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 Algebra Logika, 2015, Volume 54, Number 5, Pages 575–588 (Mi al713)

Dominions in solvable groups

A. I. Budkin

Altai State University, pr. Lenina 61, Barnaul, 656049, Russia

Abstract: 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.

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

DOI: https://doi.org/10.17377/alglog.2015.54.502

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English version:
Algebra and Logic, 2015, 54:5, 370–379

Bibliographic databases:

UDC: 512.57
Revised: 29.03.2015

Citation: A. I. Budkin, “Dominions in solvable groups”, Algebra Logika, 54:5 (2015), 575–588; Algebra and Logic, 54:5 (2015), 370–379

Citation in format AMSBIB
\Bibitem{Bud15} \by A.~I.~Budkin \paper Dominions in solvable groups \jour Algebra Logika \yr 2015 \vol 54 \issue 5 \pages 575--588 \mathnet{http://mi.mathnet.ru/al713} \crossref{https://doi.org/10.17377/alglog.2015.54.502} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3468418} \transl \jour Algebra and Logic \yr 2015 \vol 54 \issue 5 \pages 370--379 \crossref{https://doi.org/10.1007/s10469-015-9358-1} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000366155000002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84957933223} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. I. Budkin, “On $2$-closedness of the rational numbers in quasivarieties of nilpotent groups”, Siberian Math. J., 58:6 (2017), 971–982
2. A. I. Budkin, “On dominions of the rationals in nilpotent groups”, Siberian Math. J., 59:4 (2018), 598–609
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