In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane.
The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers. Since C× is abelian, it follows that T is as well. The circle group is also the group U(1) of 1×1 unitary matrices; these act on the complex plane by rotation about the origin. The circle group can be parametrized by the angle θ of rotation by
This is the exponential map for the circle group.
The circle group plays a central role in Pontryagin duality, and in the theory of Lie groups.
The notation T for the circle group stems from the fact that Tn (the direct product of T with itself n times) is geometrically an n-torus. The circle group is then a 1-torus.
Read more about Circle Group: Elementary Introduction, Topological and Analytic Structure, Isomorphisms, Properties, Representations, Group Structure
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