View source for Modular group

{{Short description|Orientation-preserving mapping class group of the torus}}
{{for|a group whose lattice of subgroups is modular|Iwasawa group}}
{{Group theory sidebar |Discrete}}

In [[mathematics]], the '''modular group'''  is the [[projective linear group|projective special linear group]] <math display=inline> \operatorname{PSL}(2,\mathbb Z)</math> of {{nowrap|2{{Hair space}}×{{Hair space}}2}} [[Matrix (mathematics)|matrices]] with [[integer]] coefficients and [[determinant]] 1. The matrices {{math|''A''}} and {{math|−''A''}} are identified. The modular group acts on the upper-half of the [[complex plane]] by [[fractional linear transformation]]s, and the name "modular group" comes from the relation to [[moduli space]]s and not from [[modular arithmetic]].

==Definition==
The '''modular group {{math|Γ}}''' is the [[Group (mathematics)|group]] of [[Möbius transformation|linear fractional transformation]]s of the [[upper half-plane|upper half of the complex plane]], which have the form

:<math>z\mapsto\frac{az+b}{cz+d}</math>

where {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} are integers, and {{math|''ad'' − ''bc'' {{=}} 1}}. The group operation is [[function composition]].

This group of transformations is isomorphic to the [[projective linear group|projective special linear group]] {{math|PSL(2, '''Z''')}}, which is the quotient of the 2-dimensional [[special linear group]] {{math|SL(2, '''Z''')}} over the integers by its [[center (group theory)|center]] {{math|{''I'', −''I''<nowiki>}</nowiki>}}.  In other words, {{math|PSL(2, '''Z''')}} consists of all matrices

:<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math>

where {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} are integers, {{math|''ad'' − ''bc'' {{=}} 1}}, and pairs of matrices {{math|''A''}} and {{math|−''A''}} are considered to be identical.  The group operation is the usual [[matrix multiplication|multiplication of matrices]].

Some authors ''define'' the modular group to be {{math|PSL(2, '''Z''')}}, and still others define the modular group to be the larger group {{math|SL(2, '''Z''')}}. 

Some mathematical relations require the consideration of the group {{math|GL(2, '''Z''')}} of matrices with determinant plus or minus one. ({{math|SL(2, '''Z''')}} is a subgroup of this group.)  Similarly, {{math|PGL(2, '''Z''')}} is the quotient group {{math|GL(2, '''Z''')/{''I'', −''I''<nowiki>}</nowiki>}}. A {{nowrap|2 × 2}} matrix with unit determinant is a [[symplectic matrix]], and thus {{math|SL(2, '''Z''') {{=}} Sp(2, '''Z''')}}, the [[symplectic group]] of {{nowrap|2 × 2}} matrices.

=== Finding elements ===
To find an explicit matrix <blockquote><math>\begin{pmatrix} a & x \\ b & y \end{pmatrix}</math></blockquote> in {{math|SL(2, '''Z''')}}, begin with two coprime integers <math>a,b</math>, and solve the determinant equation<blockquote><math>ay-bx = 1.</math></blockquote>(Notice the determinant equation forces <math>a,b</math> to be coprime since otherwise there would be a factor <math>c>1</math> such that <math>ca' = a</math>, <math>cb' = b</math>, hence<blockquote><math>c(a'y-b'x) = 1</math></blockquote>would have no integer solutions.) For example, if <math>a = 7, \text{ } b =6 </math> then the determinant equation reads<blockquote><math>7y-6x = 1</math></blockquote>then taking <math>y = -5</math> and <math>x = -6</math> gives <math>-35 - (-36) = 1</math>, hence<blockquote><math>\begin{pmatrix}
7 & -6 \\
6 & -5
\end{pmatrix}</math></blockquote>is a matrix. Then, using the projection, these matrices define elements in {{math|PSL(2, '''Z''')}}.

==Number-theoretic properties==
The unit determinant of

:<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math>

implies that the fractions <sub><big>{{math|{{sfrac|''a''|''b''}}}}</big></sub>, <sub><big>{{math|{{sfrac|''a''|''c''}}}}</big></sub>, <sub><big>{{math|{{sfrac|''c''|''d''}}}}</big></sub>, <sub><big>{{math|{{sfrac|''b''|''d''}}}}</big></sub> are all irreducible, that is having no common factors (provided the denominators are non-zero, of course).  More generally, if {{math|{{sfrac|''p''|''q''}}}} is an irreducible fraction, then

:<math>\frac{ap+bq}{cp+dq}</math>

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 {{math|{{sfrac|''p''|''q''}}}} and {{math|{{sfrac|''r''|''s''}}}} of irreducible fractions, there exist elements

:<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\operatorname{SL}(2, \mathbb Z)</math>

such that

:<math>r = ap+bq \quad \mbox{ and } \quad s=cp+dq.</math>

Elements of the modular group provide a symmetry on the two-dimensional [[period lattice|lattice]]. Let {{math|''ω''<sub>1</sub>}} and {{math|''ω''<sub>2</sub>}} be two [[complex number]]s whose ratio is not real. Then the set of points

:<math>\Lambda (\omega_1, \omega_2)=\{ m\omega_1 +n\omega_2 : m,n\in \mathbb Z \}</math>

is a lattice of parallelograms on the plane.  A different pair of vectors {{math|''α''<sub>1</sub>}} and {{math|''α''<sub>2</sub>}} will generate exactly the same lattice if and only if

:<math>\begin{pmatrix}\alpha_1 \\ \alpha_2 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix}</math>

for some matrix in {{math|GL(2, '''Z''')}}. It is for this reason that [[doubly periodic function]]s, 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 {{math|(''p'', ''q'')}} corresponding to the fraction {{math|{{sfrac|''p''|''q''}}}} (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 number|irrational]]s to the irrationals, the [[transcendental number]]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 {{math|{{sfrac|''p''<sub>''n''−1</sub>|''q''<sub>''n''−1</sub>}}}} and {{math|{{sfrac|''p''<sub>''n''</sub>|''q''<sub>''n''</sub>}}}} are two successive  convergents of a [[continued fraction]], then the matrix

:<math>\begin{pmatrix} p_{n-1} & p_{n} \\ q_{n-1} & q_{n} \end{pmatrix}</math>

belongs to {{math|GL(2, '''Z''')}}.  In particular, if {{math|''bc'' − ''ad'' {{=}} 1}} for positive integers {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} with {{math|''a'' < ''b''}} and {{math|''c'' < ''d''}} then {{math|{{sfrac|''a''|''b''}}}} and {{math|{{sfrac|''c''|''d''}}}} will be neighbours in the [[Farey sequence]] of order {{math|max(''b'', ''d'')}}. Important special cases of continued fraction convergents include the [[Fibonacci number]]s and solutions to [[Pell's equation]].  In both cases, the numbers can be arranged to form a [[semigroup]] subset of the modular group.

==Group-theoretic properties==

=== Presentation ===
The modular group can be shown to be [[generating set of a group|generated]] by the two transformations

:<math>\begin{align}
S &: z\mapsto -\frac1z \\
T &: z\mapsto z+1
\end{align}</math>

so that every element in the modular group can be represented (in a non-unique way) by the composition of powers of {{math|''S''}} and {{math|''T''}}. Geometrically, {{math|''S''}} represents inversion in the unit circle followed by reflection with respect to the imaginary axis, while {{math|''T''}} represents a unit translation to the right.

The generators {{math|''S''}} and {{math|''T''}} obey the relations {{math|''S''{{isup|2}} {{=}} 1}} and {{math|(''ST'')<sup>3</sup> {{=}} 1}}. It can be shown <ref>{{cite journal | first = Roger C. | last=  Alperin |author-link=Roger C. Alperin | title = {{math|PSL<sub>2</sub>(''Z'') {{=}} ''Z''<sub>2</sub> ∗ ''Z''<sub>3</sub>}} | journal = Amer. Math. Monthly | volume = 100 | issue= 4 | pages = 385–386 | date = April 1993 | doi = 10.2307/2324963 | jstor= 2324963 }}</ref> that these are a complete set of relations, so the modular group has the [[presentation of a group|presentation]]:

:<math>\Gamma \cong \left\langle S, T \mid S^2=I, \left(ST\right)^3=I \right\rangle</math>

This presentation describes the modular group as the rotational [[triangle group]] {{math|D(2, 3, ∞)}} (infinity as there is no relation on {{math|''T''}}), and it thus maps onto all triangle groups {{math|(2, 3, ''n'')}} by adding the relation {{math|''T{{isup|n}}'' {{=}} 1}}, which occurs for instance in the [[congruence subgroup]] {{math|Γ(''n'')}}.

Using the generators {{math|''S''}} and {{math|''ST''}} instead of {{math|''S''}} and {{math|''T''}}, this shows that the modular group is isomorphic to the [[free product]] of the [[cyclic group]]s {{math|''C''<sub>2</sub>}} and {{math|''C''<sub>3</sub>}}:

:<math>\Gamma \cong C_2 * C_3</math>

<gallery widths=300px heights=300px >
File:Sideway.gif|The action of {{math|''T'' : ''z'' ↦ ''z'' + 1}} on {{math|'''H'''}}
File:Turnovergif.gif|The action of {{math|''S'' : ''z'' ↦ −{{sfrac|1|''z''}}}} on {{math|'''H'''}}
</gallery>

===Braid group===

[[File:Braid-modular-group-cover.svg|thumb|376px|The [[braid group]] {{math|''B''<sub>3</sub>}} is the [[universal central extension]] of the modular group.]]
The [[braid group]] {{math|''B''<sub>3</sub>}} is the universal central extension of the modular group, with these sitting as lattices inside the (topological) universal covering group {{math|{{overline|SL<sub>2</sub>('''R''')}} → PSL<sub>2</sub>('''R''')}}. Further, the modular group has a trivial center, and thus the modular group is isomorphic to the [[quotient group]] of {{math|''B''<sub>3</sub>}} modulo its [[center (group theory)|center]]; equivalently, to the group of [[inner automorphism]]s of {{math|''B''<sub>3</sub>}}.

The braid group {{math|''B''<sub>3</sub>}} in turn is isomorphic to the [[knot group]] of the [[trefoil knot]].

===Quotients===
The quotients by congruence subgroups are of significant interest.

Other important quotients are the {{math|(2, 3, ''n'')}} triangle groups, which correspond geometrically to descending to a cylinder, quotienting the {{math|''x''}} coordinate [[modular arithmetic|modulo]] {{math|''n''}}, as {{math|''T{{isup|n}}'' {{=}} (''z'' ↦ ''z''&nbsp;+&nbsp;''n'')}}. {{math|(2, 3, 5)}} is the group of [[icosahedral symmetry]], and the [[(2,3,7) triangle group|{{math|(2, 3, 7)}} triangle group]] (and associated tiling) is the cover for all [[Hurwitz surface]]s.

=== Presenting as a matrix group ===
The group <math>\text{SL}_2(\mathbb{Z})</math> can be generated by the two matrices<ref>{{Cite web|title=SL(2,Z)|url=https://kconrad.math.uconn.edu/blurbs/grouptheory/SL(2,Z).pdf|last=Conrad|first=Keith}}</ref>

: <math>S = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}, \text{ }
T = \begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix}</math>

since

: <math>S^2 = -I_2, \text{ }
(ST)^3 = \begin{pmatrix}
0 & -1 \\
1 & 1
\end{pmatrix}^3 = -I_2</math>

The projection <math>\text{SL}_2(\mathbb{Z}) \to \text{PSL}_2(\mathbb{Z})</math> turns these matrices into generators of <math>\text{PSL}_2(\mathbb{Z})</math>, with relations similar to the group presentation.

==Relationship to hyperbolic geometry==
{{See also|PSL2(R)|label1={{math|PSL<sub>2</sub>('''R''')}}}}

The modular group is important because it forms a [[group (mathematics)|subgroup]] of the group of [[isometry|isometries]] of the [[hyperbolic geometry|hyperbolic plane]].  If we consider the [[upper half-plane]] model {{math|'''H'''}} of hyperbolic plane geometry, then the group of all
[[Orientation (mathematics)|orientation-preserving]] isometries of {{math|'''H'''}} consists of all [[Möbius transformation]]s of the form

:<math>z\mapsto \frac{az + b}{cz + d}</math>

where {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} are [[real number]]s.  In terms of [[projective coordinates]], the group {{math|PSL(2, '''R''')}} [[Group action (mathematics)|acts]] on the upper half-plane {{math|'''H'''}} by projectivity:

:<math>[z,\ 1]\begin{pmatrix} a & c \\ b & d \end{pmatrix} \,= \, [az + b,\ cz +d] \,\thicksim\, \left[\frac{az + b}{cz + d}, \ 1\right].</math>

This action is [[faithful group action|faithful]]. Since {{math|PSL(2, '''Z''')}} is a subgroup of {{math|PSL(2, '''R''')}}, the modular group is a subgroup of the group of orientation-preserving isometries of {{math|'''H'''}}.<ref>{{cite journal|first1=Paul R.|last1=McCreary|first2=Teri Jo|last2=Murphy|first3=Christian|last3=Carter|url=https://www.mathematica-journal.com/issue/v9i3/contents/ModularGroup/ModularGroup.pdf|title=The Modular Group|journal=The Mathematica Journal|volume=9|issue=3}}</ref>

===Tessellation of the hyperbolic plane===
[[File:ModularGroup-FundamentalDomain.svg|thumb|400px|right|A typical fundamental domain for the action of {{math|Γ}} on the upper half-plane.]]
The modular group {{math|Γ}} acts on <math display=inline>\mathbb H</math> as a [[discrete group|discrete subgroup]] of <math display=inline> \operatorname{PSL}(2,\mathbb R)</math>, that is, for each {{math|''z''}} in <math display=inline>\mathbb H</math> we can find a neighbourhood of {{math|''z''}} which does not contain any other element of the [[orbit (group theory)|orbit]] of {{math|''z''}}. This also means that we can construct [[fundamental domain]]s, which (roughly) contain exactly one representative from the orbit of every {{math|''z''}} in {{math|'''H'''}}. (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

:<math>R = \left\{ z \in \mathbb H \colon \left| z \right| > 1,\, \left| \operatorname{Re}(z) \right| < \tfrac12 \right\}</math>

bounded by the vertical lines {{math|Re(''z'') {{=}} {{sfrac|1|2}}}} and {{math|Re(''z'') {{=}} −{{sfrac|1|2}}}}, and the circle {{math|{{abs|''z''}} {{=}} 1}}. This region is a hyperbolic triangle. It has vertices at {{math|{{sfrac|1|2}} + ''i''{{sfrac|{{radic|3}}|2}}}} and {{math|−{{sfrac|1|2}} + ''i''{{sfrac|{{radic|3}}|2}}}}, where the angle between its sides is {{math|{{sfrac|π|3}}}}, and a third vertex at infinity, where the angle between its sides is&nbsp;0.

There is a strong connection between the modular group and [[elliptic curves]]. Each point <math>z</math> in the upper half-plane gives an elliptic curve, namely the quotient of <math>\mathbb{C}</math> by the lattice generated by 1 and <math>z</math>.  
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]] 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 [[tessellation|regular tessellation]] of the hyperbolic plane by congruent hyperbolic triangles known as the V6.6.∞ [[Infinite-order triangular tiling#Related polyhedra and tiling|Infinite-order triangular tiling]] is created.  Note that each such triangle has one vertex either at infinity or on the real axis {{math|Im(''z'') {{=}} 0}}. 

This tiling can be extended to the [[Poincaré_disk_model|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 [[j-invariant|{{math|''J''}}-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 {{math|(''x'', ''y'') ↦ (−''x'', ''y'')}} and taking the right half of the region {{math|''R''}} (where {{math|Re(''z'') ≥ 0}}) yields the usual tessellation. This tessellation first appears in print in {{Harv|Klein|1878/79a}},<ref name="lebruyn">{{citation
 | last = Le Bruyn | first = Lieven
 | title = Dedekind or Klein?
 | date = 22 April 2008
 | url = http://www.neverendingbooks.org/dedekind-or-klein
}}</ref> where it is credited to [[Richard Dedekind]], in reference to {{Harv|Dedekind|1877}}.<ref name="lebruyn" /><ref>{{Cite journal
| issn = 0002-9890
| volume = 108
| issue = 1
| pages = 70–76
| last = Stillwell
| first = John
| title = Modular Miracles
| journal = The American Mathematical Monthly| date = January 2001
| jstor = 2695682
| doi = 10.2307/2695682 
}}</ref>

[[File:Morphing of modular tiling to 2 3 7 triangle tiling.gif|thumb|Visualization of the map {{math|(2, 3, ∞) → (2, 3, 7)}} by morphing the associated tilings.<ref name="westendorp">{{cite web |last=Westendorp |first=Gerard |title=Platonic tessellations of Riemann surfaces |url=https://westy31.nl/Geometry/Geometry.html#Modular |website=westy31.nl}}</ref>]]
The map of groups {{math|(2, 3, ∞) → (2, 3, ''n'')}} (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.

{{Order i-3 tiling table}}

==Congruence subgroups==
{{Main|Congruence subgroup}}
Important [[subgroup]]s of the modular group {{math|Γ}}, called ''[[congruence subgroup]]s'', are given by imposing [[congruence relation]]s on the associated matrices.

There is a natural [[homomorphism]] {{math|SL(2, '''Z''') → SL(2, '''Z'''/''N'''''Z''')}} given by reducing the entries [[modulo operation|modulo]] {{math|''N''}}. This induces a homomorphism on the modular group {{math|PSL(2, '''Z''') → PSL(2, '''Z'''/''N'''''Z''')}}. The [[kernel (algebra)|kernel]] of this homomorphism is called the '''[[principal congruence subgroup]] of level {{math|''N''}}''', denoted {{math|Γ(''N'')}}. We have the following [[short exact sequence]]:

:<math>1\to\Gamma(N)\to\Gamma\to\operatorname{PSL}(2, \mathbb Z/N\mathbb Z) \to 1.</math>

Being the kernel of a homomorphism {{math|Γ(''N'')}} is a [[normal subgroup]] of the modular group {{math|Γ}}. The group {{math|Γ(''N'')}} is given as the set of all modular transformations

:<math>z\mapsto\frac{az+b}{cz+d}</math>

for which {{math|''a'' ≡ ''d'' ≡ ±1 (mod ''N'')}} and {{math|''b'' ≡ ''c'' ≡ 0 (mod ''N'')}}.

It is easy to show that the [[trace (linear algebra)|trace]] of a matrix representing an element of {{math|Γ(''N'')}} cannot be −1, 0, or 1, so these subgroups are [[torsion-free group]]s. (There are other torsion-free subgroups.)

The principal congruence subgroup of level 2, {{math|Γ(2)}}, is also called the '''modular group {{math|Λ}}'''. Since {{math|PSL(2, '''Z'''/2'''Z''')}} is isomorphic to {{math|[[symmetric group|''S''<sub>3</sub>]]}}, {{math|Λ}} is a subgroup of [[index of a subgroup|index]] 6. The group {{math|Λ}} consists of all modular transformations for which {{math|''a''}} and {{math|''d''}} are odd and {{math|''b''}} and {{math|''c''}} are even.

Another important family of congruence subgroups are the [[modular group Gamma0|modular group {{math|Γ<sub>0</sub>(''N'')}}]] defined as the set of all modular transformations for which {{math|''c'' ≡ 0 (mod ''N'')}}, or equivalently, as the subgroup whose matrices become [[upper triangular matrix|upper triangular]] upon reduction modulo {{math|''N''}}. Note that {{math|Γ(''N'')}} is a subgroup of {{math|Γ<sub>0</sub>(''N'')}}. The [[modular curve]]s associated with these groups are an aspect of [[monstrous moonshine]] – for a [[prime number]] {{math|''p''}}, the modular curve of the normalizer is [[genus (mathematics)|genus]] zero if and only if {{math|''p''}} divides the [[order (group theory)|order]] of the [[monster group]], or equivalently, if {{math|''p''}} is a [[supersingular prime (moonshine theory)|supersingular prime]].

==Dyadic monoid==
One important subset of the modular group is the '''dyadic monoid''', which is the [[monoid]] of all strings of the form {{math|''ST{{isup|k}}ST{{isup|m}}ST{{isup|n}}''...}} for positive integers {{math|''k'', ''m'', ''n'',...}}. This monoid occurs naturally in the study of [[fractal curve]]s, and describes  the [[self-similarity]] symmetries of the [[Cantor function]], [[Minkowski's question mark function]], and the [[Koch snowflake]], each being a special case of the general [[de Rham curve]]. The monoid also has higher-dimensional linear representations; for example, the {{math|''N'' {{=}} 3}} representation can be understood to describe the self-symmetry of the [[blancmange curve]].

==Maps of the torus==
The group {{math|GL(2, '''Z''')}} is the linear maps preserving the standard lattice {{math|'''Z'''<sup>2</sup>}}, and {{math|SL(2, '''Z''')}} is the orientation-preserving maps preserving this lattice; they thus descend to [[self-homeomorphism]]s of the [[torus]] (SL mapping to orientation-preserving maps), and in fact map isomorphically to the (extended) [[mapping class group]] of the torus, meaning that every self-homeomorphism of the torus is [[Homotopy#Isotopy|isotopic]] to a map of this form. The algebraic properties of a matrix as an element of {{math|GL(2, '''Z''')}} 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.<ref>{{cite book|title=Combinatorial Group Theory, Discrete Groups, and Number Theory|first1=Gerhard|last1=Rosenberger|first2=Benjamin|last2=Fine|first3=Anthony M.|last3=Gaglione|first4=Dennis|last4=Spellman|year=2006 |url=https://books.google.com/books?id=5Unmxs7yeHwC&pg=PA65|page=65|isbn=9780821839850 }}</ref>

The Hecke group {{math|''H''<sub>''q''</sub>}} with {{math|''q'' ≥ 3}}, is the discrete group generated by

:<math>\begin{align}
z &\mapsto -\frac1z \\
z &\mapsto z + \lambda_q,
\end{align}</math>

where {{math|''λ<sub>q</sub>'' {{=}} 2 cos {{sfrac|π|''q''}}}}. For small values of {{math|''q'' ≥ 3}}, one has:

:<math>\begin{align}
\lambda_3 &= 1, \\
\lambda_4 &= \sqrt{2}, \\
\lambda_5 &= \frac{1+\sqrt{5}}{2}, \\
\lambda_6 &= \sqrt{3}, \\
\lambda_8 &= \sqrt{2+\sqrt{2}}.
\end{align}</math>

The modular group {{math|Γ}} is isomorphic to {{math|''H''<sub>3</sub>}} and they share properties and applications – for example, just as one has the [[free product]] of [[cyclic group]]s

:<math>\Gamma \cong C_2 * C_3,</math>

more generally one has

:<math>H_q \cong C_2 * C_q,</math>

which corresponds to the [[triangle group]] {{math|(2, ''q'', ∞)}}. There is similarly a notion of principal congruence subgroups associated to principal ideals in {{math|'''Z'''[''λ'']}}.

==History==
The modular group and its subgroups were first studied in detail by [[Richard Dedekind]] and by [[Felix Klein]] as part of his [[Erlangen programme]] in the 1870s. However, the closely related [[elliptic function]]s were studied by [[Joseph Louis Lagrange]] in 1785, and further results on elliptic functions were published by [[Carl Gustav Jakob Jacobi]] and [[Niels Henrik Abel]] in 1827.

==See also==
{{div col|colwidth=22em}}
*[[Bianchi group]]
*[[Classical modular curve]]
*[[Fuchsian group]]
*[[J-invariant|{{math|''J''}}-invariant]]
*[[Kleinian group]]
*[[Mapping class group]]
*[[Minkowski's question-mark function]]
*[[Möbius transformation]]
*[[Modular curve]]
*[[Modular form]]
*[[Kuṭṭaka]]
*[[Poincaré half-plane model]]
*[[Uniform tilings in hyperbolic plane]]
{{div col end}}

==References==
{{Reflist}}
{{Refbegin}}
*{{cite book|first=Tom M.|last=Apostol|title=Modular Functions and Dirichlet Series in Number Theory|edition=2nd|date=1990|publisher=Springer|location=New York|isbn=0-387-97127-0|at=ch. 2}}
*{{Citation
 |last         = Klein
 |first        = Felix
 |author-link   = Felix Klein
 |ref          = {{Harvid|Klein|1878/79a}}
 |title        = Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades (On the transformation of elliptic functions and ...)
 |pages        = 13–75
 |journal      = Math. Annalen
 |volume       = 14
 |year         = 1878–1879
 |url          = http://mathdoc.emath.fr/cgi-bin/oetoc?id=OE_KLEIN__3
 |doi          = 10.1007/BF02297507
 |s2cid = 121056952
 |access-date  = 3 June 2010
 |archive-url  = https://web.archive.org/web/20110719181924/http://mathdoc.emath.fr/cgi-bin/oetoc?id=OE_KLEIN__3
 |archive-date = 19 July 2011
 |url-status   = dead
}}
* {{Citation | title = Schreiben an Herrn Borchardt über die Theorie der elliptische Modul-Functionen | first = Richard | last = Dedekind | author-link = Richard Dedekind |date=September 1877 | journal = Crelle's Journal | volume = 83 | pages =  265–292}}.
{{Refend}}
{{Use dmy dates|date=August 2019}}

{{DEFAULTSORT:Modular Group}}
[[Category:Group theory]]
[[Category:Analytic number theory]]
[[Category:Modular forms]]