G~I f, is not nilpotent. Suppose also that S

is a C -system of ~F , each member of which is either a primary 9F-group or a ~[1_group.

Then there exists an ~c$ and an f-critical subgroup M of ~ such that SZ is reduced

in M p and we have the inclusion ~0(S ~) c- ~ M (~z f]~).

Proof. Let 7:=~(~P), c~b=q6(~)f]~. By Lemma 1.3, p/qO can be written as a direct pro-

duct of certain minimal normal subgroups of G/~ b :

,z i.

433

By Lemma 1.4, 0~, (Gav/~b)=~/9 &, 77(~/r b. By Lemma 1.3 7:/~ has a complement ~7/r

in ~/~. Let ~-/r be a complement of ~ /98 in ~/q~ containing J.7/~&. Clearly,

is f-critical in ~ . If ~:f]~-= ~ for any 5, then Zi/q6 and ~n~/~ commute

element-wise, hence ZT/] ~/~/r . Then ~7~ ~:Tqb ~ ~Fr/~, and, since ~'/@6 is the Fitting

subgroup of $~:/~ , it follows that 77/q6=G~q6 , i.e., ~ ~" is nilpotent. This contra-

dicts the hypothesis. Thus, ~f] ~ is not normal in ~ for some J " It now remains to

apply Lemmas 2.2 and 2.1.

3. iT-SUPPLEMENTS

In [14] we studied supplements of a normal subgroup K of a group ~ , i.e., the smallest

(with respect to inclusion) subgroups of ~ which, together with K , generate all of ~ .

We will now give a formational generalization of this concept.

Suppose K is a fixed normal subgroup of 0 If KN ~F is not contained in 96(~),

then ~ possesses a maximal subgroup /~! not containing K n ~ F. If m/Fn ~ is not con-

tained in ~(~i) , then /~i has a maximal subgroup ~2 not containing ~]F~ ~ . Continuing

in this way, we construct a maximal chain

with the following properties:

A) ~6 does not contain /g~ O K for any ~>O ;

B) HFn ]< c__ &(,q).

The subgroup ~ is called an ~--supplement of K in ~ , and the chain (*) an F-

supplementing chain for ~ . If KO ~F~O(O), then ~=0 and the only a U -supplement of

K in ~ is O itself. The concept of supplement is obtained from this when ~ is the empty

screen, i.e., f (A)=~ for any nontrivial group

We will need a specialization of the concept of o~-supplement. Note that in construct-

ing the F-supplementing chain (*) it would be possible to use Lemma 1.8. This would enable

us to obtain a chain (*) which satisfies property B) and the following sharpened variant of

property A):

C) M~ is f -critical in ~_! relative to ~_! D K for any ~>0

If the chain (*) possesses properties B) and C), then it is called a critically ~ -

supplementing chain for K , and the subgroup ~ is called a critical ~-supplement of K in

LEMMA 3.1. If K-

in ~-I for any ~ >O

f , then C) is equivalent to the following: ~i is ~-critical

Proof. Suppose Mf is dr-critical in /~f~_/, L ~,,0. Then l~g- I contains an 2g-ultimate

normal subgroup R~ such that MgR6 ~#ff_ I . In view of eemma 1.9, we may assume that ~gc-M~l.

Applying Lemma 1.2, we see that Mfc~4f! c-~ >- . Therefore, ~c#4P_ b_/ n 0~-/~/~_f f]/(, which

434

that ~ is ~-critical in M6_! relative to ~ N ~. means

We s h o u l d emphas i ze t h a t each normal s u b g r o u p p o s s e s s e s a t l e a s t one c r i t i c a l

p l e m e n t . I f ~ i s t he empty s c r e e n , we o b t a i n t h e c o n c e p t o f c r i t i c a l s u p p l e m e n t .

f-sup-

The following definition obviously provides perspective. A subgroup ~ is called an

fl- -complement of a subgroup L in if HL=$ and ~L= f �9 We can seek a condition for

Jr -complementability of subgroups and study groups with various systems of ~-complemented

subgroups.

The next lemma follows directly from the definiton of r and Lemma 1.2.

LEMMA 3.2. Suppose H is an Jr-supplement of a normal subgroup K of G Then

HF~ G

LEMMA 3.3. Any jC-supplement of ~ s belongs to jc.

Proof. Suppose ~ is an JC-supplement of ~ p By definition, ~O~qb(~) .

eemma 3.2, //~--c~ ~ Therefore H~r , and hence ~/--/, since jc is saturated.

By

LEMMA 3.4. Suppose ~ is a critical ~-supplement of a normal subgroup ~ of

and F is a critical jC-supplement of a normal subgroup ~ of H . If XLO~F= L .

then F is a critical ~ -supplement of KL in $ .

Proof. Suppose

H=r ~@ ~ . . . ~,c" = Z ~ ; o , O )

~ = ~ =~ ~ . . . ~ H ~ = S , ~ o ,

are critically jC_supplementing chains for ~ in ~ and for

Consider the chain

in ~ , respectively.

=4 =... =Jl =H=z.

By hypothesis, for any b, O=Z.< Z , the subgroup Hi is r in #/~_/ relative to

#/~_In K. Since /~g_if]J(-cY/f_/f]/fL, it follows that //i is ~-critical in #/~_! relative to

#/c'-: ~/(/'" Also, for any 7' 0 ~j~ ~, the subgroup ~' is f-critical in ~'-i relative to ~I ~ L

and, therefore, relative to ~-I n/(L �9 Suppose KL~HF=L �9 Since PSaLm_ qb(F) and, by

Lemma 3.2, .c'c ~H F , we have F'Cfl /(L = F'CFI/'/FFI KL= FFF~ L ~ qb(F).


4. GENERALIZATION OF M. I. KARGAPOLOV'S THEOREM

By a principal f-series of a group ~ we mean a series

in which ~t =~" and the part from i to ~ is a G -principal series of ~E.

tion of subgroups ~,R 2 ..... ~+! is called an ~C_decomposition of

series (~) if the following conditions are satisfied:

(R)

A c o l l e c -

c o r r e s p o n d i n g to

435

an

2) #e (~-) ~ ~i for any Z ~ ;

4) 7~g,l...Rt+ 1 is a critical ~-supplement of ~Lr in ~ for any i=he . . . . ,~+r

When ~F=/ a principal o C -series assumes the form /=$o~ = ~ and there corresponds

-decomposition consisting of a single subgroup

THEOREM 4.1. To any principal ~-series of .~ there corresponds at least one ~--

decomposition of ~ .

Proof. Suppose ~ is a group of least order for which the theorem fails.

has a principal ~-series

~c where ~ = # -~! , but

by 97 i .

Then #

has no f-decomposition corresponding to (i).

(i)

Denote qf(~i 16_, )

Choose from among the terms of series (i) that Sm for which the following is true:

~ is not contained in ~(~), but ~_f is contained in q6(G). Such a choice is

possible, since /=~ ~(O) and ~ is not contained in qO[~) (recall that the forma-

tion ~ is saturated).

The subgroup $~ , where 0 ~ , is nilpotent and therefore possesses an ~ -subgroup

~ which is normal in ~ . Since ~n_r , it follows from Lemma 1.4 that ~ has an

~ -subgroup R~ which is normal in $ . Assume that n=~ and consider a critical ~--sup-

plement ~ of ~t in ~ . It is easy to see that the system ~,~z ..... ~, H is the

desired ~r -decomposition of G Thus, we will assume in the sequel that n<~. As before,

let H denote some critical S-supplement of ~a in ~ Consider the series

(2)

where Hi=H~n ~, i=n;~+<...,~ we have

we see that

By Lemma 1.2, H I/n= ~ , #/#

)=G #i,

Since for ~mf />/Z.

O z , (3)

It follows from the definition of ~0 that for any i=o,o~/,...,[ we have HiN~ =H n.

Using this fact and (3), we obtain an //-isomorphism:

~+I / ~ ~-- H[,t! //'/~ ' Z=n'/~t/ ..... ~-I. (4)

Since ~/~= ~ , the factor ~L§ for ~t-/ is an ~ -principal factor of ~, hence

the #/-isomorphism (4)implies that the factors of series (2) in the part from /~n to ~/t

436

are principal factors of H If we now refine series (2), we obtain a principal ~-series

of H :

4 . . . (5)

Since f i / r , the theorem is true for H Thus, H has an ff-decomposition

2~,, ~ , . . . . 3 s , / ~ + ~ , ~ , ~ . . . . . ~, , (6)

corresponding to series (5). If /'/~=/, we have 2-$ , and the system degener-

ates into the empty set. Since #/=//~c ~ c_~(~), according to the definition of f-sup-

plement, when $> 0 the subgroups Z~,2e,...,Z~ $ are contained in ~(H). In particular,

this implies that ~,+~n§ ...~+I=H . According to the definition of ff-decomposition,

the system of subgroups (6) possesses the following properties:

++: : ~ ,~_~ , i = ~ + ~ . . . . . t+/;

Therefore, in view of (3), (4), and the definition of ~Z for

~.~ ) = ~ l ~ i /0,._, ), i = / , . ~ . . . . . ~'+/.

i-~n, we obtain

.t (7)

System (6) possesses, by definition, another property: for any E=n~/,..., ~-/ the subgroup

~A~+/... ~L+ I is a critical if-supplement of #/[_! in #/ . Since ~._/ ~a =~Z_! and

~zNSZ_9~_/~ for n+/-~.< ~+7, if we apply eemma 3.4 with X=~,L=Hi_,,c=~...~, we ob-

tain the following assertion:

'~ ~-V ""/?f-! is a cducal /-supplemem

o~ G~_i in G, ~ = ~+ i .... , t-+ I. I (8)

Since //=.V~,Rn§ ~t+, is a critical ~--supplement of ~8 in $ , and Sn-~ ~qb[$), we

see that assertion (8) is true for any Z=/,2, o..,~+I �9 Theorem 4.1 is completely proved.

THEOREM 4.2. Suppose ~,Rz,..,,~+ ~ is an 3C-decomposition of a group ~ not belonging

to jc . Then ~>~/ and the following assertions are true:

l) o ; = ,~, r~ . . . R~ ;

2) ~I belongs to f and normalizes some minimal G-system of ~f;

3) if / is the empty screen, then ~t~=/ and the system of subgroups ~, % ..... ~ is

a principal decomposition of $ in the sense of Kargapolov [6].

Proof. Assertions i) and 3) follow directly from the definition of ~-decomposition

(the definition of empty screen is given in w Lemma 3.3 implies that ~+~6J C. For any

437

~+! the normalizer of ~ contains ~+/, and ~(R~) coincides with the set of prime

divisors of the order of the corresponding ~ -principal factor of ~F Let qr={~7,

~2 .... ,/~ ~ be the set of all primes ~ for which ~ is ~ -solvable. Let ~d be the

product of all subgroups among ~'~z .... ,~, which are ~-groups, / ~ a Let $ be !

the product of all subgroups among ~:,~2,...,~ which are ~ -groups. Then ~+! nor-

malizes the minimal if-system $ of ~f, where

2) S - { ~ , ~ . . . . . Pn~ , i f vT[--Og ( ~ : ) ,

3) S---{SO~J, if ~-'=.~.

The theorem is proved.

Note that if : is the empty screen, Theorem 4.1 becomes an extension of Kargapolov's

theorem, since our principal decomposition satisfies stronger requirements.

5. ~ -NORMALIZERS

We now want to study critical ~ -supplements of the Jf-coradical of a group. Among

such subgroups we find, in particular, the last member of any ~-decomposition of the

group and also the Ys -normalizers of solvable groups introduced in [8].

Definition i. By an Jff-normalizer of an arbitrary group ~ we mean any critical

-supplement of ~F in ~ .

As Lemma 3.1 shows, Definition i is equivalent to the following.

Definition 2. A subgroup ~ is called an ~-norma!izer of a group ~ if ~ej ~

there exists a chain ~=~0D~ D... ~M~-~, t~S, such that M~ is

for any ~>0 .

LEMMA 5.1. Each JY-normalizer of ~ normalizes some minimal

Proof. Suppose ~ is an ~-normalizer of ~ .

menting chain for ~ ~ in ~ :

-critical in ]~_!

-system of ~F

Consider a critically ~-supple-

and

If H=~, the lemma is trivial. Suppose Of~ / . In view of Lemma 1.9, for any g=/,2 . . . . ,~

the group Hg_/ has an f - u l t i m a t e normal subgroup ~ wi th the fo l l owing p r o p e r t i e s :

a) ~ H~ = Hi_/ ;

b> C Z,_', ,

Note also that /{.:_m /{.f for ~ > j . Thus ~- ~o . .~ / / , ~ ~ . ~ = ~ " the subgroups

5 c o = u t e pai wise, and the numbers tZi/Z ,r d iv ide the orders of 0 - p r i n -

c ipa l f a c t o r s of ~ . As in the proof of Theorem 4 .2 , by m u l t i p l y i n g a p p r o p r i a t e subgroups

f ' " " 5 we obtain a [-system S of ~r as desired.

438

THEOREM 5.1. Suppose 7/ is an 2C-normalizer of ~ . Then the following assertions

are true :

i) H covers each ~--central principal factor of ~ ;

2) ~ does not cover any non-Abelian f -eccentric principal factor of ~ ;

3) ~ isolates all Abelian 2 c-eccentric principal factors ~/K of ~ for which

the group ~GF (L//() is ~ (&/K) -solvable.

Proof. Suppose we are given a critically ~r-supplementing chain for ~[ in ~ :

~=~-~D...D~t=~ . If ~=0, then /{--~and the theorem is true. We may therefore sup-

pose that ~ >0 . Since ~ is jc -critical in ~ , we have /~I ~ = ~, where �9 is some

jC -ultimate normal subgroup of ~ . In view of Lemma 1.9, we may assume, without loss of

generality, that R- , ff (Z6)_c~(R/~), where @6=ROqE (~) . Let ~ /K be some (fixed) principal

factor of ~ We denote ~G (L /K) by ~ and consider the following cases.

I) #/I does not cover L /K .

Then Kz~,L//I = ~, Since #/I is JC-abnormal, /,/K is JC-eecentric in $ Since

#/c__~, it follows that // does not cover L//(. If L/K is Abelian, we have

#/NLc--~@/,=K, which means that H isolates L/K . Thus, we see that the theorem is true

in case i).

2) ~ covers L /K .

Then /( }(, -D L , /( (~RZ)=L, hence we have an 4 -isomorphism:

L /,# "." /_, n /.-t! / K n /-/~. (1)

We consider two alternatives.

2.1) L/K is f-central in G, i.e., G/Cef(L/K).

Since f is an inner screen, we have G/Cc ~, which implies that C-~ G #. But then

C = ~, hence ~ induces on &/,# the same group of automorphisms as HI . Therefore, in

view of (i), LNff I /KnH I is an f-central principal factor of ~ Since H is an .~-

normalizer of HI , it follows by induction that #/ covers /,r]HI/K f I!-! / , i.e.,

(Kn>/ , ) = HKnH, -=/_, n>,,.

Therefore, ~K-~K(LO~)=~n/(HI=L. i .e . , ~ covers ~/K as required.

2.2) L//< is YC-eccentric in ~ .

Assume that ~ go = ~ . Then the ~ -isomorphism (i) implies that L0 H!/K ~ ~ is an

-eccentric principal factor of ~ If ~-mL , then ~(~ ~K)=HI(n#/I-mLnHI. If

//~o~,~HNL-c~n~, this means that H isolates Z/K. Note also that ~=$ implies

GF~/C=~~,/C~Y/IF/~R~/-, and, in view of (I), CO~-c~HI (L@~//<n,~ ). From the above

argument we can conclude that if ~ C = ~, the theorem is true by induction for /41 , hence

also for ~ Therefore, we need only consider the case where H l _D ~, which implies //7--m w/

439

(since /(c-C and H! covers ~/K). Thus, we will assume in the sequel that

~ CL. (2)

Since ~c_FIG) ~C , it follows from (2) and the equality H,R~-8 that R~C=q b. There-

fore, we have a ~-isomorphism:

~/ ~ ee ~C/~. (3)

We now consider two subcases of 2.2)

2.2.1) ~/X is Abelian.

Suppose ~f~L/KI= ~J and ~PC/L ~ is ~-solvable. Then RC/C is also p-solvable.

Therefore, in view of (3), ~/~ is p-solvable. But then R/qb, hence also R , since

~(o6~_c~/qb~, is either a #,-group or a ~t-group. By Theorem VI.5.4 of [12], ~ -~C ,

which contradicts (2) and the equality HI R= G.

2.2.2) L/N is non-Abelian.

We must show that #/ does not cover ~/K. Suppose, on the contrary, that #//~

Since L //# is non-Abelian, we have 'J ~0 = ~. Therefore, we have a ~-isomorphism:

LC /C ~- L/'K. (4)

Suppose >/I has an .C-eccentric principal factor ~/T such that ~_DSDT ~ K. Since

/K is non-Abelian, $/]- is non-Abelian. By induction, #/ does not cover S/T , which

contradicts H/#-~ . Suppose now that all ~-principal factors of L/K are ~-central in

Since ~-cH I , eemma 1.5 implies that H,/0 s ~. Therefore, .~//~_cC, hence

~/C=~ Consider ~/C=HC/~'RC/~. Since ~ is not containedin ~ it follows that ~/C

is not contained in #f0/C Since ~/~_c#/C/C , we can assert that ZO/~ and RO/~, in

view of (3) and (4), areminimal normal subgroups , andcommute elementwise. This means that ~

centralizes ~/0 . In view of the ~ -isomorphism (4), ~-~ , which is impossible.

The theorem is completely proved.

In the case where I is a local screen of the formation of all nilpotent (all super-

solvable) groups, an ~-normalizer will be called a nilpotent (resp., supersolvable) nor-

malizer.

COROLLARY 5.1.1. A nilpotent normalizer of a group G is a nilpotent subgroup which

covers each central principal factor of G.

COROLLARY 5.1.2. A supersolvable normalizer of a group G is a supersolvable subgroup

which covers each cyclic principal factor of ~.

THEOREM 5.2. If ~ is a homomorphism of a group G and H is an F-normalizer of

G , then H ~ is an f-normalizer of ~

Proof Consider a critically f-supplementing chain for G F �9 in

440

If ~=O, then ~e~ and the theorem is obvious. Suppose ~>O . The subgroup #/ is an

-normalizer of ~/ By induction, /~ is an ~-normalizer of H~ If ~/~ ~ . = , the

theorem is proved. Suppose #/~ , i.e. , #/I ~ is a maximal subgroup of ~z . Let ~ he

an ~--ultimate normal subgroup of ~ such that ~=~,~-~f Denote ~096(~] by ~.

Since ~ % ~ and ~%# ~7 , it follows that ~ is not contained in ~f)~. Moreover,

q~--~C~) . Therefore, it is clear that ~/q 6~ is a principal factor of ~ Since

~-~[~%]I it follows that #/~ is an ~-critical subgroup of ~ Therefore, since /~

is an ~ -normalizer of ~ ~ , we can assert that is an ~-normalizer of ~


THEOREM 5.3. Suppose ~ and ~ are inner homogeneous screens such that ~ . Then A A

for any ~-normalizer ~ of a group ~ there exists an ~-normalizer ~ of ~ such

that ~ is an ~ -normalizer of ~. A A

P r o o f . Suppose ~ i s an & - u l t i m a t e no rma l s u b g r o u p o f ~ such t h a t R ~ ~

(see Lemma 1.8). Since $ ~ , it follows that ~ is also an ~-ultimate normal subgroup

of ~ . T h e r e f o r e , e a c h ~ - c r i t i c a l s u b g r o u p i s ~ - c r i t i c a l . C o n s e q u e n t l y , we can o b t a i n

an ~ -normalizer of ~ by extending a critical ~ -supplementing chain for ~ in ~ �9

COROLLARY 5 . 3 . 1 . Each s u p e r s o l v a b l e n o r m a l i z e r o f a g roup ~ c o n t a i n s a t l e a s t one

nilpotent normalizer of

THEOREM 5.4. Suppose ~ ~ is ~ ( ~ -solvable. Then any two ~ -normalizers of ~ are

c o n j u g a t e .

P r o o f . Suppos e ~ i s a g roup o f l e a s t o r d e r f o r which t h e t h e o r e m f a i l s . Then t h e

theorem is true for the ~-abnormal maximal s~groups of ~ , since, in view of Lenm~a 1.2,

their ~-coradicals are ~(~l-solvable.

C o n s i d e r two c r i t i c a l l y ~ - s u p p l e m e n t i n g c h a i n s f o r ~ ~ i n ~ :

c . . . cM, cG,

s u c h t h a t and a r e n o t c o n j u g a t e i n G Then, o b v i o u s l y , and a r e a l s o

n o t c o n j u g a t e .

Let ~= ~[~) . Assume that G has a normal qft-subgroup Z~/ . By Theorem 5.2,

/~/IZ/Z and H~Z/Z are ~-normalizers of G/Z, and {G/ZI~c=G~cZ/Z is ~F-solvable.

Therefore, ~Z/Z and H~Z/Z are conjugate in ~/Z �9 Since #/I and ~ are ~-groups,

it is clear that ~ and ~ are conjugate in ~ . Contradiction. Therefore, 0~ (G)-/,

hence ~ (GF-]/~(~)~ ~ ~c coincides with the product of all minimal normal subgroups of

~r/Gfnq&(~) which are contained in ~/q6[~)n ~c. Put ~=~[~)0 ~,~T/q~=~( ~F/~)"

Assume that f/96 isaminimalnormal subgroupof G/q6 .. Since ~/ and /~z are ~--critical,

/~/F=~zF = ~. Let ~g be the intersection with ~ of the core of ~i in ~, ~=l,@

Obviously, Sg F/$1~r/$~Of=Tr/~b. Therefore, ($~/~)N~/~bl=qb/~6,~'=/, 2. If ~z--~=~, then, By

441

Lemma 1.15, /~I and ~ are conjugate. Consequently, one of the subgroups ~l, Sg is not

equal to ~ Suppose g ~ q6 . Then ~I/~ contains a minimal normal subgroup ~/9 b of

~/qb Since ~/@6 is the Fitting subgroup of ~/9 ~ and a minimal normal subgroup of

~/~ , it follows that ~/q6 is a ~t-group. But then, by Lemma 1.4, ~i I~)# I, which

is impossible.

Thus, ~/~ is not a minimal normal subgroup of ~/~. Let E={Lf/q 6 I ~e~} be the

set of all minimal normal subgroups of ~/q6 contained in F/q ~ . There are two possibil-

ities: either ~! and ~ complement one of the factors in ~, or they complement two

distinct factors in ~ . Let us consider these two cases.

First case: M~ L I ---- /@~L/= ~.

Since Is1 ~ / , we see, on applying Learns I.i0, that M! contains ~ for some u#l

Suppose Ls By Lemma 1.3, L,L~/~b has complement Z~7/~ in ~/~. Let

S=~Z,o The subgroup ~ is maximal in ~ , and

G= LI=$L / . c_S,

ByLe al.13, Sn , i s f - c r i t i c a l b o t h i n $ and in , and is --critical

both in S and in M z . Therefore, conjugacy of H I and /~z follows from the validity of

the theorem for ~, ~Z and S (since in this case each P-normalizer of ~n~ is an

f -normalizer of S and of /~I , and each ~-normalizer of S n M~ is an ~-normalizer

of S and of ~ ).

Second case: ~TL~ = /~.= ~, ~j-

Obviously, Lzc~2,L/c_]~I, otherwise we are back in the first case. By Lemma 1.13,

M I f]M z is ~-critical both in ~ and in /~ . Therefore, conjugacy of ~ and /~z fol-

lows from the validity of the theorem for m I and #Ig.


E. F. Shmigirev [15] considered an ~-normalizer of a group $ with solvable ~-co-

radical tobe any jC_projector of the subgroup ~ I$), where ~ is a Sylow system of $~

The following theorem established the connection between this definition and our concept of

~- -normalizer.

Recall that a subgroup ~/ of ~ is called an ~ -projector if ~ , and if ~_c~(c_~

always implies ~/~f ~-= U.

THEOREM 5.5. Suppose H is an ~--normalizer of a group ~ and ~Y" is ~-solvable,

where ~=~(F). Suppose also that $ is a C -system of ~Y such that each member of

is either a primary r -group or a ~'-group. Then there exists ~e ~ such that ~

is an P-projector of ~G (~)"

Proof. Suppose ~ is a group of least order for which the theorem fails. Let ~ be

a minimal normal subgroup of $ contained in ~ p. Obviously, K ~ ~ ~c Assume that

is a 17' -group. In view of Lemma 2.1 and Theorem 5.2, there exists x~ such that /-/~K//(

442

is an jc -projector of /~IK/K,~=~G($) . Since H x is an f -projector of ~zK , it

follows from Lemma. VI.7.9 of [12] that ~ is an ~-proJector of ~K. Since NG~=~

and KG(NK) ~, Lemma 1.2 implies that (~X)F=N~,~< Therefore, in view of assertion

2) of Lemma 2.1, (NK) f is a direct product of a nilpotent ~ -group and a ~t-group.

But then each principal factor of NK is either a ~-group or a ~'-group. Consequently,

NK is either ~ -solvable or ~' -solvable�9 Therefore, ~x~ ~ N for some o e XK �9 Since H m~

is an ~-projector of ~K, it follows that ~xr is an /-projector of ~ . Contradiction.

Suppose ~iI~F)=/. Note that the theorem is true for ~-abnormal maximal subgroups.

Therefore, in particular, ~ ~ is not nilpotent. By Lemma 2.3, ~ has an ~C-critical sub-

group ~ such that the following condition is satisfied: S is reduced in ~ , ~ (~)_c

~M(2nM f), and an d c -normallzer T of M is an ~-normalizer of $ , hence, by Theorem

5�9 ~= T for some ~6~. Since the theorem is true for ~ , there exists E6 ~ such

that ~z is an /C-projector of ~f~SO ~f). Since N&($)~M($nM~I , it remains to show

that H ~zc - A~ G (S).

Since M is ~--critical in ~ , we have ~=~ , where ~ is an ~-ultimate normal

subgroup of & In view of Lemma 1.9, we may assume that R~GF~(~nc/)(~(/~/P~{~

Since 0~,(~)=/, it follows that ~ is a p-group, pE~. Let ;n~E={~,~,...,~},

where s is a p-g oup. Since it is clear tha ..... Since

normalizes $ ~ ~ it follows that ~z normalizes $ The theorem is completely proved

COROLLARY 5 5�9 Suppose ~ is an ~-projector of a group ~ and ~ ~ �9 is W(f)-solvable.

Then there exists a minimal C-system $ of ~ ~ such that /~ is an ~C-projector of

THEOREM 5.6. Suppose ~ = ~(~) and ~ ~ is ' ~ -solvable. Then the following asseT-

tions are true:

i) if $~(~)=~ and lis an jC-normalizer of $, then T=Hn$ for some ~ -nor-

malizer ~ of ~ ;

2) if $0~,(~) = ~, then each 3 c -normalizer of ~ is an ~-normalizer of ~ ;

3) if ~ is an JC-abnormal maximal subgroup of G , then an ~C-normalizer of

contains some jc -normalizer of ~ .

Proof�9 Note that Lemma 1.2 guarantees the preservation of 9[-solvability of the jC_

coradical for the subgroups considered in the proof. Let us prove the first assertion of

the theorem. Suppose S~= ~, where ~=~G) Consider first the case where S is maxi-

mal in ~ . If S is ~--abnormal, thenit is 3C-critical andan ~C-normalizer of this sub-

group is an JC-normalizer of G . Suppose ~ is jc -normal in ~ and A/B is an ~ -

central principal p -factor of ~ , complemented by it, B~(G),p~,~c[. We may assume,

without loss of generality, that ~ is a ~ -group. Suppose ~ has a minimal normal sub-

group ~ which is a 9~'-group. Since $ is jC-normal, we have ~-DK. Since the theorem

is true for ~/K by induction, T/( //(=///(//( f] S//(, where T and // are 3C-normalizers of

443

S and G, respectively. Then TK=IIKoS=K[IInS), hence 7"=/'/'~0~ for some ~e/( Thus,

we may assume in this case that O~, [G)=/0

Suppose /VR= G, where 7F is maximal in ~ and R is an f-ultimate normal subgroup

of G . Denote l~nq~{G) by q5 . In view of eemma 1.9, we may assume that ~G~,~Z[=/~),~

~{l~/Clb) . Since 0~,(~)=/ and ~ is ~-solvable, ~ is primary. Obviously, ~ is

covers ~/~6, and /~ covers A/~. Therefore, $ ~ R,~fB- ~A. Consequently,

$=lMnS)~?, A=BCAnM), G=SA=S(AoM), ~I= MnO =~fn ,S (AoM)= [An~)(MnS)=~(M)(MnS).

Since ~=SFI~] and ~ induces the same group of automorphisms on R/9 b, it follows that

#fO~ is maximal and ~-critical in S . Therefore, an ~-normalizer T of ~n~ is an

~- -normalizer of $ . Since the theorem is true for ~ by induction, we have /in [~fl

$)=~f7$=7 for some ~--normalizer H of /14 , which is also an ~-normalizer of ~.

Thus, we have established the validity of the first assertion of the theorem for each

maximal subgroup of G which does not contain T'=~) . Now suppose SP=~ and $ is not

maximal in ~ . Suppose $i contains $ and is a maximal subgroup of ~ . Then $I =

~($I O P). By induction, ~/I r]$=7", where ~ and T are a c-normalizers of St and

respectively. Since ~=~ and, by what has been proved, the theorem is true for ~ , we

have ~l=~n~ , where ,/-I/ is some aC-normalizer of ~. Then T=~f~$11o~=#/n~ , as required.

The first assertion of the theorem is proved.

Let us prove the second. Suppose G=SK, where K=0~, (~). Suppose 81 contains

and is maximal in ~. Obviously, $! = $I~10~), ~I is ~-critical in ~ , and the theorem

is true for ~I by induction. Then each ~--normalizer of ~ N K is an aC-normalizer both

of $ and of ~ . Since ~-normalizers are conjugate by Theorem 5.4, the second assertion

of the theorem is true.

Let us now prove the third. Suppose M is an ~--abnormal maximal subgroup of ~.

If M is ~-critical, the theorem is true. Suppose ~ is not aC-critical. Let C be

an f-critical subgroup of ~ , ~R =G , where R belongs to ~ and is an ~-ultimate

normal subgroup of ~ In view of eemma 1.9, we may assume that ~R/Rr]~[G))~Rn~6(~)).

Obviously, ~#f and, by Lemma 1.12, ~C is an ~-abnormal maximal subgroup of ~ . By

induction, an ~-normalizer X of ~C contains an ~-normalizer ~ of ~ , which is

an ~-normalizer of ~ . Since ~ is qE-solvable, ~ is either a p-group for some

~ or a ~-'-group. But then, by the first and second assertions of the theorem, an

-normalizer T of #f=/~(~f~IC) contains X . Consequently T-~ , and the theorem is com-

pletely proved.

LITERATURE CITED

i. S. A. Chunikhin, "Some trends in the development of the theory of finite groups in recent years," Usp. Mat. Nauk, 16, No. 4, 31-50 (1961).

2. S. A. Chunikhin, Subgroups of Finite Groups [in Russian], Nauka i Tekhnika, Minsk (1964). 3. M. I. Kargapolov, "Factorization of ~-separable groups," Dokl. Akad. Nauk SSSR, 114,

No. 6, 1155-1157 (1957).

444

4. M. I. Kargapolov, "On the ~ -factorization of finite groups," Uch. Zap~ Perm. Univ., 16, No. 3, 13-17 (1958).

5. M. I. Kargapolov, "Factorization of locally finite groups with finite classes of Sylow subgroups," in: Proceedings of the Third All-Union Mathematical Congress [in Russian], Vol. 4, Akad. Nauk SSSR, Moscow (1959), pp. 9-10.

6. M. I. Kargapolov, "Principal decomposition of a finite group," Uch. Zap. Perm. Univ., 17, No. 2, 5-8 (1960).

7. L. A. Shemetkov, "A theorem on indexials of finite groups," Dokl. Akad. Nauk BSSR, 12, No. ll, 969-972 (1968).

8. R. Carter and T. Hawkes, "The # -normalizers of a finite soluble group," J. Algebra, ~, No. 2, 175-202 (1967).

9. L. A. Shemetkov, "The ~'-decompositlon of a finite group," in: All-Union Algebra Sym- posium, Abstracts of Reports [in Russian], Vol. i, Gomel (1975), pp. 80-81.

i0. L. A. Shemetkov, "Graduated formations of groups," Mat. Sb., 94, No. 4, 628-648 (1974). ii. L. A. Shemetkov, "Two trends in the development of the theory of nonsimple finite groups,

Usp. Mat. Nauk, 30, No. 2, 179-198 (1975). 12. B. Huppert, Endliche Gruppen, Vol. i, Springer-Verlag, Berlin--Heidelberg--New York (1967). 13. L. A. Shemetkov, "Factorization of finite groups," Dokl. Akad. Nauk SSSR, 178, No. 3,

559-562 (1968). 14. L. A. Shemetkov, "Complements and supplements of normal subgroups of finite groups," Ukr.

Mat. Zh., No. 5, 678-689 (1971). 15. E. F. Shmigirev, " ~-Normalizers in groups with solvable f-coradicals," Dokl. Akad.

Nauk BSSR, 20, No. i, 8-11 (1976). 16. L. A. Shemetkov, "On finite solvable groups," Izv. Akad. Nauk SSSR, Ser. Mat., 3-2,

No. 3, 533-559 (1968).

SUFFICIENT CONDITIONS FOR THE EXISTENCE IN A GROUP OF INFINITE

LOCALLY FINITE SUBGROUPS

V. P. Shunkov UDC 519.45

In this paper we continue the investigations of periodic groups recently begun by the author and his students. In a 2 we find sufficient conditions for the inclusion of an element of prime order in a relatively nice infinite subgroup (with a nontrivial finite normal subgroup or in a locally finite subgroup).

As an application of Theorems i and 2 we describe the periodic conjugately biprimitively finite groups without involutions satisfying the primary minimum conditinns. It turns out that all such groups are locally finite (Theorem 3). The cases where the group .contains an involution or elements of infinite order require special consideration beyond the scope of

this paper.

Theorem 3 was proved Jointly with A. K. Shlepkin.

Our notation is standard and needs no special explanation.

i. AUXILIARY RESULTS AND NECESSARY DEFINITIONS

i. Felt--Thompson theorem [i]: A finite group of odd order is solvable.

2. A group $ and a proper subgroup H constitute a Frobenius pair if

HnH'% / (Vx Call).

Translated from Algebra i Logika, Vol. 15, No. 6, pp. 716-737, November-December, 1976.

Original article submitted November 2, 1976.

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, New York, IV. Y. 10011. No part o f this pu blica t~'on may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microJ~lming, recording or otherwise, wi thout written permission o f the publisher. A copy o f this article is

available f rom the publisher for $7.50.

445

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Factorizations of nonsimple finite groups