The cyclic subgroup separability of certain generalized free products of two groups

TL;DR

This paper investigates the conditions under which cyclic subgroups in certain generalized free products of residually finite groups are separable, providing criteria related to their conjugation properties and residual finiteness.

Contribution

It establishes a precise characterization of cyclic subgroup separability in free products with amalgamated retracts, extending understanding of residual properties in these groups.

Findings

Cyclic subgroup non-separability linked to conjugation with non-finitely separable free factor subgroups.

Results apply to separability in the class of finite p-groups.

Provides necessary and sufficient conditions for subgroup separability in the studied groups.

Abstract

Free products of two residually finite groups with amalgamated retracts are considered. It is proved that a cyclic subgroup of such a group is not finitely separable if, and only if, it is conjugated with a subgroup of a free factor which is not finitely separable in this factor. A similar result is obtained for the case of separability in the class of finite p-groups.