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

Abstract

Suppose that C is a class of groups consisting only of periodic groups and P(C) ′ is the set of prime numbers each of which does not divide the order of any element of a C-group. It is easy to see that if a subgroup Y of a group X is C-separable in this group, then it is P(C) ′ -isolated in X. Let us say that X has the property C-Sep if all its P(C) ′ -isolated subgroups are C-separable. We find a condition that is sufficient for a nilpotent group N to have the property C-Sep provided C is a root class. We also prove that if N is torsion-free, then the indicated condition is necessary for this group to have C-Sep.