Exercise 7, § 4

Let $G$ be given the filtration $(C^n G)$ and suppose that $\mathrm{gr}_1(G) = G/(G, G)$ is cyclic. Show that $\mathrm{gr}_n(G) = \{0\}$ for $n \geqslant 2$ (use Proposition 5) and deduce that $C^n G = (G, G)$ for $n \geqslant 2$.