Centralizers of subgroups in simple locally finite groups. (Q2882894)

Some twenty years ago or so Brian Hartley asked whether in a nonlinear, locally finite, simple group the centralizer of each finite subgroup is infinite. So far comparatively little is known about this question. It is known, for example, that the centralizers of the cyclic subgroups in such a group are always infinite (due to Hartley and the second author). The authors of this paper produce a further positive special case.NEWLINENEWLINE Let \(G\) be a nonlinear, locally finite, simple group that is the union of a chain of finite simple groups \(S_i\). If \(F\) is a finite subgroup of \(G\) that does not involve the natural characteristic of any of the \(S_i\) of Lie type, then the centralizer of \(F\) in \(G\) is infinite; indeed it contains a direct product of infinitely many cyclic groups of distinct prime order.