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Inmathematics, thecircle group, denoted by${\displaystyle \mathbb{T}}$ or⁠${\displaystyle {\mathbb{S}}^1}$⁠, is themultiplicative group of allcomplex numbers withabsolute value 1, that is, theunit circle in thecomplex plane or simply theunit complex numbers^[1]$${\displaystyle \mathbb{T}=\{z\in \mathbb{C}:|z|=1\}.}$$

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

A unit complex number in the circle group represents arotation of the complex plane about the origin and can be parametrized by theangle measure⁠${\displaystyle \theta }$⁠:$${\displaystyle \theta \mapsto z=e^{i\theta }=\cos \theta +i\sin \theta .}$$

This is theexponential map for the circle group.

The circle group plays a central role inPontryagin duality and in the theory ofLie 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,${\displaystyle {\mathbb{T}}^n}$ (thedirect product of${\displaystyle \mathbb{T}}$ with itself${\displaystyle n}$ times) is geometrically an${\displaystyle n}$-torus.

The circle group isisomorphic to thespecial orthogonal group⁠${\displaystyle \mathrm{S}\mathrm{O}(2)}$⁠.

One way to think about the circle group is that it describes how to addangles, 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 is150° + 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 gives420° ≡ 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 out0.4166... + 0.75, the answer is 1.1666..., but we may throw away the leading 1, so the answer (in the circle group) is just⁠${\displaystyle 0.1\overline{6}\equiv 1.1\overline{6}\equiv -0.8\overline{3}(\text{mod}\mathbb{Z})}$⁠, with some preference to 0.166..., because⁠${\displaystyle 0.1\overline{6}\in [0,1)}$⁠.

The circle group is more than just an abstract algebraic object. It has anatural topology when regarded as asubspace of the complex plane. Since multiplication and inversion arecontinuous functions on⁠${\displaystyle {\mathbb{C}}^{\times }}$⁠, the circle group has the structure of atopological group. Moreover, since the unit circle is aclosed 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 realmanifold, and multiplication and inversion arereal-analytic maps on the circle. This gives the circle group the structure of aone-parameter group, an instance of aLie group. In fact,up to isomorphism, it is the unique 1-dimensionalcompact,connected Lie group. Moreover, every${\displaystyle n}$-dimensional compact, connected, abelian Lie group is isomorphic to⁠${\displaystyle {\mathbb{T}}^n}$⁠.

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$${\displaystyle \mathbb{T}\cong \text{U}(1)\cong \mathbb{R}/\mathbb{Z}\cong \mathrm{S}\mathrm{O}(2),}$$where the slash (⁠${\displaystyle /}$⁠) denotesgroup quotient and${\displaystyle \cong }$ the existence of anisomorphism between the groups.

The set of all⁠${\displaystyle 1\times 1}$⁠unitary matrices 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 the firstunitary group⁠${\displaystyle \mathrm{U}(1)}$⁠, i.e.,$${\displaystyle \mathbb{T}\cong \text{U}(1).}$$Theexponential function gives rise to a map${\displaystyle \exp :\mathbb{R}\to \mathbb{T}}$ from the additive real numbers⁠${\displaystyle \mathbb{R}}$⁠ to the circle group⁠${\displaystyle \mathbb{T}}$⁠ known asEuler's formula$${\displaystyle \theta \mapsto e^{i\theta }=\cos \theta +i\sin \theta ,}$$where${\displaystyle \theta \in \mathbb{R}}$ corresponds to the angle (inradians) on the unit circle as measured counterclockwise from the positivex-axis. The property$${\displaystyle e^{i{\theta }_1}{e}^{i{\theta }_2}=e^{i({\theta }_1+{\theta }_2)},\mathrm{\forall }{\theta }_1,{\theta }_2\in \mathbb{R},}$$makes${\displaystyle \exp :\mathbb{R}\to \mathbb{T}}$ agroup homomorphism. While the map issurjective, it is notinjective and therefore not an isomorphism. Thekernel of this map is the set of allinteger multiples of⁠${\displaystyle 2\pi }$⁠. By thefirst isomorphism theorem we then have that$${\displaystyle \mathbb{T}\cong \mathbb{R}/2\pi \mathbb{Z}.}$$After rescaling we can also say that${\displaystyle \mathbb{T}}$ is isomorphic to⁠${\displaystyle \mathbb{R}/\mathbb{Z}}$⁠.

The unitcomplex numbers can be realized as 2×2 realorthogonal matrices, i.e.,associating thesquared modulus andcomplex conjugate with thedeterminant andtranspose, respectively, of the corresponding matrix. As theangle sum trigonometric identities imply thatwhere${\displaystyle \times }$ is matrix multiplication, the circle group isisomorphic to thespecial orthogonal group${\displaystyle \mathrm{S}\mathrm{O}(2)}$, i.e.,$${\displaystyle \mathbb{T}\cong \mathrm{S}\mathrm{O}(2).}$$This isomorphism has the geometric interpretation that multiplication by a unit complex number is a proper rotation in the complex (and real) plane, and every such rotation is of this form.

Every compact Lie group${\displaystyle \mathrm{G}}$ of dimension > 0 has asubgroup isomorphic to the circle group. This means that, thinking in terms ofsymmetry, a compact symmetry group actingcontinuously can be expected to have one-parameter circle subgroups acting; the consequences in physical systems are seen, for example, atrotational invariance andspontaneous symmetry breaking.

The circle group has manysubgroups, but its only properclosed subgroups consist ofroots of unity: For each integer⁠${\displaystyle n>0}$⁠, the${\displaystyle n}$th roots of unity form acyclic group of order ⁠${\displaystyle n}$⁠, which is unique up to isomorphism.

In the same way that thereal numbers are acompletion of theb-adic rationals${\displaystyle \mathbb{Z}[\frac{1}{b}]}$ for everynatural number⁠${\displaystyle b>1}$⁠, the circle group is the completion of thePrüfer group${\displaystyle \mathbb{Z}[\frac{1}{b}]/\mathbb{Z}}$ for⁠${\displaystyle b}$⁠, given by thedirect limit⁠${\displaystyle \underrightarrow{\lim }\mathbb{Z}/b^n\mathbb{Z}}$⁠.

Therepresentations of the circle group are easy to describe. It follows fromSchur's lemma that theirreduciblecomplex representations of an abelian group are all 1-dimensional. Since the circle group is compact, any representation$${\displaystyle \rho :\mathbb{T}\to \mathrm{G}\mathrm{L}(1,\mathbb{C})\cong {\mathbb{C}}^{\times }}$$must take values in⁠${\displaystyle \text{U}(1)\cong \mathbb{T}}$⁠. Therefore, the irreducible representations of the circle group are just thehomomorphisms from the circle group to itself.

For each integer${\displaystyle n}$ we can define a representation${\displaystyle {\varphi }_n}$ of the circle group by⁠${\displaystyle {\varphi }_n(z)=z^n}$⁠. These representations are all inequivalent. The representation${\displaystyle {\varphi }_{-n}}$ isconjugate to⁠${\displaystyle {\varphi }_n}$⁠:$${\displaystyle {\varphi }_{-n}=\overline{{\varphi }_n}.}$$

These representations are just thecharacters of the circle group. Thecharacter group of${\displaystyle \mathbb{T}}$ is clearly aninfinite cyclic group generated by⁠${\displaystyle {\varphi }_1}$⁠:$${\displaystyle \mathrm{Hom}(\mathbb{T},\mathbb{T})\cong \mathbb{Z}.}$$

The irreduciblereal representations of the circle group are thetrivial representation (which is 1-dimensional) and the representationstaking values in⁠${\displaystyle \mathrm{S}\mathrm{O}(2)}$⁠. Here we only have positive integers⁠${\displaystyle n}$⁠, since the representation${\displaystyle {\rho }_{-n}}$ is equivalent to⁠${\displaystyle {\rho }_n}$⁠.

The circle group${\displaystyle \mathbb{T}}$ is adivisible group. Itstorsion subgroup is given by the set of all${\displaystyle n}$-throots of unity for all${\displaystyle n}$ and is isomorphic to⁠${\displaystyle \mathbb{Q}/\mathbb{Z}}$⁠. Thestructure theorem for divisible groups and theaxiom of choice together tell us that${\displaystyle \mathbb{T}}$ is isomorphic to thedirect sum of${\displaystyle \mathbb{Q}/\mathbb{Z}}$ with a number of copies of⁠${\displaystyle \mathbb{Q}}$⁠.^[2]

The number of copies of⁠${\displaystyle \mathbb{Q}}$⁠ must be${\displaystyle \mathfrak{c}}$ (thecardinality of the continuum) in order for the cardinality of the direct sum to be correct. But the direct sum of${\displaystyle \mathfrak{c}}$ copies of⁠${\displaystyle \mathbb{Q}}$⁠ is isomorphic to⁠${\displaystyle \mathbb{R}}$⁠, as${\displaystyle \mathbb{R}}$ is avector space of dimension${\displaystyle \mathfrak{c}}$ over⁠${\displaystyle \mathbb{Q}}$⁠. Thus,$${\displaystyle \mathbb{T}\cong \mathbb{R}\oplus (\mathbb{Q}/\mathbb{Z}).}$$

The isomorphism$${\displaystyle {\mathbb{C}}^{\times }\cong \mathbb{R}\oplus (\mathbb{Q}/\mathbb{Z})}$$can be proved in the same way, since⁠${\displaystyle {\mathbb{C}}^{\times }}$⁠ is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of⁠${\displaystyle \mathbb{T}}$⁠.

• Group of rational points on the unit circle
• One-parameter subgroup
• n-sphere
• Orthogonal group
• Phase factor (application in quantum-mechanics)
• Rotation number
• Solenoid

• James, Robert C.; James, Glenn (1992).Mathematics Dictionary (Fifth ed.). Chapman & Hall.ISBN 9780412990410.

• Hua Luogeng (1981)Starting with the unit circle,Springer Verlag,ISBN 0-387-90589-8.

• Homeomorphism and the Group Structure on a Circle

• Lie groups

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