Exercise 5, § 3

Let G be a connected locally simply connected topological group, let $(\widetilde{G}, p)$ be a universal covering of G; we equip $\widetilde{G}$ with its topological group structure for which $p$ is a group homomorphism. We denote by Z the center of G$,e$ its identity element, $\widetilde{Z}$ the center of $\widetilde{G}$ and $\widetilde{e}$ its identity element. (a) Let $f$ be a continuous automorphism of the group G. Prove that there exists a unique continuous mapping $\widetilde{f}:\widetilde{G}\rightarrow \widetilde{G}$ such that $\widetilde{f}(\widetilde{e}) =\widetilde{e}$ and $p\circ \widetilde{f}$ = $f\circ p$. Prove that $\widetilde{f}$ is a group automorphism. Prove that the mapping $\varphi :f\mapsto \widetilde{f}$ is an injective group morphism from Aut(G) into Aut($\widetilde{G}$).

(b) We denote by Out(G) the quotient of the group Aut(G) by the subgroup of inner automorphisms; we define Out($\widetilde{G}$) in an analogous manner. Prove that the homomorphism $\varphi$ induces, by passing to the quotients, an injective group homomorphism $\psi :$ Out(G) $\rightarrow$ Out($\widetilde{G}$).