Contents

Definition

The circle group $\mathbb{T}$ is equivalently (isomorphically)

• the quotient group $\mathbb{R}/\mathbb{Z}$ of the additive group of real numbers by the additive group of integers, induced by the canonical embedding $\mathbb{Z} \hookrightarrow \mathbb{R}$;

• the unitary group $\mathrm{U}(1)$;

• the special orthogonal group $SO(2)$;

• the subgroup of the group of units $\mathbb{C}^\times$ of the field of complex numbers (its multiplicative group) given by those of any fixed positive modulus (standardly $1$).

Properties

For general abstract properties usually the first characterization is the most important one. Notably it implies that the circle group fits into a short exact sequence

the “real exponential exact sequence”.

(On the other hand, the last characterization is usually preferred when one wants to be concrete.)

A character of an abelian group $A$ is simply a homomorphism from $A$ to the circle group.

$U(1)$ is the compact real form of the multiplicative group $\mathbb{G}_m = \mathbb{C}^\times$ over the complex numbers, see at form of an algebraic group – Circle group and multiplicative group.

Related concepts

• A principal bundle with structure group the circle group is a circle bundle. The associated bundle under the standard linear representation of $U(1) \hookrightarrow \mathbb{C}$ is a complex line bundle.

• cyclic group, Q/Z

• circle 2-group

• circle, unit circle, circle type, minimal simplicial circle

• torus