Circle group

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined in a number of equivalent ways.

As a multiplicative group of complex numbers

The circle group is defined as the group, under multiplication, of complex numbers of modulus one. In other words, it is the group of complex numbers on the unit circle, under multiplication.

As reals modulo integers

An equivalent description of the circle group is as the quotient group $\mathbb{R}/\mathbb{Z}$, where $\mathbb{R}$ is the additive group of real numbers and $\mathbb{Z}$ is the subgroup of $\mathbb{R}$ defined as the additive group of integers.

Note that the isomorphism between the group $\mathbb{R}/\mathbb{Z}$ and the complex numbers of modulus one is given by:

$\theta \mapsto e^{2\pi i\theta }$.

As rotations

The circle group can also be defined in the following equivalent ways:

• It is the group of rotations in the plane (including the identity map) that fix a particular point, under composition.
• It is the special orthogonal group of degree two over the real numbers. In other words, it is the group $SO(2,\mathbb{R})$.

Note that the term circle group may be used for the more general notion of the circle group over any field or ring. The circle group over a field $k$ is the group $SO(2,k)$.