Residual properties of nilpotent groups

D. N. Azarov

Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 2, pp. 149-157

Résumé

Let $\pi $ be a set of primes. Recall that a group $G$ is said to be a residually finite $\pi $-group if for every nonidentity element $a$ of $G$ there exists a homomorphism of the group $G$ onto some finite $\pi $-group such that the image of the element $a$ differs from 1. A group $G$ will be said to be a virtually residually finite $\pi $-group if it contains a finite index subgroup which is a residually finite $\pi $-group. 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 $m$. Let $N$ be a nilpotent group and let all power subgroups in $N$ are finitely separable. It is proved that $N$ is a residually finite $\pi $-group if and only if $N$ has no nonidentity $\pi $-radicable elements. Suppose now that $\pi $ does not coincide with the set $\Pi $ of all primes. Let $\pi '$ be the complement of $\pi $ in the set $\Pi $. And let $T$ be a $\pi '$ component of $N$ i.e. $T$ be a set of all elements of $N$ whose orders are finite $\pi '$-numbers. We prove that the following three statements are equivalent: (1) the group $N$ is a virtually residually finite $\pi $-group; (2) the subgroup $T$ is finite and quotient group $N/T$ is a residually finite $\pi $-group; (3) the subgroup $T$ is finite and $T$ coincides with the set of all $\pi $-radicable elements of $N$.

Détail

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