Circle group

In mathematics, the circle group, denoted by ${\displaystyle \mathbb {T} }$ or ⁠${\displaystyle \mathbb {S} ^{1}}$⁠, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers $${\displaystyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}.}$$

The circle group forms a subgroup of ⁠${\displaystyle \mathbb {C} ^{\times }}$⁠, the multiplicative group of all nonzero complex numbers. Since ${\displaystyle \mathbb {C} ^{\times }}$ is abelian, it follows that ${\displaystyle \mathbb {T} }$ is as well.

A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure ⁠${\displaystyle \theta }$⁠: $${\displaystyle \theta \mapsto z=e^{i\theta }=\cos \theta +i\sin \theta .}$$

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 ${\displaystyle \mathbb {T} }$ for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, ${\displaystyle \mathbb {T} ^{n}}$ (the direct product of ${\displaystyle \mathbb {T} }$ with itself ${\displaystyle n}$ times) is geometrically an ${\displaystyle n}$-torus.

The circle group is isomorphic to the special orthogonal group ⁠${\displaystyle \mathrm {SO} (2)}$⁠.

One way to think about the circle group is that it describes how to add angles, where only angles between 0° and 360° or ${\displaystyle \in [0,2\pi )}$ or ${\displaystyle \in (-\pi ,+\pi ]}$ are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer is 150° + 270° = 420°, but when thinking in terms of the circle group, we may "forget" the fact that we have wrapped once around the circle. Therefore, we adjust our answer by 360°, which gives 420° ≡ 60° (mod 360°).