Circle group

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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 denoted $S^{1},T,T^{1}$ and 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 $SO(2,k)$.

Arithmetic functions

Function	Value	Similar groups	Explanation
order	cardinality of the continuum
exponent	infinite		irrational multiples of $2\pi$ have infinite order.
dimension of a real Lie group	1		Follows from definition as $\mathbb {R} /\mathbb {Z}$.
dimension of an algebraic group	1		Follows from definition as a one-dimensional variety.

Group properties

Property	Satisfied?	Explanation
abelian group	Yes
locally cyclic group	No	The subgroup generated by two elements that are not rational multiples of each other is not cyclic.

Linear representation theory

Further information: Linear representation theory of circle group