Geodesic
Residual properties of Abelian groups
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 3 (2015), pp. 5-11 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {Residual properties of {Abelian} groups},
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D. N. Azarov. Residual properties of Abelian groups. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 3 (2015), pp. 5-11. http://geodesic.mathdoc.fr/item/VTGU_2015_3_a0/

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