Exercise 31, § 9

Let $G$ and $G'$ be two Lie groups of type (N) (Exercise 29) and let $\Gamma$ be a discrete subgroup of $G$ such that $G/\Gamma$ is compact. Show that every homomorphism $f \colon \Gamma \to G'$ can be extended uniquely to a Lie group morphism of $G$ into $G'$ (begin by extending $f$ to the saturation $\bar{\Gamma}$ of $\Gamma$ and derive a homomorphism of the Lie $\mathbf{Q}$-algebra $\log(\bar{\Gamma})$ into the Lie algebra of $G'$, cf. Exercise 29).