Lie-group homomorphism, isomorphism, and automorphism

Let $G$ and $H$ be Lie groups in the sense of Lie group. A Lie-group homomorphism $F:G\to H$ is a group homomorphism in the sense of Monoid homomorphism and group homomorphism that is also smooth. Thus

$F(gh)=F(g)F(h)$

for all $g,h\in G$, while preservation of identity and inverses follows from the group-homomorphism laws.

A Lie-group isomorphism is a bijective Lie-group homomorphism whose inverse is smooth. Its set-theoretic inverse is automatically a group homomorphism by The inverse of a bijective group homomorphism is a group homomorphism, so the smoothness clause makes that inverse a Lie-group homomorphism as well. A Lie-group automorphism is a Lie-group isomorphism from $G$ to itself.

Smoothness is part of the homomorphism definition on this page. The separate automatic-regularity theorem for merely continuous group homomorphisms is not being assumed. These definitions apply without alteration in dimensions zero and one. Their Lie-group inputs are nonempty and boundaryless by the page convention, and the definitions use neither nondegeneracy nor any choice principle.