Exercise 7, § 5

We preserve the notation of Exercise 6.

(a) Let G be a nilpotent group of class $<c$ and let $(i, \overline{G})$ be a P-envelope of G. Let $\rho : F^c \to G$ be a surjective homomorphism (such a homomorphism exists if X is suitably chosen) and let $\bar{\rho}$ be the corresponding homomorphism of $F_P^c$ into $\overline{G}$ (§ 4, Exercise 15). The homomorphism $\bar{\rho}$ is surjective and maps $C^n F_P^c = F_{P,n}^c$ onto $C^n \overline{G}$. Show that $C^n \overline{G}$ is the P-envelope of $i(C^n G)$ in $\overline{G}$ and that the Lie algebra $\mathrm{gr}(\overline{G})$ is identified with $\mathbf{Z}_P \otimes \mathrm{gr}(G)$.

(b) Deduce that G is P-divisible and P-torsion-free if and only if the $C^n G/C^{n+1} G$ are.