
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 Received: 15.02.2015
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 511
\mathnet{http://mi.mathnet.ru/vtgu455}
\crossref{https://doi.org/10.17223/19988621/35/1}
\elib{http://elibrary.ru/item.asp?id=23735480}
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