PSL(2, 7)のソースを表示

{{参照方法|date=2018年7月16日 (月) 02:15 (UTC)}}
[[射影特殊線型群]]'''PSL<sub>2</sub>(7)''' (別表記: '''PSL(2, 7)''', '''PSL<sub>2</sub>(F<sub>7</sub>)''', '''PSL(2, F<sub>7</sub>)'''など)もしくはそれと[[群同型|同型]]な'''PSL<sub>3</sub>(2)''' (別表記: '''PSL(3, 2)''', '''PSL<sub>3</sub>(F<sub>2</sub>)''', '''PSL(3, F<sub>2</sub>)'''など)は、[[代数学]]、[[幾何学]]、[[数論]]といった分野で重要な役割を持つ[[有限群|有限]][[単純群]]である。PSL<sub>2</sub>(7)は{{仮リンク|クラインの平面4次曲線|en|Klein quartic}}の[[自己同型群]]と同型で、また[[ファノ平面]]の{{仮リンク|対称変換群|en|Symmetry group|label=対称性の群}}とも同型である。[[位数 (群論)|位数]]168の単純群はPSL<sub>2</sub>(7)と同型であり、位数60の[[交代群]][[五次の交代群|A<sub>5</sub>]]（PSL<sub>2</sub>(4)、PSL<sub>2</sub>(5)、[[正二十面体群]]と同型。）に次いで2番目に小さな[[アーベル群|非可換]][[単純群]]である。

== 定義 ==
{{See also|射影特殊線型群}}
[[一般線型群]]GL<sub>2</sub>(7)は、7個の要素からなる[[有限体]]'''F'''<sub>7</sub>上の[[行列式]]が0でない2次[[正方行列]]全体のなす群である。SL<sub>2</sub>(7)はGL<sub>2</sub>(7)の[[部分群]]であり、行列式が1のものだけからなる。このときPSL<sub>2</sub>(7)は[[商群]]

:<math>\mathrm{SL}_2(7) / \{ I, -I\}</math>

として定義される。ここで、''I''は[[単位行列]]である。すなわち、SL<sub>2</sub>(7)内で1倍と-1倍を同一視したものがPSL<sub>2</sub>(7)である。

== 同型 ==
以下の群はすべて同型である。

* PSL<sub>2</sub>(7)
* GL<sub>3</sub>(2)<br />'''F'''<sub>2</sub>においてGL, SL, PGL, PSLの区別はないので、ただちに次の同型もわかる。
** SL<sub>3</sub>(2)
** PGL<sub>3</sub>(2)
** PSL<sub>3</sub>(2)
* クラインの平面4次曲線の自己同型群
* ファノ平面の対称性の群

== 性質 ==
PSL<sub>2</sub>(7)は168個の要素を持つ。これは行列の取り得る列の数を数え上げることで確認できる。1列目には7<sup>2</sup>−1 = 48通りの組み合わせが存在する。2列目には7<sup>2</sup>−7 = 42通りの組み合わせが存在する。ここで行列式が1のものを取り出すために7−1 = 6で割り、''I''と-''I''を同一視するので2で割る。すると (48×42)/(6×2) = 168が得られる。

一般にPSL<sub>''n''</sub>(''q'')は ''n'', ''q'' ≥ 2 （''q''は[[素数]]の冪）のとき、 (''n'', ''q'') = (2, 2), (2, 3)という例外を除いて単純群となる。 PSL<sub>2</sub>(2)は[[対称群]][[三次の対称群|S<sub>3</sub>]]に同型であり、PSL<sub>2</sub>(3)は交代群[[四次の交代群|A<sub>4</sub>]]に同型である。PSL<sub>2</sub>(7)は交代群A<sub>5</sub>に次いで2番目に小さな非可換単純群である。

PSL<sub>2</sub>(7)は6個の[[共役類]]および非同型な[[既約表現]]をもつ。各共役類の大きさは1, 21, 42, 56, 24, 24であり、各既約表現の次元は1, 3, 3, 6, 7, 8である。

[[指標表]]

:<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>

ただし <math>\sigma = (-1+i\sqrt{7})/2</math> とする。
<!--
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
|}
-->

PSL<sub>2</sub>(7)の位数は168=3×7×8なので、位数3,7,8の[[シロー部分群]]を持つ。素数位数の群は[[巡回群]]に限られるので、前者二つが巡回群であることは容易にわかる。共役類3''A''<sub>56</sub>の任意の要素はシロー3部分群を生成する。また、共役類7''A''<sub>24</sub>, 7''B''<sub>24</sub>の任意の要素はシロー7部分群を生成する。シロー2部分群は位数8の[[二面体群]]である。これは共役類2''A''<sub>21</sub>の任意の要素の[[中心化群と正規化群|中心化群]]として記述できる。GL<sub>3</sub>(2)としての実現では、シロー2部分群は[[上三角行列]]の全体と一致する。

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This group and its Sylow 2-subgroup provide a counter-example for various [[normal p-complement]] theorems for ''p'' = 2.
-->
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== 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|collineations]] 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 &#x2014; 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 ==
{{further|Klein quartic}}
[[ファイル:Uniform_tiling_73-t0.png|サムネイル|The [[Klein quartic]] can be realized as a quotient of the [[order-3 heptagonal tiling]].]]
[[ファイル:Order-7_triangular_tiling.svg|サムネイル|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|complex numbers]] '''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 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|Hurwitz surfaces]]" 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|Hurwitz surfaces]], the Klein quartic can be given a metric of [[constant negative curvature]] and then tiled with [[Regular polygon|regular]] (hyperbolic) [[Heptagon|heptagons]], 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.
-->
== 参考文献 ==
{{reflist}}{{refbegin}}
* {{citation|title=How to Make the Mathieu Group M<sub>24</sub>|ref={{harvid|Richter}}|last=Richter|first=David A.|url=http://homepages.wmich.edu/~drichter/mathieu.htm|accessdate=2010-04-15}}
{{refend}}

== さらに詳しく ==

* {{cite journal|last1=Brown|first1=Ezra|last2=Loehr|first2=Nicholas|year=2009|title=Why is PSL (2,7)≅ GL (3,2)?|url=http://www.math.vt.edu/people/brown/doc/PSL%282,7%29_GL%283,2%29.pdf|journal=Am. Math. Mon.|volume=116|number=8|pages=727–732|doi=10.4169/193009709X460859|zbl=1229.20046}}

== 関連項目 ==

== 外部リンク ==

* [http://www.msri.org/publications/books/Book35/ The Eightfold Way: the Beauty of Klein's Quartic Curve (Silvio Levy, ed.)]
* [https://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)]

{{デフォルトソート:PSL27}}
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