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

From Groupprops

## Contents

## Statement

### Statement for algebraic groups

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

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

### Statement for Lie groups

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