Discrete subgroup - Knowlpedia

Let $G$ be a Lie group.

A subgroup $\Gamma\le G$ is discrete if it is a discrete subset in the subspace topology (equivalently, every $\gamma\in\Gamma$ is isolated in $G$ ).

Basic Lie-theoretic consequences

Any discrete subgroup is automatically closed; hence by the Closed Subgroup Theorem it is an embedded Lie subgroup.
The Lie algebra of a discrete subgroup is trivial: $\mathrm{Lie}(\Gamma)=0,$ because there are no nontrivial smooth curves in $\Gamma$ through the identity.

If $\Gamma$ is also normal, then $G/\Gamma$ is a Lie group and the projection $G\to G/\Gamma$ is a covering homomorphism.

Examples

$\mathbb Z^n$ is a discrete subgroup of $\mathbb R^n$ (additively).
The subgroup $\{\pm I\}$ is discrete in $SU(2)$ and is the kernel of the standard covering $SU(2)\to SO(3)$ (compare $SO(3)$ ).

Context. Discrete subgroups appear as “global” corrections to Lie-algebraic data: different discrete central quotients of a simply connected group yield different Lie groups with the same Lie algebra.