Dominions in quasivarieties of metabelian groups

The dominion of a subgroup $H$ of a group $A$ in a quasivariety $\mathscr M$ is the set of all $a\in A$ with equal images under all pairs of homomorphisms from $A$ into every group in $\mathscr M$ which coincide on $H$. The concept of dominion provides some closure operator on the lattice of subgroups of a given group. We study the closed subgroups with respect to this operator. We find a condition for the dominion of a divisible subgroup in quasivarieties of metabelian groups to coincide with the subgroup.