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

$\theta \mapsto z=e^{i\theta }=\cos \theta +i\sin \theta .$ 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 ${\mbox{SO}}(2)$ .