CENTRALIZERS IN SIMPLE LOCALLY FINITE GROUPS

This is a survey article on centralizers of finite subgroups in locally finite, simple groups or LFS-groups as we will call them. We mention some of the open problems about centralizers of subgroups in LFS-groups and applications of the known information about the centralizers of subgroups to the structure of the locally finite group. We also prove the following: Let G be a countably infinite nonlinear LFS-group with a Kegel sequence K = f(Gi;Ni) j i 2 N g. If there exists an upper bound for fjNij j i 2 N g, then for any finite semisimple subgroup F in G the subgroup CG(F) has elements of order pi for infinitely many distinct prime pi. In particular CG(F) is an infinite group. This answers Hartleyʹs question provided that there exists a bound on fjNij j i 2 N g: