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In mathematics, the circle group, denoted by ${\displaystyle \mathbb {T} }$ or ${\displaystyle \mathbb {S} ^{1}}$, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers^[1]

$${\displaystyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}.}$$

The circle group forms a subgroup of ${\displaystyle \mathbb {C} ^{\times }}$, the multiplicative group of all nonzero complex numbers. Since ${\displaystyle \mathbb {C} ^{\times }}$ is abelian, it follows that ${\displaystyle \mathbb {T} }$ is as well.

A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure ${\displaystyle \theta }$:

$${\displaystyle \theta \mapsto z=e^{i\theta }=\cos \theta +i\sin \theta .}$$

This is the exponential map for the circle group.

The circle group plays a central role in Pontryagin duality and in the theory of Lie groups.

The notation ${\displaystyle \mathbb {T} }$ for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, ${\displaystyle \mathbb {T} ^{n}}$ (the direct product of ${\displaystyle \mathbb {T} }$ with itself ${\displaystyle n}$ times) is geometrically an ${\displaystyle n}$-torus.

The circle group is isomorphic to the special orthogonal group ${\displaystyle \mathrm {SO} (2)}$.

Elementary introduction

One way to think about the circle group is that it describes how to add angles, where only angles between 0° and 360° or ${\displaystyle \in 0,2\pi )}$ or ${\displaystyle \in (-\pi ,+\pi }$ are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer is 150° + 270° = 420°, but when thinking in terms of the circle group, we may "forget" the fact that we have wrapped once around the circle. Therefore, we adjust our answer by 360°, which gives 420° ≡ 60° (mod 360°).

Another description is in terms of ordinary (real) addition, where only numbers between 0 and 1 are allowed (with 1 corresponding to a full rotation: 360° or ${\displaystyle 2\pi }$), i.e. the real numbers modulo the integers: ${\displaystyle \mathbb {T} \cong \mathbb {R} /\mathbb {Z} }$. This can be achieved by throwing away the digits occurring before the decimal point. For example, when we work out 0.4166... + 0.75, the answer is 1.1666..., but we may throw away the leading 1, so the answer (in the circle group) is just   ${\displaystyle 0.1{\bar {6}}\equiv 1.1{\bar {6}}\equiv -0.8{\bar {3}}\;({\text{mod}}\,\mathbb {Z} )}$   with some preference to 0.166..., because ${\displaystyle 0.1{\bar {6}}\in 0,1)}$.

Topological and analytic structureedit

The circle group is more than just an abstract algebraic object. It has a natural topology when regarded as a subspace of the complex plane. Since multiplication and inversion are continuous functions on ${\displaystyle \mathbb {C} ^{\times }}$, the circle group has the structure of a topological group. Moreover, since the unit circle is a closed subset of the complex plane, the circle group is a closed subgroup of ${\displaystyle \mathbb {C} ^{\times }}$ (itself regarded as a topological group).

One can say even more. The circle is a 1-dimensional real manifold, and multiplication and inversion are real-analytic maps on the circle. This gives the circle group the structure of a one-parameter group, an instance of a Lie group. In fact, up to isomorphism, it is the unique 1-dimensional compact, connected Lie group. Moreover, every ${\displaystyle n}$-dimensional compact, connected, abelian Lie group is isomorphic to ${\displaystyle \mathbb {T} ^{n}}$.

Isomorphismsedit

The circle group shows up in a variety of forms in mathematics. We list some of the more common forms here. Specifically, we show that

$${\displaystyle \mathbb {T} \cong {\mbox{U}}(1)\cong \mathbb {R} /\mathbb {Z} \cong \mathrm {SO} (2).}$$

Note that the slash (/) denotes here quotient group.

The set of all 1×1 unitary matrices clearly coincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value 1. Therefore, the circle group is canonically isomorphic to ${\displaystyle \mathrm {U} (1)}$, the first unitary group.

The exponential function gives rise to a group homomorphism ${\displaystyle \exp :\mathbb {R} \to \mathbb {T} }$ from the additive real numbers ${\displaystyle \mathbb {R} }$ to the circle group ${\displaystyle \mathbb {T} }$ via the map