Compact group

In mathematics, a compact (topological) group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.

In the following we will assume all groups are Hausdorff spaces.

Lie groups form a very nice class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include

The classification theorem of compact Lie groups states that up to finite extensions and finite covers this exhausts the list of examples (which already includes some redundancies).

Given any compact Lie group G one can take its identity component G₀, which is connected. The quotient group G/G₀ is the group of components π₀(G) which must be finite since G is compact. We therefore have a finite extension

Now every compact, connected Lie group G₀ has a finite covering

where $A\subset Z({\tilde {G}}_{0})$ $A\subset Z({\tilde {G}}_{0})$ is a finite abelian group and ${\tilde {G_{0}}}$ ${\tilde {G_{0}}}$ is a product of a torus and a compact, connected, simply-connected Lie group K:

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