Circle group

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 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.

Elementary introduction

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°).