Some Residual Properties of Finite Rank Groups

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.