Lie Subgroup

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

is a smooth immersion and a group homomorphism (so the group operations on $H$ are smooth).

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 quotient $G/H$ carries a natural smooth structure and is the underlying manifold of the quotient construction when $H$ is normal.

Examples

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