Locally Finite Groups and Their Subgroups with Small Centralizers

Let $p$ be a prime and $G$ a locally finite group containing an elementary abelian $p$-subgroup $A$   of rank at least $3$ such that $C_{G}(A)$ is Chernikov (that is, a locally finite group satisfying minimal condition on subgroups) and $C_{G}(a)$ involves no infinite simple groups for any $a\in A*$. In this talk, we prove that G is almost locally soluble. To prove this result, we first give a
characterization of $PSL_{p}(k)$:

Theorem: An infinite simple locally finite group $G$ admits an elementary abelian $p$-group of automorphisms $A$   such that $C_{G}(A)$ is Chernikov and $C_{G}(a)$ involves no infinite simple groups for any $a\in A^{\sharp}$ if and only if $G$   is isomorphic to $PSL_{p}(k)$ for some locally finite field $k$ of characteristic different from $p$ and $A$   has order $p^{2}$.
This is a joint work with Mahmut Kuzucuoglu and Pavel Shumyatsky.