---
tag: 5M7D1BSHVZ
kind: concept
name: Discrete subgroup
msc: ["22E40", 22D05, 20H10]
specializes: [0X1JXT47GA]
sources:
  - url: https://en.wikipedia.org/wiki/Discrete_group
---

## Summary

A discrete subgroup of a topological group is a subgroup whose elements are isolated from each other.
In a Hausdorff topological group, discrete subgroups are closed.
A discrete subgroup of a Lie group $G$ acts properly discontinuously on each quotient $G/K$ with $K$ compact.
The resulting quotient spaces are central objects of geometry and number theory.
Lattices, Fuchsian groups, Kleinian groups and crystallographic groups are all discrete subgroups.

## Explanations

### L1 human

Inside the real line, the integers form a subgroup in which each element has some room around it with no other element of the subgroup.
That is what discrete means.
The numbers $a + b\sqrt2$ with $a, b$ integers also form a subgroup, but they crowd together everywhere, so this subgroup is not discrete.
Discrete subgroups behave like crystals or grids sitting inside a continuous group.