Dimension of extension group is sum of dimensions of normal subgroup and quotient group

Statement

Statement for algebraic groups

Suppose $G$ is an algebraic group, $N$ is a closed normal subgroup and $G/N$ is the quotient group. Both $N$ and $G/N$ acquire algebraic group structures from the algebraic group structure of $G$. We then have the following relationship between the dimensions of $G,N,G/N$:

$\dim G = \dim N + \dim (G/N)$

In particular, $G$ is a finite-dimensional algebraic group if and only if both $N$ and $G/N$ are.

Statement for Lie groups

Suppose $G$ is a Lie group, $N$ is a closed normal subgroup and $G/N$ is the quotient group. Both $N$ and $G/N$ acquire Lie group structures from the Lie group structure of $G$. We then have the following relationship between the dimensions of $G,N,G/N$:

$\dim G = \dim N + \dim (G/N)$

Related facts

• Dimension of direct product is sum of dimensions