---
tag: F7H6GPR050
kind: definition
name: Maximal compact subgroup
msc: ["22E15", 22C05]
defines: T2EB8R9HEV
uses: [92700QAB5Y, 0TVJZYB6SH]
sources:
  - url: https://en.wikipedia.org/wiki/Maximal_compact_subgroup
---

## Definition

Let $G$ be a [[topological group|92700QAB5Y]].
A subgroup $K$ of $G$ is a *maximal compact subgroup* if two conditions hold.

1. The subgroup $K$ is [[compact|0TVJZYB6SH]] in the subspace topology.
2. If $L$ is a compact subgroup of $G$ with $K \subseteq L$, then $L = K$.

## Remarks

Example: $K = \mathrm{O}(n)$ in $G = \mathrm{GL}_n(\R)$.
It is closed and bounded in the space of matrices, so it is compact.
Let $L \supseteq \mathrm{O}(n)$ be a compact subgroup.
Averaging the standard inner product over $L$ with the Haar measure gives an inner product $\ip{\cdot}{\cdot}'$ that $L$ preserves.
The group $\mathrm{O}(n)$ preserves it too, and the only $\mathrm{O}(n)$-invariant inner products are the positive multiples of the standard one.
So $L$ preserves the standard inner product, and $L \subseteq \mathrm{O}(n)$.

Non-example: $\mathrm{SO}(n)$ in $\mathrm{GL}_n(\R)$ is compact, but it fails clause 2, since $\mathrm{SO}(n) \subsetneq \mathrm{O}(n)$.

A maximal compact subgroup need not exist.
In the countable direct sum $\bigoplus_{i \in \N} \Z/2$ with the discrete topology, the compact subgroups are the finite ones, and each finite subgroup lies in a larger finite subgroup.