Lie subgroup

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

is a smooth immersion and a group homomorphism (so, in particular, the multiplication and inversion on $H$ are smooth with respect to this manifold structure).

If, moreover, the inclusion $i$ is a smooth embedding , then $H$ is called an embedded Lie subgroup (equivalently, $H$ is an embedded submanifold of $G$ and a subgroup).

A fundamental existence theorem is the closed subgroup theorem: if $H\le G$ is a subgroup that is closed as a subset of the underlying topological space of $G$, then $H$ admits a unique smooth manifold structure making it an embedded Lie subgroup of $G$.

At the infinitesimal level, if $H$ is a Lie subgroup then the inclusion induces an injective linear map on tangent spaces at the identity, identifying $T_eH$ with a Lie subalgebra of the Lie algebra $T_eG$.

Examples

1. $SO(n)\subset GL(n,\mathbb{R})$ is a Lie subgroup: it is a closed subgroup of $GL(n,\mathbb{R})$, hence an embedded Lie subgroup by the closed subgroup theorem.

2. The special linear group

   $$SL(n,\mathbb{R})=\{A\in GL(n,\mathbb{R}) : \det(A)=1\}$$

   is a Lie subgroup of $GL(n,\mathbb{R})$ because it is the kernel of the determinant Lie group homomorphism $\det:GL(n,\mathbb{R})\to \mathbb{R}^\times$ and is closed.

3. The lattice $\mathbb{Z}^n\subset (\mathbb{R}^n,+)$ is a (discrete) Lie subgroup: it is a closed subgroup and becomes a $0$-dimensional smooth manifold.