#lietheo 
## Definition
A *Lie group* is a group $G$ with a smooth structure, i.e. a [[Manifold]], whose group multiplication and inversion
$$
\begin{align*}
 m: G \times G \to G, &(g_{1}, g_{2}) \mapsto g_{1}\times g_{2} \\
 i: G \to G, &g \mapsto g^{-1}
\end{align*}
$$
are [[Smooth Map|smooth]].

## Properties

The direct product and intersection of Lie groups is again a lie group.

Discrete groups, i.e. those with the discrete [[Topology]], particularly those with finite or countable elements, are lie groups.

If $G$ is a smooth manifold with a group structure such that the multiplication is smooth, the inverse must also be smooth and $G$ is a Lie group.
In particular, we do _not_ need to require a smooth inversion in the definition!