Factorization criterion for local solvability of locally finite groups with minimal condition for primary subgroups (Q1076153)

Theorem 1 of this paper states that if \(G\) is a locally finite group that contains a finite Sylow (i.e. maximal) \(p\)-subgroup for each prime \(p\), then \(G\) is locally soluble if and only if each Sylow \(p\)-subgroup is complemented. The proof is a straightforward deduction from the finite case. Theorem 2 draws the same conclusion for locally finite groups whose maximal \(p\)-subgroups satisfy the minimal condition on subgroups (i.e. are Chernikov groups). The proof uses Theorem 1, and also the fact that a locally finite group in which each maximal \(p\)-subgroup is Chernikov, has a locally soluble subgroup of finite index. It would be interesting to have a proof not depending on this deep fact.