Apologies if this is silly, but I don't quite follow the proof of Proposition 7.17 in John M. Lee's Introduction to Smooth Manifolds.
I get that we showed $F : G \rightarrow F(G)$ is a diffeomorphism onto $F(G)$ using Proposition 5.18, but I'm not sure how to prove $F : G \rightarrow F(G)$ is a group isomorphism. Many thanks in advance!
This is a consequence of the following bit of elementary group theory: If $G$ is any group and $\varphi\colon G \to H$ is a group homomorphism, then $\varphi(G)$ is a subgroup of $H$, so if $\varphi$ is injective, then $\varphi\colon G \to \varphi(G)$ is a bijection, hence a group isomorphism.