Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2015, Number 3(35), Pages 5–11
Residual properties of Abelian groups
D. N. Azarov
Ivanovo State University, Ivanovo, Russian Federation
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 $A$.
Abelian group, residually finite group.
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D. N. Azarov, “Residual properties of Abelian groups”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2015, no. 3(35), 5–11
Citation in format AMSBIB
\paper Residual properties of Abelian groups
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
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