|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], the [[projective special linear group]] '''PSL(2, 7)''' (isomorphic to '''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 second-smallest [[nonabelian group|nonabelian]] [[simple group]] after the [[alternating group]] ''A''<sub>5</sub> on five letters with 60 elements (the rotational [[icosahedral symmetry]] group), or the isomorphic PSL(2, 5).
| | One particular author is known via name of Gabrielle Lattimer though she doesn't actually like being called like this. For years she's been working due to a library assistant. To bake is something that my wife been doing for growth cycles. For years she's been living in Massachusetts. She is running and looking after a blog here: http://prometeu.net<br><br>Also visit my blog ... [http://prometeu.net clash of clans cheat no survey] |
| | |
| ==Definition==
| |
| | |
| The [[general linear group]] 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]] SL(2, 7) consists of all such matrices with unit [[determinant]]. Then 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 isomorphic to PSL(2, 7).
| |
| | |
| ==Properties==
| |
| | |
| ''G'' = PSL(2, 7) has 168 elements. This can be seen by counting the possible columns; there are 7<sup>2</sup>−1 = 48 possibilities for the first column, then 7<sup>2</sup>−7 = 42 possibilities for the second column. We must divide by 7−1 = 6 to force the determinant equal to one, and then we must divide by 2 when we identify I and −I. The result is (48×42)/(6×2) = 168. | |
| | |
| It is a general result that PSL(''n'', ''q'') is [[simple group|simple]] for ''n'', ''q'' ≥ 2 (''q'' being some power of a prime number), unless (''n'', ''q'') = (2, 2) or (2, 3). PSL(2, 2) is [[group isomorphism|isomorphic]] to the [[symmetric group]] ''S''<sub>3</sub>, and PSL(2, 3) is isomorphic to [[alternating group]] ''A''<sub>4</sub>. In fact, PSL(2, 7) is the second smallest [[nonabelian group|nonabelian]] simple group, after the [[alternating group]] A<sub>5</sub> = PSL(2, 5) = PSL(2, 4).
| |
| | |
| The number of [[conjugacy class]]es and [[irreducible representation]]s is 6. The sizes of conjugacy classes are 1, 21, 42, 56, 24, 24. The dimensions of irreducible representations 1, 3, 3, 6, 7, 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 168=3*7*8, this implies existence of [[Sylow theorems | Sylow's subgroups]] of orders 3, 7 and 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 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 | normal p-complement]] theorems for ''p'' = 2.
| |
| | |
| ==Actions on projective spaces==
| |
| | |
| ''G'' = 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>\mbox{For } \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mbox{PSL}(2, 7) \mbox{ 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 ''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 PGL(2, 7), and the group of [[collineation]]s of the projective line is the complete [[symmetric group]] of the points.
| |
| | |
| However, PSL(2, 7) is also [[group isomorphism|isomorphic]] to PSL(3, 2) (= 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, ''G'' = PSL(3, 2) acts on the [[projective plane]] '''P'''<sup>2</sup>(2) over the field with 2 elements — also known as the [[Fano plane]]:
| |
| | |
| <math>\mbox{For } \gamma = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \in \mbox{PSL}(3, 2) \mbox{ and } \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 ''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==
| |
| {{see|Klein quartic}}
| |
| [[File:Uniform tiling 73-t0.png|thumb|The [[Klein quartic]] can be realized as a quotient of the [[order-3 heptagonal tiling]].]]
| |
| [[File:Uniform tiling 73-t2.png|thumb|Dually, the [[Klein quartic]] can be realized as a quotient of the [[order-7 triangular tiling]].]]
| |
| The [[Klein quartic]] is the projective variety over the [[complex number]]s '''C''' defined by the quartic polyomial
| |
| | |
| :''x''<sup>3</sup>''y'' + ''y''<sup>3</sup>''z'' + ''z''<sup>3</sup>''x'' = 0.
| |
| | |
| It is a compact [[Riemann surface]] of genus g = 3, and is the only such surface for which the size of the conformal automorphism group attains the maximum of 84(''g''−1). This bound is due to the [[Hurwitz automorphisms theorem]], which holds for all ''g''>1. Such "[[Hurwitz surface]]s" are rare; the next genus for which any exist is ''g'' = 7, and the next after that is ''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 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 1.
| |
| | |
| ==Mathieu group==
| |
| PSL(2, 7) is a maximal subgroup of the [[Mathieu group]] M<sub>21</sub>; the Mathieu group M<sub>21</sub> and then the Mathieu group M<sub>24</sub> can be constructed as extensions of PSL(2, 7). These extensions can be interpreted in term of the tiling of the Klein quartic, but are not realized by geometric symmetries of the tiling.<ref name="richter">{{Harv|Richter}}</ref>
| |
| | |
| ==Group actions==
| |
| PSL(2, 7) acts on various sets:
| |
| * Interpreted as linear automorphisms of the projective line over '''F'''<sub>7</sub> it acts 2-transitively on a set of 8 points, with stabilizer of order 3. (PGL(2, 7) acts sharply 3-transitively, with trivial stabilizer.)
| |
| * Interpreted as automorphisms of a tiling of the Klein quartic, it acts simply transitively on the 24 vertices (or dually, 24 heptagons), with stabilizer of order 7 (corresponding to rotation about the vertex/heptagon).
| |
| * Interpreted as a subgroup of the Mathieu group M<sub>21</sub>, which acts on 21 points, it does not act transitively on the 21 points.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| {{refbegin}}
| |
| * {{citation | ref = {{harvid|Richter}} | 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}}
| |
| | |
| ==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)]
| |
| *[http://www.math.vt.edu/people/brown/doc/PSL(2,7)_GL(3,2).pdf Brown, Ezra; Loehr, Nicholas Why is PSL(2,7)≅GL(3,2)? Amer. Math. Monthly 116 (2009), no. 8, 727--732.]
| |
| | |
| [[Category:Finite groups]]
| |
| [[Category:Projective geometry]]
| |