Some Residual Properties of Finite Rank Groups

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 π of almost all primes the group G of finite rank is a residually finite nilpotent π-group, it is nilpotent. For nilpotent groups of finite rank the necessary and sufficient condition to be a residually finite π-group is obtained, where π is a set of primes.