Exercise 32, § 9

Let $G$ be a Lie group of type (N) (Exercise 29) and let $\Gamma$ be a discrete subgroup of $G$. Prove the equivalence of the following conditions: (a) $G/\Gamma$ is compact; (b) the measure of $G/\Gamma$ is finite (relative to a non-zero $G$-invariant positive measure); (c) every integral subgroup of $G$ containing $\Gamma$ is equal to $G$. (Use Exercise 29.)