I have trouble with the following corollary in John Lee's Introduction to smooth manifolds:
Corollary 7.6 A Lie group homomorphism is a Lie group isomorphism iff it is bijective.
From my understanding, a bijective homomorphism is an isomorphism and an isomorphism is always bijective. So I don't understand why we need the global rank theorem to prove this.
I've searched online and find this post very confusing, which says that "a bijective homomorphism may not be an isomorphism." Why? Am I missing something?