Geodesic
Absolute closedness of torsion-free Abelian groups in the class of metabelian groups
Algebra i logika, Tome 53 (2014) no. 1, pp. 15-25.

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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 $M$, every inclusion $H\le G$ implies that the dominion of $H$ in $G$ (in $M$) coincides with $H$. We deal with dominions in torsion-free Abelian subgroups of metabelian groups. It is proved that every nontrivial torsion-free Abelian subgroup is not absolutely closed in the class of metabelian groups. It is stated that if a torsion-free subgroup $H$ of a metabelian group $G$ and the commutator subgroup $G'$ have trivial intersection, then the dominion of $H$ in $G$ (in the class of metabelian groups) coincides with $H$.
Keywords: quasivariety, metabelian group, Abelian group, dominion, absolutely closed subgroup. 
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A. I. Budkin. Absolute closedness of torsion-free Abelian groups in the class of metabelian groups. Algebra i logika, Tome 53 (2014) no. 1, pp. 15-25. http://geodesic.mathdoc.fr/item/AL_2014_53_1_a1/

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