Let $p$ be a prime and $G$ a locally finite group containin
g an elementary abelian $p$-subgroup $A$ of rank at least $3$ such tha
t $C_{G}(A)$ is Chernikov (that is\, a locally finite group satisfying min
imal condition on subgroups) and $C_{G}(a)$ involves no infinite simple gr
oups for any $a\\in A*$. In this talk\, we prove that G is almost locally
soluble. To prove this result\, we first give a

\ncharacterization of $
PSL_{p}(k)$:

\n

\nTheorem: An infinite simple locally finite group $
G$ admits an elementary abelian $p$-group of automorphisms $A$ such th
at $C_{G}(A)$ is Chernikov and $C_{G}(a)$ involves no infinite simple grou
ps for any $a\\in A^{\\sharp}$ if and only if $G$ is isomorphic to $PS
L_{p}(k)$ for some locally finite field $k$ of characteristic different fr
om $p$ and $A$ has order $p^{2}$.

\nThis is a joint work with Mahmu
t Kuzucuoglu and Pavel Shumyatsky.

\n

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