In mathematics, the circle group, denoted by T (or in blackboard bold by $\mathbb T$), is the multiplicative group of all complex numbers with absolute value 1, i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Blackboard bold is a Typeface style often used for certain symbols in Mathematics and Physics texts in which certain lines of the symbol (usually vertical In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted e. , the unit circle in the complex plane. In Mathematics, a unit circle is

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

The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of Since C× is abelian, it follows that T is as well. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

The notation T for the circle group stems from the fact that Tn (the direct product of T with itself n times) is geometrically an n-torus. In Mathematics, one can often define a direct product of objectsalready known giving a new one In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar The circle group is then a 1-torus.

The circle group plays a central role in Pontryagin duality, and in the theory of Lie groups. In Mathematics, in particular in Harmonic analysis and the theory of Topological groups Pontryagin duality explains the general properties of the Fourier In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

## Elementary introduction

Addition on the circle group

One way to think about the circle group is that it describes how to add angles, where only angles between 0° and 360° are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer should be 150° + 270° = 420°, but when thinking in terms of the circle group, we need to "forget" the fact that we have wrapped once around the circle. Therefore we adjust our answer by 360° which gives 420° − 360° = 60°.

Another description is in terms of ordinary addition, where only numbers between 0 and 1 are allowed. To achieve this, we might need to throw away digits occurring before the decimal point. For example, when we work out 0. 784 + 0. 925 + 0. 446, the answer should be 2. 155, but we throw away the leading 2, so the answer (in the circle group) is just 0. 155.

## Topological and analytic structure

The circle group is more than just an abstract algebraic group. It has a natural topology when regarded as a subspace of the complex plane. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is Since multiplication and inversion are continuous functions on C×, the circle group has the structure of a topological group. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the Moreover, since the unit circle is a closed subset of the complex plane, the circle group is a closed subgroup of C× (itself regarded as a topological group). In Topology and related branches of Mathematics, a closed set is a set whose complement is open.

One can say even more. The circle is a 1-dimensional real manifold and multiplication and inversion are real-analytic maps on the circle. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be This article is about both real and complex analytic functions This gives the circle group the structure of a 1-dimensional Lie group. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In fact, up to isomorphism, it is the unique 1-dimensional compact, connected Lie group. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of Moreover, every n-dimensional compact, connected, abelian Lie group is isomorphic to Tn.

## Isomorphisms

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

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

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. In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^* Therefore, the circle group is canonically isomorphic to U(1), the first unitary group. In Mathematics, the unitary group of degree n, denoted U( n) is the group of n × n unitary matrices

The exponential function gives rise to a group homomorphism exp : RT from the additive real numbers R to the circle group T via the map

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

The last equality is Euler's formula. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic The real number θ corresponds to the angle on the unit circle as measured from the positive x-axis. That this map is a homomorphism follows from the fact the multiplication of unit complex numbers corresponds to addition of angles:

$e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1+\theta_2)}.\,$

This exponential map is clearly a surjective function from R to T. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every It is not, however, injective. The kernel of this map is the set of all integer multiples of 2π. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French 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 T is isomorphic to R/Z. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural

If complex numbers are realized as 2×2 real matrices (see complex number), the unit complex numbers correspond to 2×2 orthogonal matrices with unit determinant. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n Specifically, we have

$e^{i\theta} \leftrightarrow \begin{bmatrix}\cos \theta & -\sin \theta \\\sin \theta & \cos \theta \\\end{bmatrix}.$

The circle group is therefore isomorphic to the special orthogonal group SO(2). In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n This has the geometric interpretation that multiplication by a unit complex number is a proper rotation in the complex plane, and every such rotation is of this form.

## Properties

Any compact Lie group G of dimension > 0 has a subgroup isomorphic to the circle group. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of That 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, and spontaneous symmetry breaking. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or 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 In Physics, spontaneous symmetry breaking occurs when a system that is symmetric with respect to some Symmetry group goes into a Vacuum state

The circle group has many subgroups, but its only proper closed subgroups consist of roots of unity: For each integer n > 0, the nth roots of unity form a cyclic group of order n, which is unique up to isomorphism. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

## Representations

The representations of the circle group are easy to describe. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of It follows from Schur's lemma that the irreducible complex representations of an abelian group are all 1-dimensional. In Mathematics, Schur's lemma is an elementary but extremely useful statement in Representation theory of groups and Algebras In the group case In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Since the circle group is compact, any representation ρ : TGL(1, C) ≅ C×, must take values in U(1)≅ T. Therefore, the irreducible representations of the circle group are just the homomorphisms from the circle group to itself. Every such homomorphism is of the form

$\phi_n(e^{i\theta}) = e^{in\theta},\qquad n\in\mathbb Z.$

These representations are all inequivalent. The representation φ-n is conjugate to φn,

$\phi_{-n} = \overline{\phi_n}.$

These representations are just the characters of the circle group. In Mathematics, if G is a group and &rho is a representation of it over the complex Vector space V then the complex conjugate In Mathematics, a character is (most commonly a special kind of function from a group to a field (such as the Complex numbers) The character group of T is clearly an infinite cyclic group generated by φ1:

$\mathrm{Hom}(\mathbb T,\mathbb T) \cong \mathbb Z.$

The irreducible real representations of the circle group are the trivial representation (which is 1-dimensional) and the representations

$\rho_n(e^{i\theta}) = \begin{bmatrix}\cos n\theta & -\sin n\theta \\\sin n\theta & \cos n\theta \\\end{bmatrix},\quad n\in\mathbb Z^{+}.$

taking values in SO(2). In Mathematics, a character group is the group of representations of a group by complex -valued functions. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, the real numbers may be described informally in several different ways In the mathematical field of Representation theory, a trivial representation is a representation ( V, &phi) of a group Here we only have positive integers n since the representation ρ n is equivalent to ρn.

## Algebraic structure

In this section we will forget about the topological structure of the circle group and look only at its algebraic structure.

The circle group 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 Its torsion subgroup is given by the set of all nth roots of unity for all n, and is isomorphic to Q/Z. In the theory of Abelian groups the torsion subgroup AT of an abelian group A is the Subgroup of A consisting of all elements In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power The structure theorem for divisible groups tells us that T is isomorphic to the direct sum of Q/Z with a number of copies of Q. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction The number of copies of Q must be c (the cardinality of the continuum) in order for the cardinality of the direct sum to be correct. In Mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size ( Cardinality) of the set of But the direct sum of c copies of Q is isomorphic to R, as R is a vector space of dimension c over Q. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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, as C× is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of T.