Powers in finite groups.

link: http://arxiv.org/abs/0909.4639
Abstract

In this note we prove that if $G$ is a finitely generated profinite group
then the verbal subgroup $G^{q}$ is open. Equivalently in a $d$-generator
finite group every product of $q$th powers is a product of $f(d,q)$ $q$th
powers.