## 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*∈ 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*∈ ℕ} consisting of finite simple subgroups

*G*and

_{i}*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*over which the simple group

_{i}*G*is defined are splitting fields for

_{i}*L*, the inverse image of

_{i}*F*in

*Ĝ*for all

_{i}*i*∈ ℕ. The group

*Ĝ*is the inverse image of

_{i}*G*in the corresponding universal central extension group.

_{i}
## Comments (0)