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In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s with absolute value 1, that is, the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
or simply the unit complex numbers. \mathbb T = \. The circle group forms a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of \mathbb C^\times, the multiplicative group of all nonzero complex numbers. Since \mathbb C^\times is
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
, it follows that \mathbb T is as well. A unit complex number in the circle group represents a
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
of the complex plane about the origin and can be parametrized by the
angle measure In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
\theta: \theta \mapsto z = e^ = \cos\theta + i\sin\theta. This is the exponential map for the circle group. The circle group plays a central role in
Pontryagin duality In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...
and in the theory of
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
s. The notation \mathbb T for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not ...
. More generally, \mathbb T^n (the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of \mathbb T with itself n times) is geometrically an n-torus. The circle group is isomorphic to the special orthogonal group \mathrm(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 \in , 2\pi) or \in(-\pi,+\pi/math> are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer is , 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 ). 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 2\pi), i.e. the real numbers modulo the integers: This can be achieved by throwing away the digits occurring before the decimal point. For example, when we work out the answer is 1.1666..., but we may throw away the leading 1, so the answer (in the circle group) is just   0.1\bar \equiv 1.1\bar \equiv -0.8\bar\;(\text\,\Z)   with some preference to 0.166..., because


Topological and analytic structure

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 \mathbb C^\times, the circle group has the structure of a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two ...
. Moreover, since the unit circle is a closed subset of the complex plane, the circle group is a closed subgroup of \mathbb C^\times (itself regarded as a topological group). One can say even more. The circle is a 1-dimensional real
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
, and multiplication and inversion are real-analytic maps on the circle. This gives the circle group the structure of a
one-parameter group In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism :\varphi : \mathbb \rightarrow G from the real line \mathbb (as an additive group) to some other topological group G. If \varphi is in ...
, an instance of a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
. In fact, up to isomorphism, it is the unique 1-dimensional compact, connected Lie group. Moreover, every n-dimensional compact, connected, abelian Lie group is isomorphic to \mathbb T^n.


Isomorphisms

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 \mathbb T \cong \mbox(1) \cong \mathbb R/\mathbb Z \cong \mathrm(2). Note that the slash (/) denotes here
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
. 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 \mathrm(1), the first unitary group. The
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
gives rise to a group homomorphism \exp : \mathbb R \to \mathbb T from the additive real numbers \mathbb R to the circle group \mathbb T via the map \theta \mapsto e^ = \cos\theta + i \sin \theta. The last equality is
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for ...
or the complex exponential. The real number θ corresponds to the angle (in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s) on the unit circle as measured counterclockwise from the positive ''x'' axis. That this map is a homomorphism follows from the fact that the multiplication of unit complex numbers corresponds to addition of angles: e^ e^ = e^. This exponential map is clearly a surjective function from \mathbb R to \mathbb T. However, it is not injective. The
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine lea ...
of this map is the set of all
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
multiples of 2\pi. By the first isomorphism theorem we then have that \mathbb T \cong \mathbb R/2\pi\mathbb Z. After rescaling we can also say that \mathbb T is isomorphic to \mathbb R / \mathbb Z. If complex numbers are realized as 2×2 real matrices (see
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
), the unit complex numbers correspond to 2×2
orthogonal matrices In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...
with unit
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
. Specifically, we have e^ \leftrightarrow \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end = f\left(e^\right). This function shows that the circle group is isomorphic to the special orthogonal group \mathrm(2) since f\left(e^ e^\right) = \begin \cos(\theta_1 + \theta_2) & -\sin(\theta_1 + \theta_2) \\ \sin(\theta_1 + \theta_2) & \cos(\theta_1 + \theta_2) \end = f\left(e^\right) \times f\left(e^\right), where \times is matrix multiplication. 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.


Properties

Every compact Lie group \mathrm of dimension > 0 has a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
isomorphic to the circle group. This means that, thinking in terms of symmetry, a compact symmetry group acting ''continuously'' can be expected to have one-parameter circle subgroups acting; the consequences in physical systems are seen, for example, at
rotational invariance In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. Mathematics Functions For example, the function :f(x,y) = ...
and
spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
. The circle group has many
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
s, but its only proper closed subgroups consist of
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
: For each integer the n-th roots of unity form a
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
of which is unique up to isomorphism. In the same way that the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
are a completion of the ''b''-adic rationals \Z tfrac1b/math> for every
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
b > 1, the circle group is the completion of the
Prüfer group In mathematics, specifically in group theory, the Prüfer ''p''-group or the ''p''-quasicyclic group or ''p''∞-group, Z(''p''∞), for a prime number ''p'' is the unique ''p''-group in which every element has ''p'' different ''p''-th roots. ...
\Z tfrac1b\Z for b, given by the direct limit \varinjlim \mathbb/ b^n \mathbb.


Representations

The representations of the circle group are easy to describe. It follows from Schur's lemma that the irreducible complex representations of an abelian group are all 1-dimensional. Since the circle group is compact, any representation \rho: \mathbb T \to \mathrm(1, \mathbb C) \cong \mathbb C^\times must take values in \mbox(1) \cong \mathbb T. Therefore, the irreducible representations of the circle group are just the
homomorphisms In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from the circle group to itself. These representations are all inequivalent. The representation \phi_ is conjugate to \phi_: \phi_ = \overline. These representations are just the characters of the circle group. The character group of \mathbb T is clearly an infinite cyclic group generated by \phi_1: \operatorname(\mathbb T, \mathbb T) \cong \mathbb Z. The irreducible real representations of the circle group are the trivial representation (which is 1-dimensional) and the representations \rho_n(e^) = \begin \cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta \end, \quad n \in \mathbb Z^+, taking values in \mathrm(2). Here we only have positive integers n, since the representation \rho_ is equivalent to \rho_n.


Group structure

The circle group \mathbb T is a divisible group. Its torsion subgroup is given by the set of all n-th
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
for all n and is isomorphic to \mathbb Q / \mathbb Z. The structure theorem for divisible groups and the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
together tell us that \mathbb T is isomorphic to the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
of \mathbb Q / \mathbb Z with a number of copies of \mathbb Q. The number of copies of \mathbb Q must be \mathfrak c (the
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \ma ...
) in order for the cardinality of the direct sum to be correct. But the direct sum of \mathfrak c copies of \mathbb Q is isomorphic to \mathbb R, as \mathbb R is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
of dimension \mathfrak c over \mathbb Q. Thus \mathbb T \cong \mathbb R \oplus (\mathbb Q / \mathbb Z). The isomorphism \mathbb C^\times \cong \mathbb R \oplus (\mathbb Q / \mathbb Z) can be proved in the same way, since \mathbb C^\times is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of \mathbb T.


See also

* Group of rational points on the unit circle * One-parameter subgroup * -sphere *
Orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
* Phase factor (application in quantum-mechanics) * Rotation number * Solenoid


Notes


References

*


Further reading

* Hua Luogeng (1981) ''Starting with the unit circle'',
Springer Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, {{ISBN, 0-387-90589-8.


External links


Homeomorphism and the Group Structure on a Circle
Lie groups