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

From Groupprops
Jump to: navigation, search

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