This article is cited in 3 scientific papers (total in 3 papers)
Some Residual Properties of Finite Rank Groups
D. N. Azarov
Ivanovo State University, Ermaka str., 39, Ivanovo, 153025, Russia
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.
finite rank group, residually finite $p$-group.
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D. N. Azarov, “Some Residual Properties of Finite Rank Groups”, Model. Anal. Inform. Sist., 21:2 (2014), 50–55
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\paper Some Residual Properties of Finite Rank Groups
\jour Model. Anal. Inform. Sist.
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This publication is cited in the following articles:
D. N. Azarov, “Approksimatsionnye svoistva nilpotentnykh grupp”, Model. i analiz inform. sistem, 22:2 (2015), 149–157
D. N. Azarov, “Approksimatsionnye svoistva abelevykh grupp”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2015, no. 3(35), 5–11
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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