  Let $G$ be a group. We give an explicit description of the set of elements $x
\in G$ such that $x^{|G:H|} \in H$ for every subgroup of finite index $H
\leqslant G$. This is related to the following problem: given two subgroups $H$
and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?

Group Theory