HOME

TheInfoList



OR:

In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
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 fo ...
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 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 subgroup ...
of \mathbb C^\times, the multiplicative group of all nonzero complex numbers. Since \mathbb C^\times is abelian, it follows that \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 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 ar ...
\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 (mathematics), 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 numb ...
and in the theory of Lie groups. 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 tou ...
. More generally, \mathbb T^n (the direct product 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 In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that ...
when regarded as a subspace of the complex plane. Since multiplication and inversion are
continuous functions In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
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 st ...
. 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, 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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
,
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
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. The set of all 1×1
unitary matrices In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose ...
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 In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
. 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 In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
\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 fo ...
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 learn ...
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 languag ...
multiples of 2\pi. By the
first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
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 Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
(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 fo ...
), 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 a ...
. 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 subgroup ...
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) = x ...
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 subgroup ...
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 in ...
: 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 bina ...
of which is unique up to isomorphism. In the same way that the real numbers 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 ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of the circle group are easy to describe. It follows from
Schur's lemma In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ' ...
that the
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
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 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 Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of the circle group. The
character group In mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters that arise in t ...
of \mathbb T is clearly an
infinite 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 binar ...
generated by \phi_1: \operatorname(\mathbb T, \mathbb T) \cong \mathbb Z. The irreducible
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
representations of the circle group are the
trivial representation In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ...
(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 In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an ''n''th multiple for each positive in ...
. Its
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
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 in ...
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 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 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 can ...
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 In mathematics, the rational points on the unit circle are those points (''x'', ''y'') such that both ''x'' and ''y'' are rational numbers ("fractions") and satisfy ''x''2 + ''y''2 = 1. The set of such points turns out to ...
*
One-parameter subgroup 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 ...
* -sphere * Orthogonal group *
Phase factor For any complex number written in polar form (such as ), the phase factor is the complex exponential factor (). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e. not necessarily on th ...
(application in quantum-mechanics) *
Rotation number In mathematics, the rotation number is an invariant of homeomorphisms of the circle. History It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincaré later proved a theore ...
* Solenoid


Notes


References

*


Further reading

*
Hua Luogeng Hua Luogeng or Hua Loo-Keng (; 12 November 1910 – 12 June 1985) was a Chinese mathematician and politician famous for his important contributions to number theory and for his role as the leader of mathematics research and education in the Peop ...
(1981) ''Starting with the unit circle'', Springer Verlag, {{ISBN, 0-387-90589-8.


External links


Homeomorphism and the Group Structure on a Circle
Lie groups