Lie Subgroup - Knowlpedia

Let $G$ be a Lie group. A Lie subgroup of $G$ is a subgroup $H\le G$ together with a Lie group structure such that the inclusion map

i:H\hookrightarrow G

is a smooth immersion and a group homomorphism.

Embedded Lie subgroups

A Lie subgroup is called embedded if $i$ is an embedding (so $H$ is an actual submanifold of $G$ ).

Closed subgroups

A crucial fact is the closed subgroup theorem: if $H$ is a closed subgroup of $G$ (as a subset), then $H$ is an embedded Lie subgroup.

Relationship to Lie algebras

The Lie algebra $\mathfrak{h}=T_eH$ identifies with a Lie subalgebra of $\mathfrak{g}=T_eG$ ; see Lie algebra of a Lie group.

Quotients

If $H$ is closed, the coset space $G/H$ carries a natural smooth structure for which $G\to G/H$ is a submersion. When $H$ is also normal, $G/H$ is a quotient Lie group.

Examples

$\operatorname{SO}(n)\le \operatorname{GL}(n,\mathbb{R})$ .
The diagonal matrices form a Lie subgroup of $\operatorname{GL}(n,\mathbb{R})$ .