
Residual properties of nilpotent groups
D. N. Azarov^{} ^{} Ivanovo State University, Ermaka str., 39, Ivanovo, 153025, Russia
Abstract:
Let $\pi $ be a set of primes. 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 $G$ will be 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 $m$th power of an element of $G$ for every positive $\pi $number $m$. Let $N$ be a nilpotent group and let all power subgroups in $N$ are finitely separable. It is proved that $N$ is a residually finite $\pi $group if and only if $N$ 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 $N$ i.e. $T$ be a set of all elements of $N$ whose orders are finite $\pi '$numbers. We prove that the following three statements are equivalent: (1) the group $N$ is a virtually residually finite $\pi $group; (2) the subgroup $T$ is finite and quotient group $N/T$ is a residually finite $\pi $group; (3) the subgroup $T$ is finite and $T$ coincides with the set of all $\pi $radicable elements of $N$.
Keywords:
nilpotent group, finite rank group, residually finite $p$group.
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UDC:
512.543 Received: 12.03.2015
Citation:
D. N. Azarov, “Residual properties of nilpotent groups”, Model. Anal. Inform. Sist., 22:2 (2015), 149–157
Citation in format AMSBIB
\Bibitem{Aza15}
\by D.~N.~Azarov
\paper Residual properties of nilpotent groups
\jour Model. Anal. Inform. Sist.
\yr 2015
\vol 22
\issue 2
\pages 149157
\mathnet{http://mi.mathnet.ru/mais432}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3417818}
\elib{http://elibrary.ru/item.asp?id=23405824}
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