Differentiable Manifolds/Lie groups

Definition (Lie group):

A Lie group is a differentiable manifold $G$ whose group law (denoted here by juxtaposition) has the following properties:

1. The function $G\times G\to G$, $(g,h)\mapsto gh$ is differentiable
2. The function $G\to G$, $h\mapsto h^{-1}$ is differentiable