On the closedness of a locally cyclic subgroup in a metabelian group

The dominion of a subgroup $H$ in a group $G$ (in the class of metabelian groups) is the set of all elements $a\in G$ whose images are equal for all pairs of homomorphisms from $G$ into every metabelian group that coincide on $H$. The dominion is a closure operator on the lattice of subgroups of $G$. We study the closed subgroups with respect to the dominion. It is proved that if $G$ is a metabelian group, $H$ is a locally cyclic group, the commutant $G'$ of $G$ is the direct product of its subgroups of the form $H^f$ ($f\in G$), and $G'=H^G\times K$ for a suitable subgroup $K$; then the dominion of $H$ in $G$ coincides with $H$.