Exercise 12, § 4

Suppose that $G$ is nilpotent. Let $H$ be a subgroup of $G$ and let $m$ be an integer $\geq 1$ and let $x, y$ be elements of $G$ such that $x^m \in H$ and $y^m \in H$. Show that there exists an integer $N \geq 0$ such that $(xy)^{m^N} \in H$. (Apply Exercise 11 to the group generated by $\{x, y\}$ and its intersection with $H$.)