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 Algebra Logika, 2015, Volume 54, Number 2, Pages 193–211 (Mi al687)  $\mathbb Q$-completions of free solvable groups

Ch. K. Guptaa, N. S. Romanovskiibc

a Dep. Math., Univ. Manitoba, Winnipeg, R3T 2N2, Canada
b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia
c Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

Abstract: A group $G$ is said to be complete if, for any natural $n$ and any element $g\in G$, an equation $x^n=g$ is solvable in $G$. If every such equation in the group has at most one solution, then we say that the condition for uniqueness of root extraction is satisfied. A complete group with unique root extraction can be treated as a $\mathbb Q$-power group since it admits an operation of raising an element to any rational power. Let a group $G$ be embedded in a complete group $H$ with unique root extraction, and let $H$ be generated as a $\mathbb Q$-group by the set $G$. Then $H$ is called a $\mathbb Q$-completion of $G$.
We prove that every $m$-rigid group $G$ is independently embedded in a complete $m$-rigid group. Under the specified condition for independence of an embedding, the $\mathbb Q$-completion of the group $G$ in the class of rigid groups is defined uniquely up to $G$-isomorphism. It is stated that the centralizer of any element of an independent $\mathbb Q$-completion of a free solvable group which does not belong to the last nontrivial member of a rigid series of this completion is isomorphic to the additive group of a field $\mathbb Q$ of rational numbers.

Keywords: $m$-rigid group, free solvable group, $\mathbb Q$-completion.

DOI: https://doi.org/10.17377/alglog.2015.54.204  Full text: PDF file (202 kB) References: PDF file   HTML file

English version:
Algebra and Logic, 2015, 54:2, 127–139 Bibliographic databases:   UDC: 512.5

Citation: Ch. K. Gupta, N. S. Romanovskii, “$\mathbb Q$-completions of free solvable groups”, Algebra Logika, 54:2 (2015), 193–211; Algebra and Logic, 54:2 (2015), 127–139 Citation in format AMSBIB
\Bibitem{GupRom15} \by Ch.~K.~Gupta, N.~S.~Romanovskii \paper $\mathbb Q$-completions of free solvable groups \jour Algebra Logika \yr 2015 \vol 54 \issue 2 \pages 193--211 \mathnet{http://mi.mathnet.ru/al687} \crossref{https://doi.org/10.17377/alglog.2015.54.204} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3467210} \transl \jour Algebra and Logic \yr 2015 \vol 54 \issue 2 \pages 127--139 \crossref{https://doi.org/10.1007/s10469-015-9332-y} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000359424500004} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84938960189} 

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