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Definition

The circle group $S^1$ is equivalently (isomorphically)

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

• the unitary group $U(1)$;

• the special orthogonal group $\mathrm{SO}(2)$;

• the subgroup of the group ${ℂ}^{\times }$ of units of the complex numbers given by those of any fixed positive modulus.

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

Related concepts

A principal bundle with structure group the circle group is a circle bundle. The canonically corresponding associated bundle under the standard representation of $S^1\simeq U(1)\hookrightarrow ℂ$ is a complex line bundle.