Some residual properties of soluble groups of finite rank

D. N. Azarov

Čebyševskij sbornik, Tome 15 (2014) no. 1, pp. 7-18.

Résumé

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 $m$.

Détail

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

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