mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
coefficients and
determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
, such that the matrices and are identified. The modular group acts on the upper-half of the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
by
linear fractional transformation
In mathematics, a linear fractional transformation is, roughly speaking, an inverse function, invertible transformation of the form
: z \mapsto \frac .
The precise definition depends on the nature of , and . In other words, a linear fractional t ...
s. The name "modular group" comes from the relation to
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
s, and not from
modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...
function composition
In mathematics, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, applied after applying to . (g \c ...
.
This group of transformations is isomorphic to the projective special linear group , which is the quotient of the 2-dimensional special linear group by its center . In other words, consists of all matrices
:
where are integers, , and pairs of matrices and are considered to be identical. The group operation is usual
matrix multiplication
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
.
Some authors ''define'' the modular group to be , and still others define the modular group to be the larger group .
Some mathematical relations require the consideration of the group of matrices with determinant plus or minus one. ( is a subgroup of this group.) Similarly, is the quotient group .
Since all matrices with determinant 1 are symplectic matrices, then , the symplectic group of matrices.
Finding elements
To find an explicit matrix
:
in , begin with two coprime integers , and solve the determinant equation .
For example, if then the determinant equation reads
:
then taking and gives . Hence
:
is a matrix. Then, using the projection, these matrices define elements in .
Number-theoretic properties
The unit determinant of
:
implies that the fractions , , , are all irreducible, that is having no common factors (provided the denominators are non-zero, of course). More generally, if is an irreducible fraction, then
:
is also irreducible (again, provided the denominator be non-zero). Any pair of irreducible fractions can be connected in this way; that is, for any pair and of irreducible fractions, there exist elements
:
such that
:
Elements of the modular group provide a symmetry on the two-dimensional lattice. Let and be two
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 whose ratio is not real. Then the set of points
:
is a lattice of parallelograms on the plane. A different pair of vectors and will generate exactly the same lattice if and only if
:
for some matrix in . It is for this reason that doubly periodic functions, such as elliptic functions, possess a modular group symmetry.
The action of the modular group on the rational numbers can most easily be understood by envisioning a square grid, with grid point corresponding to the fraction (see Euclid's orchard). An irreducible fraction is one that is ''visible'' from the origin; the action of the modular group on a fraction never takes a ''visible'' (irreducible) to a ''hidden'' (reducible) one, and vice versa.
Note that any member of the modular group maps the projectively extended real line one-to-one to itself, and furthermore bijectively maps the projectively extended rational line (the rationals with infinity) to itself, the
irrational
Irrationality is cognition, thinking, talking, or acting without rationality.
Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
s to the irrationals, the
transcendental number
In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . ...
s to the transcendental numbers, the non-real numbers to the non-real numbers, the upper half-plane to the upper half-plane, et cetera.
If and are two successive convergents of a continued fraction, then the matrix
:
belongs to . In particular, if for positive integers , , , with and then and will be neighbours in the Farey sequence of order . Important special cases of continued fraction convergents include the
Fibonacci number
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
s and solutions to Pell's equation. In both cases, the numbers can be arranged to form a
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
subset of the modular group.
Group-theoretic properties
Presentation
The modular group can be shown to be generated by the two transformations
:
so that every element in the modular group can be represented (in a non-unique way) by the composition of powers of and . Geometrically, represents inversion in the unit circle followed by reflection with respect to the imaginary axis, while represents a unit translation to the right.
The generators and obey the relations and . It can be shown that these are a complete set of relations, so the modular group has the
presentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
:
:
This presentation describes the modular group as the rotational triangle group (infinity as there is no relation on ), and it thus maps onto all triangle groups by adding the relation , which occurs for instance in the congruence subgroup .
Using the generators and instead of and , this shows that the modular group is isomorphic to the
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, an ...
of the
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
s and :
:
File:Sideway.gif, The action of on
File:Turnovergif.gif, The action of on
Braid group
The
braid group
In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of ...
is the universal central extension of the modular group, with these sitting as lattices inside the (topological) universal covering group . Further, the modular group has a trivial center, and thus the modular group is isomorphic to the
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 ex ...
of modulo its center; equivalently, to the group of
inner automorphism
In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within thos ...
The quotients by congruence subgroups are of significant interest.
Other important quotients are the triangle groups, which correspond geometrically to descending to a cylinder, quotienting the coordinate
modulo
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation.
Given two positive numbers and , mo ...
, as . is the group of
icosahedral symmetry
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
The group can be generated by the two matrices
:
since
:
The projection turns these matrices into generators of , with relations similar to the group presentation.
Relationship to hyperbolic geometry
The modular group is important because it forms a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
upper half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
model of hyperbolic plane geometry, then the group of all
orientation-preserving isometries of consists of all
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying .
Geometrically ...
s of the form
:
where , , , are
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. In terms of
projective coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
, the group acts on the upper half-plane by projectivity:
:
This action is faithful. Since is a subgroup of , the modular group is a subgroup of the group of orientation-preserving isometries of .
Tessellation of the hyperbolic plane
The modular group acts on as a discrete subgroup of , that is, for each in we can find a neighbourhood of which does not contain any other element of the
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
of . This also means that we can construct
fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each ...
s, which (roughly) contain exactly one representative from the orbit of every in . (Care is needed on the boundary of the domain.)
There are many ways of constructing a fundamental domain, but a common choice is the region
:
bounded by the vertical lines and , and the circle . This region is a hyperbolic triangle. It has vertices at and , where the angle between its sides is , and a third vertex at infinity, where the angle between its sides is 0.
There is a strong connection between the modular group and
elliptic curves
In mathematics, an elliptic curve is a Smoothness, smooth, Projective variety, projective, algebraic curve of Genus of an algebraic curve, genus one, on which there is a specified point . An elliptic curve is defined over a field (mathematics), ...
. Each point in the upper half-plane gives an elliptic curve, namely the quotient of by the lattice generated by 1 and .
Two points in the upper half-plane give isomorphic elliptic curves if and only if they are related by a transformation in the modular group. Thus, the quotient of the upper half-plane by the action of the modular group is the so-called
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
of elliptic curves: a space whose points describe isomorphism classes of elliptic curves. This is often visualized as the fundamental domain described above, with some points on its boundary identified.
The modular group and its subgroups are also a source of interesting tilings of the hyperbolic plane. By transforming this fundamental domain in turn by each of the elements of the modular group, a regular tessellation of the hyperbolic plane by congruent hyperbolic triangles known as the V6.6.∞ Infinite-order triangular tiling is created. Note that each such triangle has one vertex either at infinity or on the real axis .
This tiling can be extended to the Poincaré disk, where every hyperbolic triangle has one vertex on the boundary of the disk. The tiling of the Poincaré disk is given in a natural way by the -invariant, which is invariant under the modular group, and attains every complex number once in each triangle of these regions.
This tessellation can be refined slightly, dividing each region into two halves (conventionally colored black and white), by adding an orientation-reversing map; the colors then correspond to orientation of the domain. Adding in and taking the right half of the region (where ) yields the usual tessellation. This tessellation first appears in print in , where it is credited to Richard Dedekind, in reference to .
The map of groups (from modular group to triangle group) can be visualized in terms of this tiling (yielding a tiling on the modular curve), as depicted in the video at right.
Congruence subgroups
Important
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
s of the modular group , called '' congruence subgroups'', are given by imposing
congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group (mathematics), group, ring (mathematics), ring, or vector space) that is compatible with the structure in the ...
s on the associated matrices.
There is a natural
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
given by reducing the entries
modulo
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation.
Given two positive numbers and , mo ...
. This induces a homomorphism on the modular group . The kernel of this homomorphism is called the principal congruence subgroup of level , denoted . We have the following
short exact sequence
In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
:
:
Being the kernel of a homomorphism is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
of the modular group . The group is given as the set of all modular transformations
:
for which and .
It is easy to show that the trace of a matrix representing an element of cannot be −1, 0, or 1, so these subgroups are torsion-free groups. (There are other torsion-free subgroups.)
The principal congruence subgroup of level 2, , is also called the modular group . Since is isomorphic to , is a subgroup of
index
Index (: indexes or indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on the Halo Array in the ...
6. The group consists of all modular transformations for which and are odd and and are even.
Another important family of congruence subgroups are the modular group defined as the set of all modular transformations for which , or equivalently, as the subgroup whose matrices become upper triangular upon reduction modulo . Note that is a subgroup of . The modular curves associated with these groups are an aspect of monstrous moonshine – for a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, the modular curve of the normalizer is
genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order
:
: = 2463205976112133171923293 ...
One important subset of the modular group is the dyadic monoid, which is the
monoid
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being .
Monoids are semigroups with identity ...
of all strings of the form for positive integers . This monoid occurs naturally in the study of
fractal curve
A fractal curve is, loosely, a mathematical curve (mathematics), curve whose shape retains the same general pattern of Pathological (mathematics), irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fract ...
de Rham curve
In mathematics, a de Rham curve is a continuous fractal curve obtained as the image of the Cantor space, or, equivalently, from the base-two expansion of the real numbers in the unit interval. Many well-known fractal curves, including the Cantor ...
. The monoid also has higher-dimensional linear representations; for example, the representation can be understood to describe the self-symmetry of the blancmange curve.
Maps of the torus
The group is the linear maps preserving the standard lattice , and is the orientation-preserving maps preserving this lattice; they thus descend to self-homeomorphisms of the
torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
(SL mapping to orientation-preserving maps), and in fact map isomorphically to the (extended)
mapping class group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mo ...
of the torus, meaning that every self-homeomorphism of the torus is isotopic to a map of this form. The algebraic properties of a matrix as an element of correspond to the dynamics of the induced map of the torus.
Hecke groups
The modular group can be generalized to the Hecke groups, named for Erich Hecke, and defined as follows.
The Hecke group with , is the discrete group generated by
:
where . For small values of , one has:
:
The modular group is isomorphic to and they share properties and applications – for example, just as one has the
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, an ...
of
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
s
:
more generally one has
:
which corresponds to the triangle group . There is similarly a notion of principal congruence subgroups associated to principal ideals in .
History
The modular group and its subgroups were first studied in detail by Richard Dedekind and by
Felix Klein
Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
as part of his Erlangen programme in the 1870s. However, the closely related
elliptic function
In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
*
Mapping class group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mo ...
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying .
Geometrically ...
Modular form
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...