Editing Modular group (section)

==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.