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 non-linear LFS-group‎ ‎with a Kegel sequence \mathcal{K}=\{(G_i,N_i)\ |\ \ i\in‎ ‎\mathbf{N}\ \}‎. ‎If there exists an upper bound for \{ |N_i| \ | ‎\ \ i\in \mathbf{N}\ \}‎, ‎then for any finite semisimple‎ ‎subgroup F in G the subgroup C_G(F) has elements of‎ ‎order p_i for infinitely many distinct prime p_i‎. ‎In‎ ‎particular C_G(F) is an infinite group‎. ‎This answers Hartley's‎ ‎question provided that there exists a bound on \{ |N_i| \ | ‎\ \ i\in \mathbf{N}\ \}

Keywords:

Centralizer , locally finite , simple groups