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 Model. Anal. Inform. Sist., 2015, Volume 22, Number 2, Pages 149–157 (Mi mais432)

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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Bibliographic databases:

UDC: 512.543

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 149--157 \mathnet{http://mi.mathnet.ru/mais432} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3417818} \elib{http://elibrary.ru/item.asp?id=23405824}