# Circle group

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## 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 $\R/\Z$, where $\R$ is the additive group of real numbers and $\Z$ is the subgroup of $\R$ defined as the additive group of integers.

Note that the isomorphism between the group $\R/\Z$ 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 $SO(2,\R)$.

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 $\R/\Z$.
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.