Exercise 11, § 4

Let $H$ be a subgroup of $G$ and let $m$ be an integer $\geq 1$. Suppose that $G$ is generated by a family $(x_i)_{i \in I}$ such that $x_i^m \in H$ for all $i$. (a) Let $H$ be given the filtration induced by the filtration $(C^n G)$ on $G$ and let $\operatorname{gr}(H)$ be identified with a graded Lie subalgebra of $\operatorname{gr}(G)$, cf. Exercise 2. Show that, for all $n \geq 0$,

Deduce that, for all $z \in H$, $C^n G$, $z^m \in H$, $C^{n+1} G$. (b) If $G$ is nilpotent, show that there exists an integer $N \geq 0$ (depending only on the nilpotency class of $G$) such that $z^{m^N} \in H$ for all $z \in G$.