This article is cited in 3 papers

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

A. I. Budkin

Altai State University, Barnaul, Russia

Abstract: 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 $H$.

Keywords: metabelian group, abelian group, dominion, closed subgroup.

UDC: 512.57

Received: 28.02.2014

 English version: Siberian Mathematical Journal, 2014, 55:6, 1009–1016

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