Invariable generation of prosoluble groups

A group G is invariably generated by a subset S of G if G = 〈sg(s) | s ∈ S〉 for each choice of g(s) ∈ G, s ∈ S. Answering two questions posed by Kantor, Lubotzky and Shalev in [8], we prove that the free prosoluble group of rank d ≥ 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. © 2016, Hebrew University of Jerusalem.