Invariable generation of prosoluble groups

TL;DR

This paper investigates the invariable generation properties of prosoluble and solvable profinite groups, establishing specific conditions under which these groups can or cannot be invariably generated by finite sets.

Contribution

It proves that free prosoluble groups of rank at least 2 cannot be finitely invariably generated, while free solvable profinite groups are generated by a precise number of elements depending on their rank and derived length.

Findings

Free prosoluble groups of rank ≥ 2 are not finitely invariably generated.

Free solvable profinite groups are invariably generated by exactly l(d-1)+1 elements.

Answers two open questions posed by Kantor, Lubotzky, and Shalev.

Abstract

A group $G$ is invariably generated by a subset $S$ of $G$ if $G= s^{g(s)} \mid s\in S$ for each choice of $g(s) \in G$, $s \in S$. Answering two questions posed by Kantor, Lubotzky and Shalev, we prove that the free prosoluble group of rank $d \ge 2$ cannot be invariably generated by a finite set of elements, while the free solvable profinite group of rank $d$ and derived length $l$ is invariably generated by precisely $l(d-1)+1$ elements.