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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

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English version:
Algebra and Logic, 2015, 54:2, 127–139

Bibliographic databases:

UDC: 512.5
Received: 23.01.2015

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
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\by Ch.~K.~Gupta, N.~S.~Romanovskii
\paper $\mathbb Q$-completions of free solvable groups
\jour Algebra Logika
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\vol 54
\issue 2
\pages 193--211
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\crossref{https://doi.org/10.17377/alglog.2015.54.204}
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\jour Algebra and Logic
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\vol 54
\issue 2
\pages 127--139
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