Circle group

Description

The circle group is the set $\mathbb{T} = \{z \in \mathbb{C} : |z| = 1\}$ of the complex numbers of absolute value $1$, with complex multiplication. It is an abelian group with identity element $1$, and the inverse of $z$ is its complex conjugate $\bar{z} = 1/z$. Each element of $\mathbb{T}$ is $e^{i\theta}$ for a real number $\theta$, and $e^{i\theta} e^{i\varphi} = e^{i(\theta + \varphi)}$.

Remarks

The map $t \mapsto e^{2\pi i t}$ is a surjective homomorphism from $(\mathbb{R}, +)$ to $\mathbb{T}$ with kernel $\mathbb{Z}$, thus $\mathbb{T}$ is isomorphic to the quotient $\mathbb{R}/\mathbb{Z}$. The circle group is also isomorphic to the group $\mathrm{SO}(2)$ of rotations of the plane about the origin. For each positive integer $n$, the group of $n$th roots of unity is a finite subgroup of $\mathbb{T}$.

Explanations

L1 human

The circle group is the unit circle in the complex plane, where the product of two points is their complex product. If you write the points as angles, the product adds the angles. Thus the circle group is the group of rotations of a circle.