Exercise 33, § 9

Let $\Gamma$ be a group. Prove the equivalence of the following conditions: (a) $\Gamma$ is nilpotent, torsion-free and finitely generated; (b) there exists a Lie group of type (N) (Exercise 29) containing $\Gamma$ as a discrete subgroup; (c) there exists a Lie group $G$ of type (N) (Exercise 29) containing $\Gamma$ as a discrete subgroup and such that $G/\Gamma$ is compact. (The equivalence of (b) and (c) follows from Exercise 29. The implication (c) $\Rightarrow$ (a) is proved by induction on $\dim(G)$ by the method of Exercise 29 (c). To prove that (a) $\Rightarrow$ (c), show first that the Lie $\mathbf{Q}$-algebra attached to the saturation of $\Gamma$ (cf. Chapter II, § 6, Exercise 4) is finite-dimensional; then take the tensor product of this algebra with $\mathbf{R}$ and the corresponding group of type (N).)