Exercise 6, § 3

So let $G$ and $H$ be connected real Lie groups and let $f: G\rightarrow H$ be a surjective morphism of Lie groups. We denote by $Z_G$ the center of G and by $Z_H$ that of H.

a) Prove the equivalence of the following assertions: (i) The mapping $f$ makes G a covering of H; (ii) Ker($f$) is discrete; (iii) The morphism $L(f): L(G)\rightarrow L(H)$ deduced from $f$ by passing to the Lie algebras is an isomorphism.

b) Prove that then $f(Z_G) = Z_H$ and that $f^{-1}(Z_H) = Z_G$.