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 $\mathcal{C}\mbox{-}\mathfrak{Sep}$.

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