Root class residuality of HNN-extensions with central cyclic associated subgroups

Abstract

Suppose that G is a group, H and K are proper isomorphic central subgroups of G, and G is an HNN-extension of G with the associated subgroups H and K. We prove necessary and sufficient conditions for G to be residually a C-group, where C is a class of groups closed under taking subgroups, extensions, homomorphic images, 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 .

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

“…The notion of a root class was introduced in [10] and allows one to prove many statements at once using the same reasoning. It turns out to be especially useful in studying the residual properties of free constructions of groups (see, e.g., [22][23][24][25][26][27][28][29][30][31]). If X is such a construction and C is a root class of groups, then the C-separability of some subgroups of X is quite often one of the necessary and/or sufficient conditions for X to be residually a C-group.…”