On the root-class residuality of generalized free products with a normal amalgamation

Abstract

We obtain both necessary and sufficient conditions for the free product of two groups with normal amalgamated subgroups to be a residually C-group, where C is a root class of groups, which must be homomorphically closed in most cases.

On the separability of subgroups of nilpotent groups by root classes of groups

“…At the same time, the approximability by other classes of groups is also considered in the literature, and many of these classes are root classes of groups.In accordance with one of the equivalent definitions (see Proposition 3.2 below), a class of groups C is called a root class if it contains non-trivial groups and is closed under taking subgroups, extensions, and Cartesian products of the form y∈Y X y , where X, Y ∈ C and X y is an isomorphic copy of X for each y ∈ Y . The concept of a root class was introduced by K. Gruenberg [5] and turned out to be very useful in studying the approximability of the fundamental groups of various graphs of groups [1,4,[13][14][15][16][17][18]21,22]. Thanks to its use, it became possible, in particular, to make significant progress in the study of the residual p-finiteness (where p is a prime number) and the residual solvability of such groups.Everywhere below, it is assumed that Γ = (V, E) is a non-empty connected undirected graph with a vertex set V and an edge set E (loops and multiple edges are allowed).…”