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 Chebyshevskii Sb., 2014, Volume 15, Issue 1, Pages 7–18 (Mi cheb321)

This article is cited in 4 scientific papers (total in 4 papers)

Some residual properties of soluble groups of finite rank

D. N. Azarov

Ivanovo State University

Abstract: The generalization of one classical Smel'kin's theorem for polycyclic groups is obtained. A. L. Smelkin proved that if $G$ is a polycyclic group, then it is a virtually residually finite $p$-group for any prime $p$. Recall that a group $G$ is said to be a residually finite $p$-group if for every nonidentity element $a$ of $G$ there exists a homomorphism of the group $G$ onto some finite $p$-group such that the image of the element $a$ differs from 1. A group $G$ will be said to be a virtually residually finite $p$-group if it contains a finite index subgroup which is a residually finite $p$-group.
One of the generalizations of the notation of polycyclic group is a notation of soluble finite rank group. Recall that a group $G$ is said to be a group of finite rank if there exists a positive integer $r$ such that every finitely generated subgroup in $G$ is generated by at most $r$ elements. For soluble groups of finite rank the following necessary and sufficient condition to be a residually finite $\pi$-group for some finite set $\pi$ of primes is obtained.
If $G$ is a group of finite rank, then the group $G$ is a residually finite $\pi$-group for some finite set $\pi$ of primes if and only if $G$ is a reduced poly-(cyclic, quasicyclic, or rational) group. Recall that a group $G$ is said to be a reduced group if it has no nonidentity radicable subgroups. A group $H$ is said to be a radicable group if every element $h$ in $H$ is an $m$th power of an element of $H$ for every positive number $m$.
It is proved that if a soluble group of finite rank is a residually finite $\pi$-group for some finite set $\pi$ of primes, then it is a virtually residually finite nilpotent $\pi$-group. We prove also the following generalization of Smel'kin's theorem.
Let $\pi$ be a finite set of primes. If $G$ is a soluble group of finite rank, then the group $G$ is a virtually residually finite $\pi$-group if and only if $G$ is a reduced poly-(cyclic, quasicyclic, or rational) group and $G$ has no $\pi$-radicable elements of infinite order. 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 $m$.

Keywords: finite rank group, soluble group, polycyclic group, nilpotent group, residually finite $p$-group.

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UDC: 512.543
Received: 31.01.2014

Citation: D. N. Azarov, “Some residual properties of soluble groups of finite rank”, Chebyshevskii Sb., 15:1 (2014), 7–18

Citation in format AMSBIB
\Bibitem{Aza14}
\by D.~N.~Azarov
\paper Some residual properties of soluble groups of finite rank
\jour Chebyshevskii Sb.
\yr 2014
\vol 15
\issue 1
\pages 7--18
\mathnet{http://mi.mathnet.ru/cheb321}

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This publication is cited in the following articles:
1. D. N. Azarov, “Approksimatsionnye svoistva nilpotentnykh grupp”, Model. i analiz inform. sistem, 22:2 (2015), 149–157
2. D. N. Azarov, “Approksimatsionnye svoistva abelevykh grupp”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2015, no. 3(35), 5–11
3. D. V. Goltsov, “Approksimiruemost fundamentalnoi gruppy konechnogo grafa grupp kornevym klassom grupp”, Chebyshevskii sb., 17:3 (2016), 64–71
4. D. N. Azarov, N. S. Romanovskii, “Finite homomorphic images of groups of finite rank”, Siberian Math. J., 60:3 (2019), 373–376
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