On some groups with finite involution saturated with finite simple groups. (Q1810204)

By definition, a group \(G\) is saturated with finite simple subgroups if every finite subgroup of \(G\) is contained in a finite simple subgroup of \(G\). An involution \(t\in G\) is said to be finite if, for every \(g\in G\), the subgroup generated by \(t\) and \(t^g\) is finite. A proper subgroup \(H\) of \(G\) is said to be strongly embedded (in \(G\)) if \(H\) contains an involution, but \(H\cap H^g\) does not contain an involution for every \(g\not\in H\). The authors prove the following results: Suppose that an infinite group \(G\) contains a strongly embedded subgroup and a finite involution. If \(G\) is saturated with finite simple subgroups and the centralizer of some involution in \(G\) is a 2-group then \(G\) is locally finite and isomorphic to \(L_2(Q)\) or \(Sz(Q)\) for some locally finite field \(Q\) of characteristic two (Theorem 1). Suppose that an infinite periodic group \(G\) is saturated with finite simple subgroups and contains an Abelian Sylow 2-subgroup. Then \(G\) is locally finite and isomorphic to \(L_2(Q)\) or \(^2F_4(Q)\) for some locally finite field \(Q\) (Theorem 2).