# Circle group

In mathematics, the circle group, denoted by $\mathbb {T}$ or $\mathbb {S} ^{1}$ , 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

$\mathbb {T} =\{z\in \mathbb {C$ :|z|=1\}.} The circle group forms a subgroup of $\mathbb {C} ^{\times }$ , the multiplicative group of all nonzero complex numbers. Since $\mathbb {C} ^{\times }$ is abelian, it follows that $\mathbb {T}$ 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 $\theta$ :

$\theta \mapsto z=e^{i\theta }=\cos \theta +i\sin \theta .$ 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 $\mathbb {T}$ for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, $\mathbb {T} ^{n}$ (the direct product of $\mathbb {T}$ with itself $n$ times) is geometrically an $n$ -torus.

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