On groups of odd order admitting an elementary 2-group of automorphisms

On Groups of Odd Order Admitting an Elementary 2-Group of Automorphisms

KARISEG. OLIVEIRA(*) - PAVELSHUMYATSKY(**) - CARMELASICA(***)

ABSTRACT- LetGbe a finite group of odd order with derived lengthk. We show that ifGis acted on by an elementary abelian groupAof order2ⁿandCG(A) has exponent e, then G has a normal series GˆG0T0G1T1 GnTnˆ1such that the quotientsGi=Tihavefk;e;ng-bounded exponent and the quotientsTi=Gi‡1are nilpotent offk;e;ng-bounded class.

Dedicated to Professor Said Sidki on the occasion of his 70th birthday

1. Introduction

LetGbe a group andAa group of automorphisms ofG. The subgroup of all elements ofGfixed byAis usually denoted byC_G(A). It is well-known that very often the structure of C_G(A) has strong influence over the structure of the whole groupG. The influence seems especially strong in the case where Gis a finite group of odd order and Ais an elementary abelian 2-group. It was shown in [5] that if Gis a finite group of derived lengthkon which an elementary abelian groupAof order 2ⁿ acts fixed- point-freely, thenGhas a normal seriesGˆN₀ N_{n 1}N_nˆ1 all of whose quotients are nilpotent and the class ofN_{i 1}=N_iis bounded with a

(*) Indirizzo dell'A.: Instituto Federal de EducacËaÄo, CieÃncia e Tecnologia de GoiaÂs, Av. UniversitaÂria, s/n, Vale das Goiabeiras, CEP: 75400-000 Inhumas GO, Brazil.

E-mail: [email protected]

(**) Indirizzo dell'A.: Department of Mathematics, Universidade de Brasilia, 70.919 Brasilia - DF, Brazil.

E-mail: [email protected]

(***) Indirizzo dell'A.: Dipartimento di Matematica, UniversitaÁ di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy.

E-mail: [email protected]

function of k and i (see also [7] for a short proof of this result). In the present paper we further exploit the techniques developed in [7]. One of the obtained results is the following theorem.

THEOREM 1.1. Let G be a finite group of odd order and of derived length k. Suppose that G admits an elementary abelian group A of auto- morphisms of order 2ⁿ such that C_G(A) has exponent e. Then G has a normal series

GˆG₀T₀G₁T₁ G_nT_nˆ1

such that the quotients G_i=T_i have fk;e;ng-bounded exponent and the quotients Ti=Gi‡1are nilpotent offk;e;ng-bounded class.

The other result deals with the situation whereg_c(C_G(a)) has exponent efor all automorphismsa2A^#. HereA^# denotes the set of non-trivial elements of A and g_i(H) stands for the i-th term of the lower central series.

THEOREM 1.2. Let G be a finite group of odd order and of derived length k. Suppose that a four-group A acts on G in such way thatg_c(C_G(a)) has exponent e for all a2A^#. Then G has a normal series

GˆT4T3T2T1T0ˆ1

such that the quotients T4=T3and T2=T1are nilpotent offe;c;kg-bounded class and the quotients T₃=T₂and T₁havefe;c;kg-bounded exponent.

Throughout the paper we say that a group G is nilpotent of class c meaning that the class ofGis at mostcand we say thatGhas exponente meaning that the exponent of G divides e. The expression ``(a;b;. . .)- bounded'' stands for ``bounded from above by a function depending only on the parametersa,b,. . .''.

2. Auxiliary Results

The following lemmas are well-known (see for example Theorem 6.2.2, Theorem 6.2.4 and Theorem 10.4.1 in [1]).

LEMMA2.1. Let G be a finite group admitting a coprime group of au- tomorphisms A. Then we have

a) CG=N(A)ˆCG(A)N=N for any A-invariant normal subgroup N of G;

b) GˆC_G(A)[G;A];

c) [G;A]ˆ[G;A;A];

d) If G is abelian, then GˆC_G(A)[G;A].

LEMMA 2.2. Let G be a finite group admitting an abelian non-cyclic coprime group of automorphisms A. Then Gˆ hC_G(a)ja2A^#i.

LEMMA 2.3. Let G be a finite group of odd order admitting an auto- morphism a of order 2such that Gˆ[G;a]. Suppose that N is an a-in- variant normal subgroup of G such that C_N(a)ˆ1. Then NZ(G).

Our next lemma is immediate from [3, Lemma 2.6].

LEMMA2.4. Let G be a finite group admitting a coprime group of au- tomorphisms A. Suppose that G is generated by a family fH_iji2Igof normal A-invariant subgroups. Then

C_G(A)ˆ hCHi(A)ji2Ii:

LEMMA2.5. Let G be a metabelian finite group of odd order admitting an automorphism a of order 2 such that C_G(a) has exponent e and Gˆ[G;a]. Then G⁰has exponent e.

PROOF. Since G=G⁰ is abelian and Gˆ[G;a], by Lemma 2.1 C_G=G⁰(a)ˆ1 and soC_G(a)G⁰. LetNbe the normal closure ofC_G(a) inG.

SinceG⁰is abelian we conclude thatNhas exponente. On the other hand, G=Nis abelian becauseaacts onG=Nfixed-point-freely. ThereforeG⁰ˆN

and the result follows. p

PROPOSITION2.6. Let G be a finite group of odd order and of derived length k admitting an automorphism a of order 2 such that C_G(a) has exponent e and Gˆ[G;a]. Then G⁰hasfe;kg-bounded exponent.

PROOF. In view of Lemma 2.5 the result is obvious ifk2. Suppose that k3 and let MˆG^{(k 1)}. By induction we conclude that G⁰=M has fe;kg-bounded exponent. Hence, it is enough to show thatM hasfe;kg- bounded exponent. Working with the quotientG=hC_M(a)^Giwe can simply assume thatC_M(a)ˆ1. Lemma 2.3 now shows thatMZ(G). Therefore G^{(k 2)}is nilpotent of class 2. SinceG^{(k 2)}=Mhasfe;kg-bounded exponent, it follows that G^{(k 2)} has fe;kg-bounded exponent [4, Theorem 2.5.2]. In

particular,Mhasfe;kg-bounded exponent. p

A well-known theorem of Hall says that ifGis a soluble group of de- rived lengthkand all metabelian sections ofGare nilpotent of class at most c, then Gis nilpotent of fk;cg-bounded class [2]. We will require the fol- lowing related result obtained in [6]. We denote byf(g;c) the expression (g 1)c(c‡1)

2 ‡c.

THEOREM2.7. Let G be a group and N a nilpotent normal subgroup of G of class g such thatg_c‡1(G=N⁰)has exponent e. Suppose thatg_c‡1(G)is soluble of derived length d. Theng_f(g;c)‡1(G)has finitefg;e;c;dg-bounded exponent.

The next proposition was obtained in [7, Corollary 3.2]. It plays a crucial role in the subsequent proofs.

PROPOSITION2.8. There exists a number sˆs(k;n)depending only on k and n with the following property. Suppose that G is a finite group of odd order that is soluble with derived length at most k. Assume that an elementary group A of order 2ⁿ acts on G and let R be a normal A- invariant subgroup of G such that CR(A)ˆ1. Set Nˆ T

a2A^#[G;a]. Then [R;N;|‚‚‚‚‚‚{z‚‚‚‚‚‚}. . .;N

s

]ˆ1.

3. Main Results

PROPOSITION3.1. Let G be a finite group of odd order and of derived length k. Suppose that G admits an elementary abelian group of auto- morphisms A of order2ⁿsuch that C_G(A)has exponent e. Then T

a2A^#[G;a]

is an extension of a group of fk;e;ng-bounded exponent by a nilpotent group offk;e;ng-bounded class.

PROOF. SetNˆ T

a2A^#[G;a] and use induction on the derived length of N. IfNis abelian, the result is trivial. Suppose that the derived length ofN is greater than or equal to 2 and letLbe the metabelian term of the derived series ofN. PutMˆL⁰\C_G(A) and letDbe the normal closure ofMinG.

SinceDis abelian with generators of ordere, we see thatDhas exponente.

Considering the quotient G=D we can simply suppose that C_L⁰(A)ˆ1.

Proposition 2.8 shows thatL⁰Z_s(L) and soLis nilpotent of classs‡1. By

induction,N=L⁰is an extension of a group offk;e;ng-bounded exponent by a nilpotent group offk;e;ng-bounded class, sayc. Since the derived length ofg_c‡1(N) is at mostk, by Theorem 2.7 we conclude thatg_f(s‡1;c)‡1(N) has fk;e;ng-bounded exponent, as required. p Given a groupHand positive integerseandc, we denote byV(H) the verbal subgroup ofHcorresponding to the group-word [x1;. . .;xc]^eand by W(H) the verbal subgroup of H corresponding to the group-word [x₁;. . .;x_{3c 2}]^e3.

LEMMA3.2. Let G be a finite metabelian group of odd order admitting a four-group of automorphisms A such thatg_c(CG(a))has exponent e for all a2A^#. Then there exists a normal subgroup M of G such that C_M(A)ˆ1 andg_{3c 2}(G=M)has exponent e³.

PROOF. For every a2A^# put G_aˆG⁰C_G(a). It is easy to see that [Ga;a]ˆ[G⁰;a] is normal inGa. SinceGa=[G⁰;a] is isomorphic to a quotient ofC_G(a), it follows thatV(Ga) is contained in [G⁰;a]. SinceG⁰is abelian, by Lemma 2.1 we have [G⁰;a]\C_G(a)ˆ1. It follows thatV(Ga)\C_G(a)ˆ1.

LetMˆ Q

a2A^#V(G_a), obviouslyMis a normal subgroup ofG. By Lemma 2.4, C_M(A)ˆ hC_V(Ga)(A)ja2A^#iand so C_M(A)ˆ1. By Lemma 2.2 we have Gˆ hG_aja2A^#i. Thus,G=Mis product of three normal subgroups each of which is an extension of a group of exponenteby a nilpotent group of class c 1. ThusG=Mis an extension of a group of exponente³ by a nilpotent

group of class 3c 3. p

LEMMA 3.3. Let G be a metabelian group such that g_c(G=Z(G)) has exponent e. Theng_c‡1(G)has exponent e.

PROOF. Let Eˆg_c(G). Then E=(Z(G)\E) has exponente. For arbi- trary elementsx1;x2;. . .;xc;xc‡12Gwe have

[x₁;x₂;. . .;x_c]2E:

Therefore

[x₁;x₂;. . .;x_c]^e2Z(G)\E;

whence

[[x1;x2;. . .;xc]^e;xc‡1]ˆ1:

Since [x1;x2;. . .;xc]2G⁰, which is a normal abelian subgroup of G, we

obtain that

[x₁;x₂;. . .;x_c;x_c‡1]^eˆ[[x₁;x₂;. . .;x_c]^e;x_c‡1]ˆ1:

Hence,g_c‡1(G) has exponente. p

PROPOSITION3.4. Let G be a finite group of odd order and of derived length k admitting a four-group of automorphisms A such thatg_c(C_G(a)) has exponent e for all a2A^#. Then T

a2A^#[G;a]is an extension of a group of fe;c;kg-bounded exponent by a nilpotent group of fe;c;kg-bounded class.

PROOF. LetNˆ T

a2A^#[G;a]. We use induction on the derived length of N. IfNis abelian the result is trivial. Suppose that the derived length ofNis greater than or equal to 2 and letLbe the metabelian term of the derived series ofN. First we will show thatLis an extension of a group offe;c;kg- bounded exponent by a nilpotent group of fe;c;kg-bounded class. By Lemma 3.2 there exists a normal subgroupMofLsuch thatC_M(A)ˆ1 and g_{3c 2}(L=M) has exponente³. ThusW(L)Mand soW(L)\C_L(A)ˆ1. If W(L)ˆ1, then g_{3c 2}(L) has exponent e³. Suppose that W(L)6ˆ1. Since W(L)\C_G(A)ˆ1 it follows from Proposition 2.8 that there exists a boun- ded numberssuch that

[W(L);N;|‚‚‚‚‚‚{z‚‚‚‚‚‚}. . .;N

s

]ˆ1:

Let t be the smallest number such that W(L)Zt(N). We know that ts. It will be shown by induction ontthatLis an extension of a group of fe;c;kg-bounded exponent by a nilpotent group of fe;c;kg-bounded class. For tˆ1 we have the inclusion W(L)Z(N). In this case L=(L\Z(N)) is an extension of a group of exponent e³ by a nilpotent group of class 3c 3. By Lemma 3.3 we deduce that g_{3c 1}(L) has ex- ponent e³, as required.

Suppose now that t2 and the result is valid for t 1. Let Kˆ[W(L);N;|‚‚‚‚‚‚{z‚‚‚‚‚‚}. . .;N

t 1

]. It is clear thatKZ(N). By induction it follows that L=K is an extension of a group offe;c;kg-bounded exponent by a nilpotent group of fe;c;kg-bounded class and by Lemma 3.3 also L is an extension of a group of fe;c;kg-bounded exponent by a nilpotent group of fe;c;kg-bounded class, say c1. Put Hˆg_c1‡1(L). So H has fe;c;kg-bounded exponent. Passing to the quotientG=Hwe can simply

assume that Hˆ1. By induction on the derived length of N, we deduce that N=L⁰ is an extension of a group of fe;c;kg-bounded exponent by a nilpotent group of fe;c;kg-bounded class, say c2. B y Theorem 2.7 we conclude that g_f(c1;c2)‡1(N) has fe;c;kg-bounded

exponent, as required. p

Now we are in a position to prove our main results.

PROOF OF THEOREM 1.1. If nˆ1, Proposition 2.6 guarantees that [G;a]⁰hasfe;kg-bounded exponent and so

GˆG0[G;a]ˆT0[G;a]⁰ˆG1T1ˆ1

is the required series. Suppose thatn2 and use induction onn. For every normalA-invariant subgroupNof Gthe group Ainduces a group of au- tomorphisms ofG=N. In particularAinduces a group of automorphismsBa

ofG=[G;a] for everya2A^#withjBaj 2^{n 1}andC_G=[G;a](Ba) of exponente.

Let Bbe the elementary abelian group of order 2^{n 1}. We can define an action of B on G=[G;a] as follows. For all a2A^# we consider B_a as a subgroup ofBand we can writeBˆB_aD_afor a suitable subgroupD_aof B. For any b2Bthere exists a unique pair (b1;b2)2BaDa such that bˆb1b2. Then for any x2G=[G;a] we put x^bˆx^b1. This action is well defined andC_G=[G;a](B) has exponente. LetK ˆ Q

a2A^#G=[G;a], we have an action ofBonK that extends the action ofBon every factorG=[G;a] and C_K(B) has exponente.

By induction, K has a series of length 2(n 1) with the required properties. By Proposition 3.1 the subgroupNˆ T

a2A^#[G;a] is an extension of a group offk;e;ng-bounded exponent by a nilpotent group offk;e;ng- bounded class. SinceG=Nembeds inK the result follows. p PROOF OFTHEOREM1.2. Each factorG=[G;a] is isomorphic to a quo- tient ofC_G(a), so it is an extension of a group of exponenteby a nilpotent group of classc 1. ThereforeKˆ Q

a2A^#G=[G;a] is an extension of a group of exponent e³ by a nilpotent group of class 3c 3. Let Nˆ T

a2A^#[G;a].

Since G=N embeds in K, we conclude that g_{3c 2}(G=N) has exponent e³. Proposition 3.4 ensures that N is an extension of a group of fe;c;kg- bounded exponent by a nilpotent group of fe;c;kg-bounded class. The

proof is complete. p

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Manoscritto pervenuto in redazione il 24 febbraio 2011.