Siberian Federal University

We are interested in the following questions of B. Hartley: (1) Is it true that, in an infinite, simple locally finite group, if the centralizer of a finite subgroup is linear, then G is linear? (2) For a finite subgroup F of a non-linear simple locally finite group is the order jCG(F)j infinite? We prove the following: Let G be a non-linear simple locally finite group which has a Kegel sequence K = f(Gi; 1) : i 2 Ng consisting of finite simple subgroups. Let p be a fixed prime and s 2 N. Then for any finite p�����subgroup F of G, the centralizer CG(F) contains subgroups isomorphic to the homomorphic images of SL(s;Fq). In particular CG(F) is a non-linear group. We also show that if F is a finite p-subgroup of the infinite locally finite simple group G of classical type and given s 2 N and the rank of G is sufficiently large with respect to jFj and s, then CG(F) contains subgroups which are isomorphic to homomorphic images of SL(s;K)