Editing Möbius transformation (section)

== Subgroups of the Möbius group ==
If we require the coefficients <math>a,b,c,d</math> of a Möbius transformation to be real numbers with {{tmath|1= ad-bc=1 }}, we obtain a subgroup of the Möbius group denoted as [[PSL2(R)|{{nowrap|PSL(2, '''R''')}}]]. This is the group of those Möbius transformations that map the [[upper half-plane]] {{nowrap|1=''H'' = {''x'' + i''y'' : ''y'' > 0} }} to itself, and is equal to the group of all [[biholomorphic]] (or equivalently: [[bijective]], [[conformal map|conformal]] and orientation-preserving) maps {{nowrap|''H'' → ''H''}}. If a proper [[Riemannian metric|metric]] is introduced, the upper half-plane becomes a model of the [[hyperbolic geometry|hyperbolic plane]] ''H''{{i sup|2}}, the [[Poincaré half-plane model]], and {{nowrap|PSL(2, '''R''')}} is the group of all orientation-preserving isometries of ''H''{{i sup|2}} in this model.

The subgroup of all Möbius transformations that map the open disk {{nowrap|1=''D'' = {''z'' : {{abs|''z''}} < 1} }} to itself consists of all transformations of the form
<math display="block">f(z) = e^{i\phi} \frac{z + b}{\bar{b} z + 1}</math>
with {{nowrap|<math>\phi</math> ∈ '''R''', ''b'' ∈ '''C'''}} and {{nowrap|{{abs|''b''}} < 1}}. This is equal to the group of all biholomorphic (or equivalently: bijective, angle-preserving and orientation-preserving) maps {{nowrap|''D'' → ''D''}}. By introducing a suitable metric, the open disk turns into another model of the hyperbolic plane, the [[Poincaré disk model]], and this group is the group of all orientation-preserving isometries of ''H''{{i sup|2}} in this model.

Since both of the above subgroups serve as isometry groups of ''H''{{i sup|2}}, they are isomorphic. A concrete isomorphism is given by [[inner automorphism|conjugation]] with the transformation
<math display="block">f(z)=\frac{z+i}{iz+1}</math>
which bijectively maps the open unit disk to the upper half plane.

Alternatively, consider an open disk with radius ''r'', centered at ''r''{{hsp}}''i''. The Poincaré disk model in this disk becomes identical to the upper-half-plane model as ''r'' approaches&nbsp;∞.

A [[maximal compact subgroup]] of the Möbius group <math>\mathcal{M}</math> is given by {{Harvard citation|Tóth|2002}}{{sfn |Tóth|2002|loc=Section 1.2, Rotations and Möbius Transformations, [https://books.google.com/books?id=i76mmyvDHYUC&pg=PA22 p.&nbsp;22]}}
<math display="block">\mathcal{M}_0 := \left\{z \mapsto \frac{uz - \bar v}{vz + \bar u} : |u|^2 + |v|^2 = 1\right\},</math>
and corresponds under the isomorphism <math>\mathcal{M} \cong \operatorname{PSL}(2,\Complex)</math> to the [[projective special unitary group]] {{nowrap|PSU(2, '''C''')}} which is isomorphic to the [[special orthogonal group]] SO(3) of rotations in three dimensions, and can be interpreted as rotations of the Riemann sphere. Every finite subgroup is conjugate into this maximal compact group, and thus these correspond exactly to the polyhedral groups, the [[point groups in three dimensions]].

[[Icosahedral group]]s of Möbius transformations were used by [[Felix Klein]] to give an analytic solution to the [[quintic equation]] in {{Harvard citation|Klein|1913}}; a modern exposition is given in {{Harvard citation|Tóth|2002}}.{{sfn|Tóth|2002|loc=Section 1.6, Additional Topic: Klein's Theory of the Icosahedron, [https://books.google.com/books?id=i76mmyvDHYUC&pg=PA66 p.&nbsp;66]}}

If we require the coefficients ''a'', ''b'', ''c'', ''d'' of a Möbius transformation to be [[integer]]s with {{nowrap|1=''ad'' − ''bc'' = 1}}, we obtain the [[modular group]] {{nowrap|PSL(2, '''Z''')}}, a discrete subgroup of {{nowrap|PSL(2, '''R''')}} important in the study of [[Lattice (order)|lattice]]s in the complex plane, [[elliptic function]]s and [[elliptic curve]]s. The discrete subgroups of {{nowrap|PSL(2, '''R''')}} are known as [[Fuchsian group]]s; they are important in the study of [[Riemann surface]]s.