Exercise 26, § 4

Let $G$ be a group and let $(H_n)_{n \geq 1}$ be a decreasing sequence of subgroups of $G$; suppose that $H_1 = G$ and that if we write $(x, y) = xyx^{-1}y^{-1}$, the relations $x \in H_i,\ y \in H_j$ imply $(x, y) \in H_{i+j}$.

(a) Let $G_i = H_i/H_{i+1}$; show that $G_i$ is commutative and that the mapping $x, y \mapsto (x, y)$ defines on passing to the quotient a $\mathbf{Z}$-bilinear mapping of $G_i \times G_j$ into $G_{i+j}$.

(b) Let $\operatorname{gr}(G) = \sum_{i=1}^\infty G_i$ and extend by linearity the mappings $G_i \times G_j \to G_{i+j}$ defined in (a) to a $\mathbf{Z}$-bilinear mapping of $\operatorname{gr}(G) \times \operatorname{gr}(G)$ into $\operatorname{gr}(G)$. Show that $\operatorname{gr}(G)$ is a Lie $\mathbf{Z}$-algebra with this mapping (to verify the Jacobi identity, use the following formula:

where $x^y$ denotes $yxy^{-1}$ and $e$ denotes the identity element of $G$.

(c) Suppose that there exists $n$ such that $H_n = \{e\}$. Show that $\operatorname{gr}(G)$ is a nilpotent Lie $\mathbf{Z}$-algebra.