Centralizers of subgroups in simple locally finite groups

Abstract

Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite K-semisimple subgroups. Namely let G be a non-linear simple locally finite group which has a Kegel sequence K = {(G(i), 1) : i is an element of N} consisting of finite simple subgroups. Then for any finite subgroup F consisting of K-semisimple elements in G, the centralizer C-G(F) has an infinite abelian subgroup A isomorphic to a direct product of Z(pi) for infinitely many distinct primes p(i). Moreover we prove that if G is a non-linear simple locally finite group which has a Kegel sequence K = {(G(i), 1) : i is an element of N} consisting of finite simple subgroups G(i) and F is a finite K-semisimple subgroup of G, then C-G(F) involves an infinite simple non-linear locally finite group provided that the finite fields k(i) over which the simple group G(i) is defined are splitting fields for L-i, the inverse image of F in (G) over cap (i) for all i is an element of N. The group (G) over cap (i) is the inverse image of G(i) in the corresponding universal central extension group.