View source for PSL(2,7)

{{Short description|Automorphism group of the Klein quartic}}
In [[mathematics]], the [[projective special linear group]] {{nowrap|'''PSL(2, 7)'''}}, isomorphic to {{nowrap|'''GL(3, 2)'''}}, is a [[finite group|finite]] [[simple group]] that has important applications in [[algebra]], [[geometry]], and [[number theory]]. It is the [[automorphism group]] of the [[Klein quartic]] as well as the [[symmetry group]] of the [[Fano plane]]. With 168 elements, PSL(2, 7) is the smallest [[nonabelian group|nonabelian]] [[simple group]] after the [[alternating group]] A<sub>5</sub> with 60 elements, isomorphic to {{nowrap|PSL(2, 5)}}.

== Definition ==

The [[general linear group]] {{nowrap|GL(2, 7)}} consists of all invertible 2×2 [[matrix (mathematics)|matrices]] over '''F'''<sub>7</sub>, the [[finite field]] with 7 elements.  These have nonzero determinant.  The [[subgroup]] {{nowrap|SL(2, 7)}} consists of all such matrices with unit [[determinant]]. Then {{nowrap|PSL(2, 7)}} is defined to be the [[quotient group]]
: SL(2, 7) / {''I'', −''I''}
obtained by identifying ''I'' and −''I'', where ''I'' is the [[identity matrix]]. In this article, we let ''G'' denote any group that is isomorphic to {{nowrap|PSL(2, 7)}}.

== Properties ==
''G'' = {{nowrap|PSL(2, 7)}} has 168 elements. This can be seen by counting the possible columns; there are {{nowrap|1=7<sup>2</sup> − 1 = 48}} possibilities for the first column, then {{nowrap|1=7<sup>2</sup> − 7 = 42}} possibilities for the second column. We must divide by {{nowrap|1=7 − 1 = 6}} to force the determinant equal to one, and then we must divide by 2 when we identify ''I'' and&nbsp;−''I''. The result is {{nowrap|1=(48 × 42) / (6 × 2) = 168}}.

It is a general result that {{nowrap|PSL(''n'', ''q'')}} is [[simple group|simple]] for {{nowrap|''n'', ''q'' ≥ 2}} (''q'' being some power of a prime number), unless {{nowrap|1=(''n'', ''q'') = (2, 2)}} or&nbsp;{{nowrap|(2, 3)}}. {{nowrap|PSL(2, 2)}} is [[group isomorphism|isomorphic]] to the [[symmetric group]] S<sub>3</sub>, and {{nowrap|PSL(2, 3)}} is isomorphic to [[alternating group]] A<sub>4</sub>. In fact, {{nowrap|PSL(2, 7)}} is the second smallest [[nonabelian group|nonabelian]] simple group, after the [[alternating group]] {{nowrap|1=A<sub>5</sub> = PSL(2, 5) = PSL(2, 4)}}.

The number of [[conjugacy class]]es and [[irreducible representation]]s is&nbsp;6. The sizes of conjugacy classes are 1, 21, 42, 56, 24,&nbsp;24. The dimensions of irreducible representations 1, 3, 3, 6, 7,&nbsp;8.

Character table
: <math>\begin{array}{r|cccccc}
         & 1A_{1} & 2A_{21} & 4A_{42} &  3A_{56} & 7A_{24} & 7B_{24}  \\  \hline
\chi_1  &   1  & 1  & 1 &  1 & 1  & 1  \\ 
\chi_2  &   3  & -1 & 1 &  0 & \sigma & \bar \sigma  \\ 
\chi_3  &   3  & -1 & 1 &  0 & \bar \sigma  & \sigma  \\ 
\chi_4  &   6  & 2  & 0 &  0 & -1  & -1  \\ 
\chi_5  &   7  & -1 &-1 &  1 & 0  & 0  \\ 
\chi_6  &   8  & 0  & 0 & -1 & 1  & 1  \\ 
\end{array},</math>
where
: <math>\sigma = \frac{-1+i\sqrt{7}}{2}.</math>

The following table describes the conjugacy classes in terms of the order of an element in the class, the size of the class, the minimum polynomial of every representative in GL(3, 2), and the function notation for a representative in PSL(2, 7). Note that the classes 7A and 7B are exchanged by an automorphism, so the representatives from GL(3, 2) and PSL(2, 7) can be switched arbitrarily.

{| class="wikitable sortable"
|-
! Order !! Size  !! Min Poly !! Function
|-
| 1 || 1 ||  ''x'' + 1 || ''x''
|-
| 2 || 21 || ''x''<sup>2</sup> + 1 || −1/''x''
|-
| 3 || 56 || ''x''<sup>3</sup> + 1 || 2''x''
|-
| 4 || 42 || ''x''<sup>3</sup> + ''x''<sup>2</sup> + ''x'' + 1 || 1/(3 − ''x'')
|-
| 7 || 24 || ''x''<sup>3</sup> + ''x'' + 1 || ''x'' + 1
|-
| 7 || 24 || ''x''<sup>3</sup> + ''x''<sup>2</sup> + 1 || ''x'' + 3
|}

The order of group is {{nowrap|1=168 = 3 × 7 × 8}}, this implies existence of [[Sylow theorems|Sylow's subgroups]] of orders 3, 7 and&nbsp;8. It is easy to describe the first two, they are cyclic, since [[Cyclic group#Properties|any group of prime order is cyclic]]. Any element of conjugacy class 3''A''<sub>56</sub> generates Sylow 3-subgroup. Any element from the conjugacy classes 7''A''<sub>24</sub>, 7''B''<sub>24</sub> generates the Sylow 7-subgroup. The Sylow 2-subgroup is a [[dihedral group of order 8]]. It can be described as [[Centralizer and normalizer|centralizer]] of any element from the conjugacy class 2''A''<sub>21</sub>. In the {{nowrap|GL(3, 2)}} representation, a Sylow 2-subgroup consists of the upper triangular matrices.

This group and its Sylow 2-subgroup  provide a counter-example for various [[normal p-complement]] theorems for {{nowrap|1=''p'' = 2}}.

== Actions on projective spaces ==

''G'' = {{nowrap|PSL(2, 7)}} acts via [[Möbius transformation|linear fractional transformation]] on the [[projective line]] '''P'''<sup>1</sup>(7) over the field with 7 elements:

:<math>\text{For  } \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{PSL}(2, 7) \text{  and  } x \in \mathbf{P}^1\!(7),\ \gamma \cdot x = \frac{ax+b}{cx+d} .</math>

Every orientation-preserving automorphism of '''P'''<sup>1</sup>(7) arises in this way, and so {{nowrap|1=''G'' = PSL(2, 7)}} can be thought of geometrically as a group of symmetries of the projective line '''P'''<sup>1</sup>(7); the full group of possibly orientation-reversing projective linear automorphisms is instead the order 2 extension {{nowrap|PGL(2, 7)}}, and the group of [[collineation]]s of the projective line is the complete [[symmetric group]] of the points.

However, {{nowrap|PSL(2, 7)}} is also [[group isomorphism|isomorphic]] to {{nowrap|PSL(3, 2)}} ({{nowrap|1== SL(3, 2) = GL(3, 2)}}), the special (general) linear group of 3×3 matrices over the field with 2 elements. In a similar fashion, {{nowrap|1=''G'' = PSL(3, 2)}} acts on the [[projective plane]] '''P'''<sup>2</sup>(2) over the field with 2 elements &mdash; also known as the [[Fano&nbsp;plane]]:
: For <math> \gamma = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \in \text{PSL}(3, 2)\ \ </math> and <math>\ 
 \ \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \in \mathbf{P}^2\!(2),\ \ \gamma \ \cdot \ \mathbf{x} = \begin{pmatrix} ax+by+cz \\ dx+ey+fz \\ gx+hy+iz \end{pmatrix}</math>

Again, every automorphism of '''P'''<sup>2</sup>(2) arises in this way, and so {{nowrap|1=''G'' = PSL(3, 2)}} can be thought of geometrically as the [[symmetry group]] of
this projective plane.  The [[Fano plane]] can be used to describe multiplication of [[octonions]], so ''G'' acts on the set of octonion multiplication tables.

== Symmetries of the Klein quartic ==
{{further|Klein quartic}}
[[File:Klein quartic with dual graphs.svg|thumb|The [[Klein quartic]] can be realized as a quotient of the [[order-3 heptagonal tiling|order-3 heptagonal]] or the [[order-7 triangular tiling]].]]
The [[Klein quartic]] is the projective variety over the [[complex number]]s '''C''' defined by the quartic polynomial
: ''x''<sup>3</sup>''y'' + ''y''<sup>3</sup>''z'' + ''z''<sup>3</sup>''x'' = 0.

It is a compact [[Riemann surface]] of genus {{nowrap|1=''g'' = 3}}, and is the only such surface for which the size of the conformal automorphism group attains the maximum of {{nowrap|84(''g'' − 1)}}. This bound is due to the [[Hurwitz automorphisms theorem]], which holds for all {{nowrap|''g'' > 1}}. Such "[[Hurwitz surface]]s" are rare; the next genus for which any exist is {{nowrap|1=''g'' = 7}}, and the next after that is {{nowrap|1=''g'' = 14}}.

As with all [[Hurwitz surface]]s, the Klein quartic can be given a metric of [[constant negative curvature]] and then tiled with [[Regular polygon|regular]] (hyperbolic) [[heptagon]]s, as a quotient of the [[order-3 heptagonal tiling]], with the symmetries of the surface as a Riemannian surface or algebraic curve exactly the same as the symmetries of the tiling. For the Klein quartic this yields a tiling by 24 heptagons, and the order of ''G'' is thus related to the fact that {{nowrap|1=24 × 7 = 168}}. Dually, it can be tiled with 56 equilateral triangles, with 24 vertices, each of degree 7, as a quotient of the [[order-7 triangular tiling]]. 
 
Klein's quartic arises in many fields of mathematics, including representation theory, homology theory, octonion multiplication, [[Fermat's Last Theorem]], and [[Stark–Heegner theorem|Stark's theorem]] on imaginary quadratic number fields of class number&nbsp;1.

== Mathieu group ==
{{nowrap|1=PSL(2, 7)}} is a maximal subgroup of the [[Mathieu group]] M<sub>21</sub>; the groups M<sub>21</sub> and  M<sub>24</sub> can be constructed as extensions of {{nowrap|PSL(2, 7)}}. These extensions can be interpreted in terms of the tiling of the Klein quartic, but are not realized by geometric symmetries of the tiling.<ref name="richter">{{Harv|Richter}}</ref>

== Permutation actions ==
The group {{nowrap|PSL(2, 7)}} acts on various finite sets:
* In its original interpretation as {{nowrap|PSL(2, 7)}}, orientation-preserving linear automorphisms of the projective line P<sup>1</sup>('''F'''<sub>7</sub>), it acts transitively on the 8 points with a stabilizer of order 21 fixing a given point. It also acts 2-transitively with stabilizer of order 3 on each pair of points; and it has two orbits on triples of points, with trivial stabilizer on each triple. (The larger group {{nowrap|PGL(2, 7)}} acts sharply 3-transitively.)
* Interpreted as {{nowrap|PGL(3, 2)}}, linear automorphisms of the Fano plane P<sup>2</sup>('''F'''<sub>2</sub>), it acts 2-transitively on the 7 points, with stabilizer of order 24 fixing each point, and stabilizer of order 4 fixing each pair of points.
* Interpreted as automorphisms of a tiling of the Klein quartic, it acts transitively on the 24 vertices (or dually, 24 heptagons), with stabilizer of order 7 (corresponding to a rotation about the vertex/heptagon).
* Interpreted as a subgroup of the Mathieu group M<sub>21</sub>, the subgroup acts non-transitively on 21 points.

== References ==
{{reflist}}
{{refbegin}}
* {{citation | first = David A. | last = Richter | url = http://homepages.wmich.edu/~drichter/mathieu.htm | title = How to Make the Mathieu Group M<sub>24</sub> | accessdate = 2010-04-15 }}
{{refend}}

== Further reading ==
* {{cite journal | last1=Brown | first1=Ezra | last2=Loehr | first2=Nicholas | title=Why is PSL (2,7)≅ GL (3,2)? | zbl=1229.20046 | journal=Am. Math. Mon. | volume=116 | number=8 | pages=727–732 | year=2009 | doi=10.4169/193009709X460859 | url=http://www.math.vt.edu/people/brown/doc/PSL%282,7%29_GL%283,2%29.pdf | access-date=2014-09-27 | archive-url=https://web.archive.org/web/20161009000204/http://www.math.vt.edu/people/brown/doc/PSL%282,7%29_GL%283,2%29.pdf | archive-date=2016-10-09 | url-status=dead }}

== External links ==
* [http://www.msri.org/publications/books/Book35/ The Eightfold Way: the Beauty of Klein's Quartic Curve (Silvio Levy, ed.)]
* [http://math.ucr.edu/home/baez/week214.html This Week's Finds in Mathematical Physics - Week 214 (John Baez)]
* [http://www.msri.org/publications/books/Book35/files/elkies.pdf The Klein Quartic in Number Theory (Noam Elkies)]
* [http://groupprops.subwiki.org/wiki/Projective_special_linear_group:PSL(3,2) Projective special linear group: PSL(3, 2)]

[[Category:Finite groups]]
[[Category:Projective geometry]]
[[Category:Octonions]]