On power subgroups of profinite groups

Abstract:

In this paper we prove that if G is a finitely generated pro-(finite nilpotent) group, then every subgroup ${G^n}$, generated by nth powers of elements of G, is closed in G. It is also obtained, as a consequence of the above proof, that if G is a nilpotent group generated by m elements ${x_1}, \ldots ,{x_m}$, then there is a function $f(m,n)$ such that if every word in $x_i^{ \pm 1}$ of length $\leq f(m,n)$ has order n, then G is a group of exponent n. This question had been formulated by Ol’shansky in the general case and, in this paper, is proved in the solvable case and the problem is reduced to the existence of such function for finite simple groups.