INVARIABLE GENERATION OF PROSOLUBLE GROUPS

A group G is invariably generated by a subset S of G if G = < s(g(s)) vertical bar s is an element of S > for each choice of g(s) is an element of G, s is an element of 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.