SO(2)
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 or simply the unit complex numbers
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 complex-valued 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, with the standard topology (see below), the circle group is a 1-torus. More generally Tn (the direct product of T with itself n times) is geometrically an n-torus.
One way to think about the circle group is that it describes how to add angles, where only angles between 0° and 360° are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer should be 150° + 270° = 420°, but when thinking in terms of the circle group, we need to "forget" the fact that we have wrapped once around the circle. Therefore, we adjust our answer by 360° which gives 420° = 60° (mod 360°).
Another description is in terms of ordinary addition, where only numbers between 0 and 1 are allowed (with 1 corresponding to a full rotation). To achieve this, we might need to throw away digits occurring before the decimal point. For example, when we work out 0.784 + 0.925 + 0.446, the answer should be 2.155, but we throw away the leading 2, so the answer (in the circle group) is just 0.155.
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