Exercise 2, § 4

Let $(G_\alpha)$ be a real filtration on a group G and let $v$ be the corresponding order function. Let H be a subgroup of G.

(a) Let $H_\alpha = H \cap G_\alpha$. Show that $(H_\alpha)$ is a real filtration of H and that the corresponding order function is the restriction of $v$ to H. $H_\alpha^+ = H \cap G_\alpha^+$ and $gr(H)$ is identified with a graded subgroup of $gr(G)$.

(b) Suppose that H is normal and that $v(G) \cap \mathbf{R}$ is a discrete subset of $\mathbf{R}$. We write $(G/H)_\alpha = (G_\alpha H)/H$. Show that $((G/H)_\alpha)$ is a real filtration of $G/H$ and that the corresponding order function $v_{G/H}$ is given by the formula

Show that, for all $\alpha \in \mathbf{R}$, there is an exact sequence

(c) With the hypotheses of (b), suppose that $(G_\alpha)$ is a central filtration. Show that so are the filtrations on $H$ and $G/H$ induced by $(G_\alpha)$ and that the Lie algebra $\operatorname{gr}(G/H)$ is identified with the quotient of $\operatorname{gr}(G)$ by the ideal $\operatorname{gr}(H)$.