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

TL;DR

This paper investigates the conditions under which cyclic subgroups in a free product of groups with commuting subgroups are separable in classes of finite groups, especially focusing on finitely generated nilpotent and free groups.

Contribution

It characterizes C-separable cyclic subgroups in such free products and establishes separability results for specific classes of groups and subgroups.

Findings

C-separable cyclic subgroups are described explicitly in residually C-groups.

All p'-isolated cyclic subgroups are separable in finite p-groups for certain group classes.

Separable cyclic subgroups are characterized for free and nilpotent groups with p'-isolated subgroups.

Abstract

Let G be the free product of groups A and B with commuting subgroups H \leqslant A and K \leqslant 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.