Exercise 15, § 4

Preserving the notation of the above exercise, suppose that G is nilpotent. Let $i : G \to \overline{G}$ be a homomorphism of G into a nilpotent group $\overline{G}$. $(i, \overline{G})$ is called a P-envelope of G if the following conditions hold:

(i) $\overline{G}$ is P-divisible and P-torsion-free. (ii) The kernel of $i$ is the set of P-torsion elements of G. (iii) The P-saturation of $i(G)$ in $\overline{G}$ is equal to $\overline{G}$.

(a) Let $(i, \overline{G})$ be a P-envelope of G and let L be a P-divisible P-torsion-free nilpotent group. Show that, for every homomorphism $f : G \to L$, there exists one and only one homomorphism $\tilde{f} : \overline{G} \to L$ such that $\tilde{f} \circ i = f$ (reduce it to the case where G is P-torsion-free and use Exercise 14 (d)). Deduce that, if G has a P-envelope,\footnote{† In fact, every nilpotent group has a P-envelope, cf. § 5, Exercise 6; see also M. Lazard, Annales E. N. S., 71 (1954), pp. 101–190, Chapter II, § 3.} this is unique up to isomorphism.

(b) Let $(i, \overline{G})$ be a P-envelope of G, let H be a subgroup of G, let $\overline{H}$ be the P-envelope of $i(H)$ in $\overline{G}$ and let $i_H : H \to \overline{H}$ the homomorphism induced by $i$. Show that $(i_H, \overline{H})$ is a P-envelope of H.

Suppose that H is normal in G. Then $\overline{H}$ is normal in $\overline{G}$, cf. Exercise 14 (b); if $i_{G/H}: G/H \to \overline{G}/\overline{H}$ is the homomorphism induced by i, show that $(i_{G/H}, \overline{G}/\overline{H})$ is a P-envelope of G/H.