Editing Projective linear group (section)

=== Modular group ===
{{main|Modular group}}
The groups {{nowrap|PSL(2, '''Z'''{{hsp}}/{{hsp}}''n'''''Z''')}} arise in studying the [[modular group]], {{nowrap|PSL(2, '''Z''')}}, as quotients by reducing all elements mod ''n''; the kernels are called the [[principal congruence subgroup]]s.

A noteworthy subgroup of the projective ''general'' linear group {{nowrap|PGL(2, '''Z''')}} (and of the projective special linear group {{nowrap|PSL(2, '''Z'''[''i''])}}) is the symmetries of the set {{nowrap|{{mset|0, 1, ∞}} ⊂ '''P'''<sup>1</sup>('''C''')}}<ref group="note">In projective coordinates, the points {{mset|0, 1, ∞}} are given by [0:1], [1:1], and [1:0], which explains why their stabilizer is represented by integral matrices.</ref> which is known as the [[anharmonic group]], and arises as the symmetries of the [[six cross-ratios]]. The subgroup can be expressed as [[fractional linear transformation]]s, or represented (non-uniquely) by matrices, as:
: {| class="wikitable"
|-
| align="center" | <math>x</math>
| align="center" | <math>1/(1-x)</math>
| align="center" | <math>(x-1)/x</math>
|-
| align="center" | <math>\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}</math>
| align="center" | <math>\begin{pmatrix}
0 & 1\\
-1 & 1
\end{pmatrix}</math>
| align="center" | <math>\begin{pmatrix}
1 & -1\\
1 & 0
\end{pmatrix}</math>
|-
| colspan="3" |
|-
| align="center" | <math>1/x</math>
| align="center" | <math>1-x</math>
| align="center" | <math>x/(x-1)</math>
|-
| align="center" |<math>\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}</math>
| align="center" |<math>\begin{pmatrix}
-1 & 1\\
0 & 1
\end{pmatrix}</math>
| align="center" |<math>\begin{pmatrix}
1 & 0\\
1 & -1
\end{pmatrix}</math>
|-
| align="center" |<math>\begin{pmatrix}
0 & i\\
i & 0
\end{pmatrix}</math>
| align="center" |<math>\begin{pmatrix}
-i & i\\
0 & i
\end{pmatrix}</math>
| align="center" |<math>\begin{pmatrix}
i & 0\\
i & -i
\end{pmatrix}</math>
|}
Note that the top row is the identity and the two 3-cycles, and are orientation-preserving, forming a subgroup in {{nowrap|PSL(2, '''Z''')}}, while the bottom row is the three 2-cycles, and are in {{nowrap|PGL(2, '''Z''')}} and {{nowrap|PSL(2, '''Z'''[''i''])}}, but not in {{nowrap|PSL(2, '''Z''')}}, hence realized either as matrices with determinant −1 and integer coefficients, or as matrices with determinant 1 and [[Gaussian integer]] coefficients.

This maps to the symmetries of {{nowrap|{{mset|0, 1, ∞}} ⊂ '''P'''<sup>1</sup>(''n'')}} under reduction mod ''n''. Notably, for {{nowrap|1=''n'' = 2}}, this subgroup maps isomorphically to {{nowrap|1=PGL(2, '''Z'''{{hsp}}/{{hsp}}2'''Z''') = PSL(2, '''Z'''{{hsp}}/{{hsp}}2'''Z''') ≅ S<sub>3</sub>}},<ref group="note">This isomorphism can be seen by removing the minus signs in matrices, which yields the matrices for {{nowrap|PGL(2, 2)}}</ref> and thus provides a splitting {{nowrap|PGL(2, '''Z'''{{hsp}}/{{hsp}}2'''Z''') <math>\hookrightarrow</math> PGL(2, '''Z''')}} for the quotient map {{nowrap|PGL(2, '''Z''') <math>\twoheadrightarrow</math> PGL(2, '''Z'''{{hsp}}/{{hsp}}2'''Z''')}}.

[[File:PGL2 stabilizer of 3 points on line.svg|thumb|300px|The subgroups of the stabilizer of {{mset|0, 1, ∞}} further stabilize the points {{mset|−1, 1/2, 2}} and {{mset|''ζ''<sub>−</sub>, ''ζ''<sub>+</sub>}}.]]

The fixed points of both 3-cycles are the "most symmetric" cross-ratios, <math>e^{\pm i\pi/3} = \tfrac{1}{2} \pm \tfrac{\sqrt{3}}{2}i</math>, the solutions to {{nowrap|''x''<sup>2</sup> − ''x'' + 1}} (the [[primitive root of unity|primitive]] sixth [[roots of unity]]). The 2-cycles interchange these, as they do any points other than their fixed points, which realizes the quotient map {{nowrap|S<sub>3</sub> → S<sub>2</sub>}} by the group action on these two points. That is, the subgroup {{nowrap|C<sub>3</sub> < S<sub>3</sub>}} consisting of the identity and the 3-cycles, {{nowrap|{{mset|(), (0 1 ∞), (0 ∞ 1)}}}}, fixes these two points, while the other elements interchange them.

The fixed points of the individual 2-cycles are, respectively, −1, 1/2, 2, and this set is also preserved and permuted by the 3-cycles. This corresponds to the action of S<sub>3</sub> on the 2-cycles (its [[Sylow subgroup|Sylow 2-subgroups]]) by conjugation and realizes the isomorphism with the group of [[inner automorphisms]], {{nowrap|S<sub>3</sub> {{overset|lh=0.5|~|→}} Inn(S<sub>3</sub>) ≅ S<sub>3</sub>}}.

Geometrically, this can be visualized as the [[rotation group]] of the [[triangular bipyramid]], which is isomorphic to the [[dihedral group]] of the triangle {{nowrap|D<sub>3</sub> ≅ S<sub>3</sub>}}; see [[anharmonic group]].