Centralizers in simple locally finite groups

Abstract

This is a survey article on centralizers of finitesubgroups in locally finite, simple groups or LFS-groups as wewill call them. We mention some of the open problems aboutcentralizers of subgroups in LFS-groups and applications of theknown information about the centralizers of subgroups to thestructure of the locally finite group. We also prove thefollowing: Let $G$ be a countably infinite non-linear LFS-groupwith a Kegel sequence $mathcal{K}={(G_i,N_i) | iinmathbf{N} }$. If there exists an upper bound for ${ |N_i| |iin mathbf{N} }$, then for any finite semisimplesubgroup $F$ in $G$ the subgroup $C_G(F)$ has elements oforder $p_i$ for infinitely many distinct prime $p_i$. Inparticular $C_G(F)$ is an infinite group. This answers Hartley'squestion provided that there exists a bound on ${ |N_i| | iin mathbf{N}$.

Keywords

• Centralizer
• locally finite
• simple groups