Diffeomorphism in Lie Group

Question

$G$ is a Lie group and consider $L_{g}: G \rightarrow G$ ($L_g(h)=gh$). What i need to show that $L_{g}$ is diffeomorphism. Is it something obvious? Can someone explain it to me?

Answer 1

The fact that it is a diffeomorphism comes from the very definition of Lie Group: the operation $G\times G \to G$ is a smooth map, and $L_g$ is a restriction of this map, so it is smooth. The inverse is $L_{g^{-1}}$, and that's smooth by the same argumet.