Circle group

In mathematics, the circle group, denoted by T {\displaystyle \mathbb {T} } or ⁠ S 1 {\displaystyle 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 T = { z ∈ C : | z | = 1 } . {\displaystyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}.} The circle group plays a fundamental role in many areas of mathematics. When a given unit complex number u {\displaystyle u} multiplies the other points of the circle, the effect is to rotate them through an angle determined by u {\displaystyle u} . In this way, the circle group becomes the group of symmetries of the circle which preserve its orientation (do not flip it). Composition of two rotations is the ordinary multiplication of complex numbers. Multiplication is commutative, ⁠ z 1 z 2 = z 2 z 1 {\displaystyle z_{1}z_{2}=z_{2}z_{1}} ⁠, making the circle group commutative (an abelian group); correspondingly several rotations of the plane can be composed in any order with the same result. Rotations can alternately be parametrized by the angle measure ⁠ θ {\displaystyle \theta } ⁠, which is related to ⁠ z {\displaystyle z} ⁠ by the complex exponential function: θ ↦ z = e i θ = cos ⁡ θ + i sin ⁡ θ . {\displaystyle \theta \mapsto z=e^{i\theta }=\cos \theta +i\sin \theta .} The circle group is sometimes denoted by U ( 1 ) {\displaystyle U(1)} , which is the unitary group of 1 × 1 {\displaystyle 1\times 1} complex matrices. It is structurally the same as (i.e., isomorphic to) the group of 2-dimensional rotation matrices, i.e., the special orthogonal group ⁠ S O ( 2 ) {\displaystyle \mathrm {SO} (2)} ⁠. The angle measure...