Fundamental Groups of Graphs of Cyclic Subgroup Separable and Weakly Potent Groups

We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form [Formula: see text], where [Formula: see text] has infinite order and [Formula: see text] is finite.

Abelian Subgroup Separability of Certain Generalized Free Products of Groups

We prove that generalized free products of certain abelian subgroup separable groups are abelian subgroup separable. Applying this, we show that tree products of polycyclic-by-finite groups, amalgamating central subgroups or retracts are abelian subgroup separable.

Abelian subgroup separability of certain HNN extensions

In this paper, we prove that certain HNN extensions of finitely generated abelian subgroup separable groups are finitely generated abelian subgroup separable. Using this, we show that certain HNN extensions of finitely generated nilpotent groups are finitely generated abelian subgroup separable.

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

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.