Editing Compact group (section)

{{Short description|Topological group with compact topology}}
{{About|mathematics|astronomy of galaxies|galaxy group}}

[[Image:Circle as Lie group.svg|right|thumb|The [[circle]] of center 0 and radius 1 in the [[complex plane]] is a compact Lie group with complex multiplication.]]

In [[mathematics]], a '''compact''' ('''topological''') '''group''' is a [[topological group]] whose [[topology]] realizes it as a [[compact space|compact topological space]] (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of [[finite group]]s with the [[discrete topology]] and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to [[Group action (mathematics)|group action]]s and [[representation theory]].

In the following we will assume all groups are [[Hausdorff space]]s.