3-Characterizations of finite groups

18. 19. 20.

S. N. Chernikov, Groups with Given Properties of a System of Subgroups [in Russian], Nauka, Moscow (1980). A. N. Ostylovskii, "On the local finiteness of certain groups with a minimality condition for Abelian subgroups," Algebra Logika, 16, No. I, 63-73 (1977). N. Blackburn, "Some remarks on Chernikov p~groups," Illinois J. Math., 6, 421-433 (1962).

3-CHARACTERIZATIONS OF FINITE GROUPS A. A. Makhnev

UDC 512.542

Odd characterizations of finite groups are an important direction in the theory of finite groups.

Thus, one of the final steps in the identification of a simple group of char-

acteristic 2 type in the existing program of classification is the characterization by the standard component for an odd prime p . The sporadic groups 0 ~ ~(8~=ExK, property

and ~

K~a-A 6 for any e ~ E ~

[7].

contain an elementary subgroup E

S. A. Syskin characterized the groups indicated by this

We consider a more general situation.

THEOREM I.

Let the finite simple group ~ contain an elementary subgroup ~

such that C(E) is a group of odd order and for an involution ~ of ~(~

of order 9 such that

of order 9

~(E))~G$~(~{~))

and

contains a subgroup of index no greater than 2, in which the centralizers of elements of

order 3 are 3-decomposable.

Su ,oW.

Then

To reconstruct the structure of centralizers of involutions of ~{~)

we need the follow-

ing result, which generalizes a theorem of Maier [3]. THEOREM 2.

Let ~ be a finite group with noncyclic Sylow B-subgroup, ~ r ( ~ ) = { and the

centralizer of any element of order 3 have nilpotent Hall

{2,3}-subgroup.

Then either

is a solvable group without elements of order 6, of 2-rank no greater than 1, or

~-e(&) _~

In the proof of the theorem one uses the description of simple groups with a standard subgroup of type Proposition.

~3(~)

.

Then

Proof.

~

f#), Let ~ be a standard subgroup of the finite simple group ~ and ~ / ~ { ~ )

~"~/"/e, ,,,~Z

O')V.

See the corollary of [5].

We prove Theorem 2. LEMMA 1.

Let ~

be a counterexample of smallest order to Theorem 2.

If some involution ~ of ~

izes an ~3-subgroup of ~ Proof.

or

~---

centralizes an element of order 3, then ~ central-

.

Let # be an element of order 3 of £(g) . By the hypothesis of the theorem,

centralizes some ~3-subgroup of ~ ) .

Now ~ centralizes an element ~

of order 3 of the

Translated from Algebra i Logika, Vol. 24, No. 2, pp. 173-180, March-April, Original article submitted June 26, 1984.

0002-5232/85/2402-0105509.50

© 1986 Plenum Publishing Corporation

1985.

105

center of tralizes

an ~3 -subgroup of the group ~ . Again by the hypothesis of the theorem ~ cenan S3 -subgroup of ~(~,

LEMMA 2. Proof. Z ~ ) #,

If a noncyclic 3-subgroup of ~ normalizes a 2-subgroup of ~ , then C ~ B ] m ~ , Let U be an elementary subgroup of order 9 of ~ ,

Then ~ = < ~ ( U ) } ~ U e >

E~,~=#.

Now ~

(~)is

containing an element • of

and by the hypothesis of the theorem C 6 ~ ( U ) , ~ = / . a nilpotent group and

So

E~,BJ=J,

LEMMA 3. ~ is a simple group. Proof.

Let us assume that ~ ( ~ ) ~ /

without elements of order 6. for an involution 2 of ~ .

.

If ~ ( ~ e ( ~ ) ,

thenby Lemma i, ~ is a group

From this the 2-rank of ~ is not greater than i and ~ = O ( ~ ) C ( Z ) If ~ is a nonsolvable group, then

group with dihedral ~ -subgroup.

~(Z~/<'Z>is

But it follows from considerations of fusion of involu-

tions that a 3'-group with dihedral ~ -subgroup is 2'-closed. of ~ . Thus, ~ contains some component ~ .

If ~

Contradiction with the choice

centralizes an element of order 3, then

an ~-subgroup of ~ centralizes an S2 -subgroup ~ of ~ .

But in this case ~ contains a non-

identity Abelian strongly closed subgroup and in view of [15], ~ - ~ ( ~ ) with the fact that ~ t ( ~ ) = ~ . Let # # ~ , ~ S ~ 3 ( ~ ) or ~j (~) .

.

If I D N ~

is a noncyclic group, then by induction L ~ 2 ( ~ Hence, ~ O ~ : ~ N ~

ment ~----~'~(~) and ~ = ~ ( ~ ( / ~ ( ~ ) ~ ( S 0 ) of generality some element # of soC$o,e]:/

) , so

--~_ normalizes St.

~)

is a cyclic group and

some 2-element of ~ inverts~# . Let SO be an ~ -subgroup of ~ C ~ ) .

tradiction.

. Contradiction

Hence ~ = ~ ( ~ ) ,

Contradiction with the choice of ~ .

#So:SonClpo)l: ,

a nonsolvable

By the Frattini argu-

~=~Z~/(~)~f~/(~OI!_.Without But e centralizes

P• NOW0 by the hypothesis of the theorem

loss

~ ~ C (~ _) and

.. COo,~J = I. Con-

The lemma is proved.

If the sectional 2-rank of ~ is no greater than 4, then from the structure of centralizers of involutions in groups it follows from the conclusion of the basic theorem of [16] that ~--~ ~2 (~n) or ~3(~)

Hence the sectional 2-rank of ~ is greater than 4, an S2-

subgroup of ~ contains a normal elementary subgroup of order 8.

Let ~ S ~ { # ) ,

~E

SgI2CC(P)). By [6], RY:/. LEMMA 4. Froof.

~.

For some involution ~

of ~ ( ~ ) ,

Let us assume that

(C(~)))'~L~(~~) or /_4~(/f~,

is a solvable group f o r any involution t of

Then the possible 2-components of centralizers of involutions are solvable Suzuki

groups and by Theorem A of [17], ~{~(~)):/ by Lemma 2, ~ ( ~ ( ~ ) ) for ~ # . Thus, for some involution ~ o f ~

~o(C(~)))~(. ~n} or

~3(~.

we shall have an isomorphism

for any involution ~ of ~ . On the other hand,

, ~(~) is an unsolvable group.

By the structure of

of index no greater than 2 of CR{~)

106

~*~(~)/~r

A~I~ n) and

By induction, ~-~C(~}/

A~4{~)

asubgroup

X

lies in ~, (~(~)) . Now for an involution ~ o f ~ O ~ I ~

~m~)/~rI~(%)))~---~{$n)or ~8(#). Thelearns is proved.

)

J , e a -component of

C(l~:;P0) ,

2-component of

ontatning

R0=C

and L bea

containlngP .

L E M M A 5. ~/0~(~)--~3{~)0

Proof. ofR

~K

La

Let us assume that ~ / 0 5 p I ~ ) ~ 2 1 3 ~ ) 2-component of <~(~)~C~)~ .

~u.t~3(~) does ~/O~t (~) ~!

.

Then

RO----R.

if~/03t{~)~/O3t(~)

Let ~ be an involution

, then ~ / 0 3 t ( ~ ) ~ j ~ ) ,

not contain involutive automorphisms centralizing ~2~).

If ~/O{~)=e~2~,

But

Hence, ~/O3t(~) -~

then by [11], ~ is a Chevalley group of odd characteristic or

But then the hypotheses of Theorem 2 do not hold in ~ . Thus, ~/0(~)-~-~2(3 n) . Let ~ b e

of//,

andX

bed 2-component of ~H(~).

ing group for ~ )

or ~3 ~ ) "

a subgroup of ~ , Q and b be commuting involutions If X isnot of type ~ [ £ ) ,

But in the latter case

then ~ I ~ ( ~ )

is a cover-

~OI~H~))N~(~))/~]~OI~ll(~) .

Thus, the hypotheses of the theorem of [13] hold and by [13] the 2-rank of ~ is equal to i. It follows from [II] that ~ is a cyclic group.

By the hypotheses of the theorem it follows

that there are no 2-elements in ~#Z) , which induce involutive field automorphisms on ~ (~n) . Further, an ~2 -subgroup of ~(~) contains a normal elementary subgroup of order 8, so ~ - - + ~ (~) . If Ig is even, then an $2-subgroup of ~ is a dihedral subgroup of order 8 and from [19] we get a contradiction with the hypotheses of the theorem. Now either ~ ( ~ )

Hence /g is odd.

or ~ = < ~ , f

We have shown that 021~{~)/~[~[~))) --~-~2(~), that ~---2 ~

(~).

Now it follows from [2, 12]

But then the hypotheses of our theorem do not hold in ~ .

L E M M A 6. ~3s (~C~a)))=~

Proof.

/~ is odd.

for any involution a of 0 .

For an involution ~ of ~ we let ~ G ) )

= 03p 101~))).

We show that for any

commuting involutions ~ and ~ of ~ the condition of being balanced, ~ I ~ a ) ) f ] ~ ( ~ ) ~ ( ~ ) , holds.

Let X=CI~)

and ~ a ) ) N ~ O I X )

.

We recall that by Lemma 5 the possible 2-compo-

nents of centralizers of involutions are the solvable groups ~ ( ~ ) 3.12 of [18], 0 { ~ ) N X ~ 0 ( X ) f But then ~

and ~3(#) .

, where f is the 2-component of X and

By Theorem

K/~t{/~)~--~3~If).

induces a unitary automorphism on ~ ~#) and ~¿~a))~]~(6# ~< 01~{8)) •

But it

follows from Lemmas 1 and 5 that the order of an ~-subgroup of ~ is equal to 9 and ~ ) ) f ~

Let

8(C~))~-[

for some involution CZ of & .

One can show by routine modification of

the arguments of Sec. 4 of [17] that ~ contains a proper 2-generated core. a contradiction with the choice of ~ By Lemma 6, ~ is a quasisimple ~3 (#)

for any involution ~ of ~

osition ~--~H~, S L ~

or ~ .

.

The lemma is proved.

group,

I~)

From [8] we get

and by Lemma 5, < ~

is a covering group for

From this ~ is a standard subgroup and by the prop-

But these groups do not satisfy the hypotheses of the theorem.

Theorem 2 is proved. Until the end of the paper let ~ be a counterexample to Theorem I, ~ be an involution of

~(_.C')

. By Theorem 2, either ~(~)/03~I~))

is a solvable group, or

~(~)/~3tI~(~))),'~ 107

L E n A 7. Proof.

is a aolvable group for any involution t of C{E). Let us assume the contrary.

Let • be a 2-component of C(~ , containing E •

the hypothesis of the theorem £ centralizes 0(C(~)) , so Z is a quasisimpl, group. above, L"~C(G)

any

for

rem 2, L ~-- 0J(C(L)) /r~C~)=L xh~

Contradiction with the simplicity of ~ .

from [19] that ~ , ,~Z Let

If [L,L~ = {

, then by Theo-

Thus, the subgroup ~ is standard.

in view of [11], ~'v~S(8), ~/@(~) •

~4 {3) orH~$

or ~ ' ~

If ~ 2 ( S ) ,

L/Z~)~{@)

Finally, if

then it follows

by the proposition

The lemma is proved.

~"~(~(~)),

LEMMA 8.

for some ~ E &

As

. Now the hypotheses of Theorem 5 of [9] hold and by this theorem,

/.,~-~L2(8) , then

If ~-~

involution • of C(L) .

By

~S~

I~),

The subgroup ~ is densely imbedded in ~ and ~ is a cyclic or elementary group.

Let ~ be an involution of ~ ~ ~ # . From Lamas 7 and the hypotheses of the theorem we conclude that E = ~

(~(~)) . From this ~ E ~ ( ~ )

and

~=~#.

Hence the subgroup ~ is

densely imbedded in ~ . Let us assume

that the

rank of ~ is greater than 1 and ~ is not an elementary group.

By Theorem 1 of [i0], # ~ ~ ) # ~

for some ~

. By Theorem 1 of [4], ~ is an Abelian

group of period 4 or ~ is the direct product of ~

and an elementary group, where in the

latter case ~ does not contain Abelian subgroups of index 2 which are normal in N ( ~ ) , and E

is not a SyJ.ow 3-subgroup__._e.of ~(~') . Hence, ~ the structure of A

~-~Z# xZ2

or

Z~ xZ4.

subgroup of ~ # ~

{~)'

E,

and

is a subgroup

(~), ~ gJ.

.

a polydihedral group of order 16 and

Nowfor some involution ~ of ~ f ] ~ ( ~ )

l= ~

centralizes f ~

isan Abelian group and ~ R ( ~ ~) == ~ ( ~ )

, G(~)

is a nonidentity

But now a subgroup of index not greater than 2 of

. Contradiction.

If ~ is the group of quaternions, then by [11] ~ is a Chevalley group of odd characteristic, ~f{

or ~{2 " But then ~ does not contain the subgroup ~ required.

LEMMA 9. ~ Proof. one has

is a cyclic group.

Let ~ be an elementary. O~ noncyclic group.

E~--'<~(jE) IX6/~/D{~aJ@~=~

~E&-~(~)

Contradiction.

, ~ is weakly closed" in ~(~) and N(~)

view of [15], ~

is not strongly closed in N(~).

Then ~ , ( ~ $ ) ~ ( ~ )

and ~ i s a four-group.

If I~R ( ~ ) I

I~/D(~a)]<~2 for

any

is a subgroup of~odd index in ~ .

Let ~ e ~ # , ~ 6 ~

In

and ~ 6 ~ ( ~ ) - ~

•

By Theorem 3 of [10], & - ~ 6 , A F , ~ 3 ( 2 )

or

(~) K. Again ~ does not contain the subgroup F required. Let ~ be an involution from ~ .

Hence

>~.. o.--f°r some ~ E ~ - ~ ( ~ )

The lemma is proved.

In view of the structure of the centralizers of in-

volutions in groups, from the conclusion of the basic theorem of [16] we conclude that the sectional 2-rank of ~ is greater than 4 and an $~-subgroup $ of ~[Z) contains a normal elementary subgroup of order 8. %"

If ~ = < Z >

I~I>/~.

108

Hence an ~-subgroup of ~ ( ~ ) / ~ ( ~

is isomorphic to ~# or

then by [i] the sectional m-rank of ~ is not greater than 4.

If the group ~ isweakly closed in

C[~) , then

Hence

by [14] an ~2-subgroup of ~ is

,

a dihedral or polydihedral group. Thus, ~

In particular, the sectional 2-rank of ~ i s

is not weakly closed in C ( R )

morphic to ~ 8

, IRI'~

and an ~ - s u b g r o u p of N I E ) / C I F )

Let ~ be a Sylow 2-subgroup of ~ ,

sectional 2-rank of ~

is not greater than 4.

containing $.

less than 4. is iso-

Then I~: $I = ~

and the

The contradiction found proves Theorem I.

LITERATURE CITED i. 2. 3. 4. 5. 6. 7. 8. 9. I0. 11. 12. 13. 14. 15. 16. 17. 18. 19.

V. V. Kabanov and A. I. Starostln, "Finite groups in which a Sylow 2-subgroup of the centralizer of some involution is of order 16," Mat. Zametki, 18, No. 6, 869-876 (1975). V. D. Mazurov, "Centralizers of involutions in simple g r o u p s , " ~ t . Sb., 93, No. 4, 529-539 (1974). V. R. Maier, "Finite groups with nilpotent centralizers of elements of order 3," Mat. Sb., 114, No. 4, 643-651 (1981). A. A. Makhnev, "Densely imbedded subgroups of finite groups," Mat. Sb., 121, No. 4, 523-532 (1983). A. A. Makhnev, "Finite simple groups with standard subgroup of type ~3(~) ," Mat. Zametki, 37, No. i, 7-12 (1985). N. D. Podufalov, "Finite simple groups without elements of order 6," Algebra Logika, 16, No. 2, 200-203 (1977). S. A. Syskin, "3-characterization of the O'Nan-Sims group," Mat. Sb., ii, No. 3, 471478 (1981). M. Aschbacher, "Finite groups with a proper 2-generated core," Trans. Am. Math. Soc., 197, 87-112 (1974). M. Aschbacher, "On finite groups of component type," IIi. J. Math., 19, No. i, 87-115 (1975). M. Aschbacher, "Tightly embedded subgroups of finite groups," J. Algebra, 42, No. I, 85-101 (1976). M. Aschbacher, "A characterization of Chevalley groups over fields of odd order," Ann. Math., 106, Nos. 2-3, 353-468 (1977). E. Bombieri, "Thompson's problem ( ~t = 3 )," Invent. Math., 58, No. 1, 77-100 (1980). R. Foote, "Finite groups with maximal 2-component of type L~), ~odd ,"Proc. London Math. Soc., 37, No. 3, 422-458 (1978). H. Fukushima, "Weakly closed cyclic 2-subgroups in finite groups," J. Math. Soc. Jpn., 30, No. i, 133-138 (1978). D. Goldschmidt, "2-fusion in finite groups," Ann. Math., 99, No. i, 70-117 (1974). D. Gorenstein and K. Harada, "Finite groups whose 2-subgroups are generated by at most 4 elements," Mem. Am. Math. Soc., 147, 1-464 (1974). D. Gorenstein and J. Walter, "Centralizers of involutions in balanced groups," J. Algebra, 20, No. 2, 284-319 (1972). D. Gorenstein and J. Walter, "Balance and generation in finite groups," J. Algebra, 3_~3, No. 2, 224-287 (1975). M. Harris, "Finite groups having an involution centralizer with a 2-component of dihedral type," IIi. J. Math., 21, No. 3, 621-647 (1977).

109