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

Abstract

Suppose that $\mathcal{C}$ is a class of groups consisting only of periodic groups and $\mathfrak{P}(\mathcal{C})^{\prime}$ is the set of prime numbers each of which does not divide the order of any element of a $\mathcal{C}$-group. A subgroup $Y$ of a group $X$ is called a) $\mathcal{C}$-separable in this group if, for each $x \in X \setminus Y$, there exists a homomorphism $\sigma$ of $X$ onto a group from $\mathcal{C}$ such that $x\sigma \notin Y\sigma$; b) $\mathfrak{P}(\mathcal{C})^{\prime}$-isolated in $X$ if, for any $x \in X$, $q \in \mathfrak{P}(\mathcal{C})^{\prime}$, the inclusion $x^{q} \in Y$ implies that $x \in Y$. It is easy to see that if $Y$ is $\mathcal{C}$-separable in $X$, then it is $\mathfrak{P}(\mathcal{C})^{\prime}$-isolated in this group. Let us say that $X$ has the property $\mathcal{C}\mbox{-}\mathfrak{Sep}$ if all its $\mathfrak{P}(\mathcal{C})^{\prime}$-isolated subgroups are $\mathcal{C}$-separable. We find a condition that is sufficient for a nilpotent group $N$ to have the property $\mathcal{C}\mbox{-}\mathfrak{Sep}$ provided $\mathcal{C}$ is a root class (i.e., it contains non-trivial groups and is closed under taking subgroups, extensions, and Cartesian products of the form $\prod_{v \in V}U_{v}$, where $U, V \in \mathcal{C}$ and $U_{v}$ is an isomorphic copy of $U$ for each $v \in V$). We also prove that if $N$ is torsion-free, then the indicated condition is necessary for this group to have $\mathcal{C}\mbox{-}\mathfrak{Sep}$.