# Definition Intuitively speaking, a *Lie group* is a [[group]] that can be [[Differentiable Function|smoothly]] parameterized by one or more [[Continuous Map|continuous]] variables. The number of independent variables is the [[dimension]] of the Lie group. If the [[group]] doesn't accommodate such a parameterization, it is known as a [[Discrete Group|discrete group]]. If the elements of the group are [[matrix|matrices]], then it is known as a [[Matrix Lie Group|matrix Lie group]]. ## As a Manifold A Lie group is a [[group]] which is also a [[manifold]] with [[Differentiable Manifold|differentiability class]] of $C^\infty$ , with the restriction that the group operation induces a $C^\infty$ map from the manifold into itself. Pick out any element of the group $a$. This element induces a [[map]] of $G$ into itself, taking any element $b \in G$ into $ba \in G$, i.e. $b \mapsto ba$. This map must be $C^\infty$ to fit the definition of the Lie Group above. Concretely, this means that in whatever [[Coordinate Transformations on Manifolds|coordinates]] used on the manifold $G$, the coordinates of $ba$ must be $C^\infty$ function of those of $b$. The demand for such a [[map]] is a compatibility requirement, to ensure that the manifold property is compatible with the group property. $\mathbb{R}^n$ is the simplest Lie group (note that all [[vector space|vector spaces]] are [[group|groups]] *and* [[manifold|manifolds]], and are thus Lie groups since a $C^\infty$ map exists).