Exercise 9, § 4

Let $G$ be given the filtration $(C^n G)$.

(a) Show that the ideal of $\mathrm{gr}(G)$ generated by $\mathrm{gr}_2(G)$ is $\sum_{q \geqslant 2} \mathrm{gr}_q(G)$.

† We do not know whether $G_n = C^n G$ for all $n$. It is true in any case when $G$ is a free group, cf. § 5, Exercise 1. ‡ It can be shown that $\xi$ is of order 12.

(b) Let $(x_i)_{i \in I}$ be a generating family of $G$ and let $m \geq 1$. Suppose that, for all $(i, j) \in I^2$, $(x_i, x_j)^m \in C^3 G$. Show that, for all $q \geq 2$ and all $u \in C^q G$, $u^m \in C^{q+1} G$ (use (a)).