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 GG0T0G1T1 GnTn1such that the quotientsGi=Tihavefk;e;ng-bounded exponent and the quotientsTi=Gi1are 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 byCG(A). It is well-known that very often the structure of CG(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 seriesGN0 Nn 1Nn1 all of whose quotients are nilpotent and the class ofNi 1=Niis 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: karisemat@gmail.com
(**) Indirizzo dell'A.: Department of Mathematics, Universidade de Brasilia, 70.919 Brasilia - DF, Brazil.
E-mail: pavel@mat.unb.br
(***) Indirizzo dell'A.: Dipartimento di Matematica, UniversitaÁ di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy.
E-mail: csica@unisa.it
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 CG(A) has exponent e. Then G has a normal series
GG0T0G1T1 GnTn1
such that the quotients Gi=Ti have fk;e;ng-bounded exponent and the quotients Ti=Gi1are nilpotent offk;e;ng-bounded class.
The other result deals with the situation wheregc(CG(a)) has exponent efor all automorphismsa2A#. HereA# denotes the set of non-trivial elements of A and gi(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 thatgc(CG(a)) has exponent e for all a2A#. Then G has a normal series
GT4T3T2T1T01
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)CG(A)N=N for any A-invariant normal subgroup N of G;
b) GCG(A)[G;A];
c) [G;A][G;A;A];
d) If G is abelian, then GCG(A)[G;A].
LEMMA 2.2. Let G be a finite group admitting an abelian non-cyclic coprime group of automorphisms A. Then G hCG(a)ja2A#i.
LEMMA 2.3. Let G be a finite group of odd order admitting an auto- morphism a of order 2such that G[G;a]. Suppose that N is an a-in- variant normal subgroup of G such that CN(a)1. 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 fHiji2Igof normal A-invariant subgroups. Then
CG(A) hCHi(A)ji2Ii:
LEMMA2.5. Let G be a metabelian finite group of odd order admitting an automorphism a of order 2 such that CG(a) has exponent e and G[G;a]. Then G0has exponent e.
PROOF. Since G=G0 is abelian and G[G;a], by Lemma 2.1 CG=G0(a)1 and soCG(a)G0. LetNbe the normal closure ofCG(a) inG.
SinceG0is abelian we conclude thatNhas exponente. On the other hand, G=Nis abelian becauseaacts onG=Nfixed-point-freely. ThereforeG0N
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 CG(a) has exponent e and G[G;a]. Then G0hasfe;kg-bounded exponent.
PROOF. In view of Lemma 2.5 the result is obvious ifk2. Suppose that k3 and let MG(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=hCM(a)Giwe can simply assume thatCM(a)1. 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(c1)
2 c.
THEOREM2.7. Let G be a group and N a nilpotent normal subgroup of G of class g such thatgc1(G=N0)has exponent e. Suppose thatgc1(G)is soluble of derived length d. Thengf(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 ss(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)1. Set N T
a2A#[G;a]. Then [R;N;|{z}. . .;N
s
]1.
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 CG(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. SetN 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. PutML0\CG(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 CL0(A)1.
Proposition 2.8 shows thatL0Zs(L) and soLis nilpotent of classs1. 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 ofgc1(N) is at mostk, by Theorem 2.7 we conclude thatgf(s1;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;. . .;x3c 2]e3.
LEMMA3.2. Let G be a finite metabelian group of odd order admitting a four-group of automorphisms A such thatgc(CG(a))has exponent e for all a2A#. Then there exists a normal subgroup M of G such that CM(A)1 andg3c 2(G=M)has exponent e3.
PROOF. For every a2A# put GaG0CG(a). It is easy to see that [Ga;a][G0;a] is normal inGa. SinceGa=[G0;a] is isomorphic to a quotient ofCG(a), it follows thatV(Ga) is contained in [G0;a]. SinceG0is abelian, by Lemma 2.1 we have [G0;a]\CG(a)1. It follows thatV(Ga)\CG(a)1.
LetM Q
a2A#V(Ga), obviouslyMis a normal subgroup ofG. By Lemma 2.4, CM(A) hCV(Ga)(A)ja2A#iand so CM(A)1. By Lemma 2.2 we have G hGaja2A#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 gc(G=Z(G)) has exponent e. Thengc1(G)has exponent e.
PROOF. Let Egc(G). Then E=(Z(G)\E) has exponente. For arbi- trary elementsx1;x2;. . .;xc;xc12Gwe have
[x1;x2;. . .;xc]2E:
Therefore
[x1;x2;. . .;xc]e2Z(G)\E;
whence
[[x1;x2;. . .;xc]e;xc1]1:
Since [x1;x2;. . .;xc]2G0, which is a normal abelian subgroup of G, we
obtain that
[x1;x2;. . .;xc;xc1]e[[x1;x2;. . .;xc]e;xc1]1:
Hence,gc1(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 thatgc(CG(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. LetN 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 thatCM(A)1 and g3c 2(L=M) has exponente3. ThusW(L)Mand soW(L)\CL(A)1. If W(L)1, then g3c 2(L) has exponent e3. Suppose that W(L)61. Since W(L)\CG(A)1 it follows from Proposition 2.8 that there exists a boun- ded numberssuch that
[W(L);N;|{z}. . .;N
s
]1:
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 t1 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 g3c 1(L) has ex- ponent e3, as required.
Suppose now that t2 and the result is valid for t 1. Let K[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 Hgc11(L). So H has fe;c;kg-bounded exponent. Passing to the quotientG=Hwe can simply
assume that H1. 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 gf(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 n1, Proposition 2.6 guarantees that [G;a]0hasfe;kg-bounded exponent and so
GG0[G;a]T0[G;a]0G1T11
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 2n 1andCG=[G;a](Ba) of exponente.
Let Bbe the elementary abelian group of order 2n 1. We can define an action of B on G=[G;a] as follows. For all a2A# we consider Ba as a subgroup ofBand we can writeBBaDafor a suitable subgroupDaof B. For any b2Bthere exists a unique pair (b1;b2)2BaDa such that bb1b2. Then for any x2G=[G;a] we put xbxb1. This action is well defined andCG=[G;a](B) has exponente. LetK Q
a2A#G=[G;a], we have an action ofBonK that extends the action ofBon every factorG=[G;a] and CK(B) has exponente.
By induction, K has a series of length 2(n 1) with the required properties. By Proposition 3.1 the subgroupN 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 ofCG(a), so it is an extension of a group of exponenteby a nilpotent group of classc 1. ThereforeK Q
a2A#G=[G;a] is an extension of a group of exponent e3 by a nilpotent group of class 3c 3. Let N T
a2A#[G;a].
Since G=N embeds in K, we conclude that g3c 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.