Concept

Compact group

Summary
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a 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 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. Compact Lie groups Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include
  • the circle group T and the torus groups Tn,
  • the orthogonal group O(n), the special orthogonal group SO(n) and its covering spin group Spin(n),
  • the unitary group U(n) and the special unitary group SU(n),
  • the compact forms of the exceptional Lie groups: G2, F4, E6, E7, and
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