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 CG(A) is Chernikov and CG(a) involves no infinite simple groups for any a∈A#. We show that G is almost locally soluble (Theorem 1.1). The key step in the proof is the following characterization of PSLp(k): An infinite simple locally finite group G admits an elementary abelian p-group of automorphisms A such that CG(A) is Chernikov and CG(A) involves no infinite simple groups for any a∈A# if and only if G is isomorphic to PSLp(k) for some locally finite field k of characteristic different from p and A has order p2.