---
tag: NM8XCPS1Q3
kind: concept
name: Lie subgroup
msc: ["22E15", "22E60"]
specializes: [0X1JXT47GA]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_group
---

## Summary

A Lie subgroup of a [[Lie group|7JBRGCVJRB]] $G$ is a [[subgroup|0X1JXT47GA]] with its own Lie group structure.
The inclusion into $G$ must be a smooth [[immersion|6RW68JGBHC]].
Its topology may be finer than the subspace topology, so a Lie subgroup need not be closed or embedded.
The connected Lie subgroups of $G$ correspond exactly to the Lie subalgebras of the Lie algebra of $G$.

## Explanations

### L1 human

A Lie subgroup is a smaller Lie group sitting inside a bigger one.
It can be a nice closed piece, like the rotations about one axis inside all rotations of space.
It can also wind around in a complicated way.
In a torus, a line with irrational slope wraps around forever and comes close to every point, yet it is a Lie subgroup, a copy of $\R$.
Its own smooth structure is that of the line, not the one it would inherit as a subset of the torus.