Dominions in solvable groups

A. I. Budkin

Algebra i logika, Tome 54 (2015) no. 5, pp. 575-588

Résumé

The dominion of a subgroup $H$ of a group $G$ in a class $M$ is the set of all elements $a\in G$ whose images are equal for all pairs of homomorphisms from $G$ to each group in $M$ that coincide on $H$. A group $H$ is absolutely closed in a class $M$ if, for any group $G$ in $M$ and any inclusion $H\le G$, the dominion of $H$ in $G$ (with respect to $M$) coincides with $H$ (i.e., $H$ is closed in $G$). We prove that every torsion-free nontrivial Abelian group is not absolutely closed in $\mathcal{AN}_c$. It is shown that if a subgroup $H$ of $G$ in $\mathcal N_c\mathcal A$ has trivial intersection with the commutator subgroup $G'$, then the dominion of $H$ in $G$ (with respect to $\mathcal N_c\mathcal A$) coincides with $H$. It is stated that the study of closed subgroups reduces to treating dominions of finitely generated subgroups of finitely generated groups.

Détail

Keywords: quasivariety, nilpotent group, extension of Abelian group by nilpotent group, dominion, closed subgroup.