Invariable generation of prosoluble groups

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.

References in corpus (2)

• Invariable generation with elements of coprime prime-power order
• Invariable Generation of Infinite Groups