---
tag: T2EB8R9HEV
kind: concept
name: Maximal compact subgroup
msc: ["22E15", 22C05]
sources:
  - url: https://en.wikipedia.org/wiki/Maximal_compact_subgroup
same_as:
  wikidata: Q6795635
  wikipedia: Maximal compact subgroup
---

## Summary

A maximal compact subgroup of a [[topological group|92700QAB5Y]] is a compact subgroup that is not contained in any larger compact subgroup.
In a connected [[Lie group|7JBRGCVJRB]], maximal compact subgroups exist and are all conjugate.
The group is then, as a space, the product of a maximal compact subgroup with a Euclidean space.
So a maximal compact subgroup carries the topology of the whole group.

## Explanations

### L1 human

Inside a big group of symmetries, look for the largest piece that is bounded.
For the invertible real $n \times n$ matrices, this piece is the group $\mathrm{O}(n)$ of rotations and reflections.
Every invertible matrix is a rotation or reflection times a stretching along perpendicular axes.
The stretchings form a space with no holes, so all the interesting shape of the group comes from $\mathrm{O}(n)$.