Generation of polycyclic groups

TL;DR

This paper provides a simpler proof of a theorem relating the minimal number of generators of polycyclic groups to their profinite completions, with implications for virtually abelian groups.

Contribution

It offers an alternative, simpler proof of an existing theorem and identifies conditions where the minimal number of generators equals that of the profinite completion.

Findings

Simpler proof of Linnell and Warhurst's theorem

Conditions for equality of generators in virtually abelian groups

Verification method for the generator count equality

Abstract

In this note we give an alternative proof of a theorem of Linnell and Warhurst that the number of generators d(G) of a polycyclic group G is at most d(\hat G), where d(\hat G) is the number of generators of the profinite completion of G. While not claiming anything new we believe that our argument is much simpler that the original one. Moreover our result gives some sufficient condition when d(G)=d(\hat G) which can be verified quite easily in the case when G is virtually abelian.