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[1]

$${\displaystyle \mathbb {T} =\{z\in \mathbb {C}$$ :|z|=1\}~.}

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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. The circle group is also the group ${\displaystyle {\mbox{U}}(1)}$ 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 ${\displaystyle \theta }$ of rotation by

$${\displaystyle \theta \mapsto z=e^{i\theta }=\cos \theta +i\sin \theta .}$$

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 ${\displaystyle {\mbox{SO}}(2)}$.