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 Model. Anal. Inform. Sist., 2014, Volume 21, Number 2, Pages 50–55 (Mi mais370)

Some Residual Properties of Finite Rank Groups

D. N. Azarov

Ivanovo State University, Ermaka str., 39, Ivanovo, 153025, Russia

Abstract: The generalization of one classical Seksenbaev theorem for polycyclic groups is obtained. Seksenbaev proved that if $G$ is a polycyclic group which is residually finite $p$-group for infinitely many primes $p$, it is nilpotent. 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 a finite $p$-group such that the image of the element $a$ differs from 1. One of the generalizations of the notation of a polycyclic group is the notation of a 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. We prove the following generalization of Seksenbaev theorem: if $G$ is a group of finite rank which is a residually finite $p$-group for infinitely many primes $p$, it is nilpotent. Moreover, we prove that if for every set $\pi$ of almost all primes the group $G$ of finite rank is a residually finite nilpotent $\pi$-group, it is nilpotent. For nilpotent groups of finite rank the necessary and sufficient condition to be a residually finite $\pi$-group is obtained, where $\pi$ is a set of primes.

Keywords: finite rank group, residually finite $p$-group.

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Citation: D. N. Azarov, “Some Residual Properties of Finite Rank Groups”, Model. Anal. Inform. Sist., 21:2 (2014), 50–55

Citation in format AMSBIB
\Bibitem{Aza14} \by D.~N.~Azarov \paper Some Residual Properties of Finite Rank Groups \jour Model. Anal. Inform. Sist. \yr 2014 \vol 21 \issue 2 \pages 50--55 \mathnet{http://mi.mathnet.ru/mais370} 

• http://mi.mathnet.ru/eng/mais370
• http://mi.mathnet.ru/eng/mais/v21/i2/p50

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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. N. Azarov, “A criterion for the $\mathscr F_\pi$-residuality of free products with amalgamated cyclic subgroup of nilpotent groups of finite ranks”, Siberian Math. J., 57:3 (2016), 377–384
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