---
tag: 04EAFVFTF7
kind: object
name: Circle group
msc: [22A05]
instance_of: [082GHNPJ53]
uses: [071DDFVG4J]
sources:
  - url: https://en.wikipedia.org/wiki/Circle_group
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Circle_group
    locator: Isomorphisms and conventions
same_as:
  wikidata: Q105356493
  wikipedia: 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|082GHNPJ53]] 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|0WX11EA3F3]] 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.