Algebra and Logic, Vol. 52, No. 5, November, 2013 (Russian Original Vol. 52, No. 5, September-October, 2013)

LOCALLY FINITE GROUPS WITH BOUNDED

CENTRALIZER CHAINS

A. A. Buturlakin1∗ and A. V. Vasil’ev2∗ UDC 512.544.5

Keywords: locally finite group, non-Abelian simple group, lattice of centralizers, c-dimension.

The c-dimension of a group G is the maximal length of a chain of nested centralizers in G. We prove that a locally finite group of finite c-dimension k has less than 5k non-Abelian composition factors.

INTRODUCTION

Let G be a group and CG(X) the centralizer of a subset X of G. Since CG(X) < CG(Y ) iff

CG(CG(X)) > CG(CG(Y )), it follows that the minimal and maximal conditions for centralizers are equivalent. Thus the length of every chain of nested centralizers in a group with the minimal condition for centralizers is finite. If a uniform bound for the lengths of chains of centralizers in a group G exists, then we refer to the maximal such length as the c-dimension of the group G, as in [1]. The same notion is also known as the height of the lattice of centralizers. It is worth observing that a class of groups of finite c-dimension includes Abelian groups, torsion-free hyperbolic groups, linear groups over fields, and so on. In addition, the class is closed under taking subgroups and finite direct products, but the c-dimension of a homomorphic image of a group in this class is not necessarily finite.

In [2], it was proved that a periodic locally soluble group with the minimal condition for centralizers is soluble. In [3], in particular, it was shown that the derived length of a periodic locally soluble group of finite c-dimension k is bounded in terms of k. The same paper contains a conjecture of A. V. Borovik which asserts that the number of non-Abelian composition factors of ∗Supported by RFBR, project No. 13-01-00505. 1Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia; Novosibirsk State

University, ul. Pirogova 2, Novosibirsk, 630090 Russia; buturlakin@math.nsc.ru. 2Sobolev Institute of

Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia; Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia; vasand@math.nsc.ru. Translated from Algebra i Logika, Vol. 52, No. 5, pp. 553-558, September-October, 2013. Original article submitted October 29, 2013. 0002-5232/13/5205-0367 c© 2013 Springer Science+Business Media New York 367 a locally finite group of finite c-dimension k is bounded in terms of k. The purpose of this paper is to prove that conjecture.

THEOREM. Let G be a locally finite group of c-dimension k. Then the number of non-Abelian composition factors of G is less than 5k. 1. PRELIMINARIES

For a locally finite group G, we denote by η(G) the number of non-Abelian composition factors of G.

The following well-known fact (see, e.g., [4, Cor. 3.5]) allows us to reduce the proof of the theorem to the case of a finite group.

LEMMA 1.1. If G is a locally finite locally soluble simple group, then G is cyclic.

Recall that the factor group of a finite group G with respect to its soluble radical R is a subgroup of the automorphism group of a direct product of non-Abelian simple groups. Thus if the socle Soc(G/R) is a direct product of non-Abelian simple groups S1, S2, . . . , Sn, then G/R is a subgroup of the semidirect product (Aut(S1)×Aut(S2)× . . .×Aut(Sn)) Symn, where Symn permutes S1, S2, . . . , Sn. By the classification of finite simple groups, the outer automorphism group of a finite simple group is soluble. Therefore, every non-Abelian composition factor of G is either a composition factor of Soc(G/R) or a composition factor of a corresponding subgroup of Symn.

The next three lemmas give an upper bound for the number of non-Abelian composition factors of an arbitrary subgroup of Symn. We denote by μ(G) the degree of a minimal faithful permutation representation of a finite group G.

LEMMA 1.2 [5, Thm. 2]. Let G be a finite group and L be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. If N is a maximal normal L-subgroup of

G, then μ(G) μ(G/N).

LEMMA 1.3 [6, Thm. 3.1]. Let S1, S2, . . . , Sr be simple groups. Then μ(S1×S2× . . .×Sr) = μ(S1) + μ(S2) + . . . + μ(Sr).

LEMMA 1.4. If G is a subgroup of a symmetric group Symn, then η(G) (n − 1)/4.

Proof. We proceed by induction on n. If R is the soluble radical of G, then Lemma 1.2 implies that μ(G/R) does not exceed μ(G). Hence we may assume that the soluble radical of G is trivial. Let the socle Soc(G) of G be a direct product of non-Abelian simple groups S1, S2, . . . , Sl. By Lemma 1.3, we have l n/5. The group G is a subgroup of the semidirect product (Aut(S1)×Aut(S2)× . . .×Aut(Sl)) Syml. By the induction hypothesis, η(G) n/5 + (n/5− 1)/4 = (n− 1)/4.

Remark. The group Sn, where n = 5k with k 1, contains a subgroup G isomorphic to a permutation wreath product (. . . ((Alt5 Alt5) Alt5) . . .), where the wreath product is taken k− 1 times. We have η(G) = 5 k−1 5−1 = n−1 4 . 368

The following lemma is crucial in bounding the number of composition factors of Soc(G/R) for a finite group G.

LEMMA 1.5 [3, Lemma 3]. If an elementary Abelian p-group E of order pn acts faithfully on a finite nilpotent p′-group Q, then there exists a series of subgroups E = E0 > E1 > E2 > . . . >

En = 1 for which all inclusions CQ(E0) < CQ(E1) < . . . < CQ(En) are strict.

As usual, Op(G) stands for the largest normal p-subgroup of a finite group G, while Op′(G) denotes the largest normal p′-subgroup of G. If a series of commutator subgroups of a group G stabilizes, then we denote by G(∞) the last subgroup of this series. A quasisimple group is a perfect central extension of a non-Abelian simple group. The layer E(G) is the subgroup of G generated by all subnormal quasisimple subgroups of G, which are called components of G. Recall that the layer is a central product of components of G. 2. PROOF OF THE THEOREM