Exercise 10, § 5

Let $p$ be a prime number and let G be a group with a real central filtration $(G_\alpha)$. Suppose that the relation $x \in G_\alpha$ implies $x^p \in G_{p\alpha}$, in which case the filtration $(G_\alpha)$ is called restricted. (a) Show that the Lie algebra $\mathrm{gr}(G)$ associated with $(G_\alpha)$ is such that $p \cdot \mathrm{gr}(G) = 0$ and hence can be given an algebra structure over $\mathbf{F}_p$. (b) Let $\xi \in \mathrm{gr}_\alpha(G)$ and let $x$ be a representative of $\xi$ in $G_\alpha$. Show that the image of $x^p$ in $\mathrm{gr}_{p\alpha}(G)$ does not depend on the choice of $x$ (use Exercise 9 (c)). If this image is denoted by $\xi^{[p]}$, prove that $\xi \mapsto \xi^{[p]}$ is a linear mapping of $\mathrm{gr}_\alpha(G)$ into $\mathrm{gr}_{p\alpha}(G)$ and that $(\xi + \xi')^{[p]} = \xi^{[p]} + {\xi'}^{[p]} + \Delta_p(\xi, \xi')$. (Same method.)

Show that, if $\xi \in \mathrm{gr}_\alpha(G)$ and $\eta \in \mathrm{gr}_\beta(G)$, then

(Use Exercise 9 (d).)

(c) Show that there exists one and only one $p$-mapping of $\mathrm{gr}(G)$ into itself (Chapter I, § 1, Exercise 20) which extends the mappings $\xi \mapsto \xi^{[p]}$ defined on the $\mathrm{gr}_\alpha(G)$. With this structure, $\mathrm{gr}(G)$ is a Lie $p$-algebra, said to be associated with the restricted filtration $(G_\alpha)$.