On the subgroup separability of the free product of groups

Abstract

Suppose that $\mathcal{C}$ is a root class of groups (i.e., a class of groups that contains non-trivial groups and is closed under taking subgroups and unrestricted wreath products), $G$ is the free product of residually $\mathcal{C}$-groups $A_i$ ($i\in \mathcal{I}$), and $H$ is a subgroup of $G$ satisfying a non-trivial identity. We prove a criterion for the $\mathcal{C}$-separability of $H$ in $G$. It follows from this criterion that, if $\{{\mathcal{V}}_j\mid j\in \mathcal{J}\}$ is a family of group varieties, each ${\mathcal{V}}_j$ ($j\in \mathcal{J}$) is distinct from the variety of all groups, and $\mathcal{V}={\bigcup }_{j\in \mathcal{J}}{\mathcal{V}}_j$, then one can give a description of $\mathcal{C}$-separable $\mathcal{V}$-subgroups of $G$ provided such a description is known for every group $A_i$ ($i\in \mathcal{I}$).

8 pages; the English version of the previously published Russian original