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.
UDC:
512.543 Received: 15.02.2015
DOI:
10.17223/19988621/35/1
Fulltext:
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