Exercise 2, § 3

Suppose that K is locally compact. Let G be a Lie group of finite dimension n.

(a) Let $\phi$ be an étale endomorphism of G with finite kernel. Let $\mu$ be a left Haar measure of G. Then $\phi$ is proper and

(Use Proposition 55.)

(b) Suppose that G is compact. Let $\phi$ be an étale endomorphism of G. Then Ker $\phi$ is finite, $\phi(G)$ is of finite index in G and

(Use (a) to transform $\mu(G)$.) (c) Suppose that G is compact and commutative. Let $r \in \mathbf{Z}$ be such that $r.1 \neq 0$ in K and let $\phi$ be the endomorphism $x \mapsto x^r$ of G. Then (Observe that $L(\phi)$ is the homothety of ratio r by § 2, Corollary to Proposition 7.)