This article is cited in 3 scientific papers (total in 3 papers)
Absolute closedness of torsion-free Abelian groups in the class of metabelian groups
A. I. Budkin
Pavlovskii road, 60a-168, Barnaul, 656064, Russia
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 $H$.
quasivariety, metabelian group, Abelian group, dominion, absolutely closed subgroup.
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Algebra and Logic, 2014, 53:1, 9–16
A. I. Budkin, “Absolute closedness of torsion-free Abelian groups in the class of metabelian groups”, Algebra Logika, 53:1 (2014), 15–25; Algebra and Logic, 53:1 (2014), 9–16
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\paper Absolute closedness of torsion-free Abelian groups in the class of metabelian groups
\jour Algebra Logika
\jour Algebra and Logic
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This publication is cited in the following articles:
A. I. Budkin, “Dominions in solvable groups”, Algebra and Logic, 54:5 (2015), 370–379
A. I. Budkin, “On $2$-closedness of the rational numbers in quasivarieties of nilpotent groups”, Siberian Math. J., 58:6 (2017), 971–982
A. I. Budkin, “On dominions of the rationals in nilpotent groups”, Siberian Math. J., 59:4 (2018), 598–609
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