CENTRALIZERS OF p-SUBGROUPS IN SIMPLE LOCALLY FINITE GROUPS

Abstract

In Ersoy et al. [J. Algebra 481 (2017), 1-11], we have proved that if G is a locally finite group with an elementary abelian p-subgroup A of order strictly greater than p(2) such that C-G(A) is Chernikov and for every non-identity alpha is an element of A the centralizer C-G(alpha) does not involve an infinite simple group, then G is almost locally soluble. This result is a consequence of another result proved in Ersoy et al. [J. Algebra 481 (2017), 1-11], namely: if G is a simple locally finite group with an elementary abelian group A of automorphisms acting on it such that the order of A is greater than p(2), the centralizer C-G(A) is Chernikov and for every non-identity alpha is an element of A the set of fixed points C-G(alpha) does not involve an infinite simple groups then G is finite. In this paper, we improve this result about simple locally finite groups: Indeed, suppose that G is a simple locally finite group, consider a finite non-abelian subgroup P of automorphisms of exponent p such that the centralizer C-G(P) is Chernikov and for every non-identity alpha is an element of P the set of fixed points C-G(alpha) does not involve an infinite simple group. We prove that if Aut(G) has such a subgroup, then G approximately equal to PSLp(k) where char k not equal p and P has a subgroup Q of order p(2) such that C-G(P) = Q.