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This group is denoted 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
Note that the isomorphism between the group and the complex numbers of modulus one is given by:
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 .
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 is the group .
|order||cardinality of the continuum|
|exponent||infinite||irrational multiples of have infinite order.|
|dimension of a real Lie group||1||Follows from definition as .|
|dimension of an algebraic group||1||Follows from definition as a one-dimensional variety.|
|locally cyclic group||No||The subgroup generated by two elements that are not rational multiples of each other is not cyclic.|