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 $\mathcal{ C}\mbox{ -}\mathfrak{ Sep}$.
Categories: math.GR