On centralizers of locally finite simple groups (Q2325701)

\textit{B. Hartley} [Pac. J. Math. 152, No. 1, 101--118 (1992; Zbl 0713.20023)] asked the following long-standing open problem: Let \(G\) be a non-linear simple locally finite group. Is the centralizer of every finite subgroup infinite? In this interesting paper, the authors contribute the discussion about centralizers of finite subgroups in simple locally finite groups by proving a nice characterization of \(\mathrm{PSL}(2,F)\). Namely, they prove the following result: Theorem. (Brescia-Russo) Let \(G\) be an infinite simple locally finite group. Then, either \(G\) is isomorphic to \(\mathrm{PSL}(2,F)\), where \(F\) is an infinite locally finite field, or \(G\) contains a subgroup which is the direct product of an infinite abelian subgroup of prime exponent \(p\) and a finite non-abelian \(p\)-subgroup.