Centralizers in simple locally finite groups

Document Type: Ischia Group Theory 2012

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Abstract

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 $\{ |N_i| \ | ‎\ \ i\in \mathbf{N}\ \}$

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