On the cyclic subgroup separability of the free product of two groups with commuting subgroups

Abstract

Let G be the free product of groups A and B with commuting subgroups H ≤ A and K ≤ B, and let 𝒞 be the class of all finite groups or the class of all finite p-groups. We derive the description of all 𝒞-separable cyclic subgroups of G provided this group is residually a 𝒞-group. We prove, in particular, that if A, B are finitely generated nilpotent groups and H, K are p′-isolated in the free factors, then all p′-isolated cyclic subgroups of G are separable in the class of all finite p-groups. The same statement is true provided A, B are free and H, K are p′-isolated and cyclic.