Lie Groups
Definition1
A group $G$ is called a Lie group if it satisfies the following conditions:
It has a differentiable structure.
The binary operation $\cdot : G \times G \to G$ defined in $G$ is differentiable.
The inverse ${}^{-1} : G \to G$ is differentiable.
Explanation
Simply put, a Lie group is a differentiable group.
Examples
$(\mathbb{R}, +)$
Euclidean space has a differentiable structure.
$f : (x,y) \mapsto x+y \in C^{\infty}$
$g : x \mapsto -x \in C^{\infty}$
Manfredo P. Do Carmo, Riemannian Geometry (Eng Edition, 1992), p39-40 ↩︎