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 Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2015, Number 3(35), Pages 5–11 (Mi vtgu455)

MATHEMATICS

Residual properties of Abelian groups

D. N. Azarov

Ivanovo State University, Ivanovo, Russian Federation

Abstract: Let $\pi$ be a set of primes. For Abelian groups, the necessary and sufficient condition to be a virtually residually finite $\pi$-group is obtained, as well as a characterization of potent Abelian groups. Recall that a group $G$ is said to be a residually finite $\pi$-group if for every nonidentity element a of $G$ there exists a homomorphism of the group $G$ onto some finite $\pi$-group such that the image of the element a differs from 1. A group $G$ is said to be a virtually residually finite $\pi$-group if it contains a finite index subgroup which is a residually finite $\pi$-group. Recall that an element $g$ in $G$ is said to be $\pi$-radicable if g is an mth power of an element of $G$ for every positive $\pi$-number $m$. Let $A$ be an Abelian group. It is well known that $A$ is a residually finite $\pi$-group if and only if $A$ has no nonidentity $\pi$-radicable elements. Suppose now that $\pi$ does not coincide with the set $\Pi$ of all primes. Let $\pi'$ be the complement of $\pi$ in the set $\Pi$. And let $T$ be a $\pi'$-component of $A$, i.e., $T$ be a set of all elements of $A$ whose orders are finite $\pi'$-numbers. We prove that the following three statements are equivalent to each other: (1) the group $A$ is a virtually residually finite $\pi$-group; (2) the subgroup $T$ is finite and the quotient group $A/T$ is a residually finite $\pi$-group; (3) the subgroup $T$ is finite and $T$ coincides with the set of all $\pi$-radicable elements of $A$.

Keywords: Abelian group, residually finite group.

DOI: https://doi.org/10.17223/19988621/35/1

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UDC: 512.543

Citation: D. N. Azarov, “Residual properties of Abelian groups”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2015, no. 3(35), 5–11

Citation in format AMSBIB
\Bibitem{Aza15} \by D.~N.~Azarov \paper Residual properties of Abelian groups \jour Vestn. Tomsk. Gos. Univ. Mat. Mekh. \yr 2015 \issue 3(35) \pages 5--11 \mathnet{http://mi.mathnet.ru/vtgu455} \crossref{https://doi.org/10.17223/19988621/35/1} \elib{https://elibrary.ru/item.asp?id=23735480}