Topological groups, Lie groups Locally compact abelian groups and Pontryagin duality Object

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 $\mathbb{T}$ .

Reading path

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Binary operation definition
Group concept
Commutativity definition
Abelian group definition
Circle group object

Where this leads

14 pages build on this one.

Tag	`04EAFVFTF7`
URI	`m2://04EAFVFTF7`
Kind	object
Writing	W1 stated
MSC	22A05
Instance of	Abelian group
Uses	Abelian group
Same as	Wikidata Q105356493 Wikipedia Circle group

Circle group

Description

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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