---
tag: EVHCE5VR6Y
kind: concept
name: Compact group
msc: [22C05]
specializes: [AHV05ZBCYW]
sources:
  - url: https://en.wikipedia.org/wiki/Compact_group
same_as:
  wikidata: Q1887083
  wikipedia: Compact group
---

## Summary

A compact group is a topological group whose topology is compact and Hausdorff.
Compact groups carry a Haar measure of total mass $1$, so one can average over the group as over a finite group.
So their representation theory looks much like that of finite groups.
Every finite-dimensional representation splits into irreducible ones, and the Peter–Weyl theorem decomposes $L^2(G)$.

## Explanations

### L1 human

A compact group is a group that is also a "closed and bounded" space, with no way to run off to infinity.
The circle of rotations of the plane is the simplest infinite example, and the group of rotations of space is another.
Every finite group is a compact group with the discrete topology.
The key feature is that one can take the average of a function over all elements of the group, just as one averages over a finite set.