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## A Criterion Concerning Lie Groups

Differential Geometry
Tsakanikas Nickos
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### A Criterion Concerning Lie Groups

Definition: A smooth manifold $G$ with a group structure such that the multiplication map $\cdot \ \colon G \times G \longrightarrow G \ , \ (g,h) \mapsto g \cdot h$ and the inversion map $G \longrightarrow G \ , \ g \mapsto g^{-1}$ are smooth is called a Lie Group.

Let $G$ be a smooth manifold with a group structure such that the map $G \times G \longrightarrow G \ , \ (g,h) \mapsto gh^{-1}$ is smooth. Show that $G$ is a Lie Group.