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

Abstract

The dominion of a subgroup H in a group G (in the class of metabelian groups) is the set of all elements a ∈ 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 Hf (f ∈ G), and G′ = HG × K for a suitable subgroup K; then the dominion of H in G coincides with H.