Editing Projective linear group (section)

==== Action on ''p'' points ====
While {{nowrap|PSL(''n'', ''q'')}} naturally acts on {{nowrap|1=(''q''<sup>''n''</sup> − 1)/(''q'' − 1) = 1 + ''q'' + ... + ''q''<sup>''n''−1</sup>}} points, non-trivial actions on fewer points are rarer. Indeed, for {{nowrap|PSL(2, ''p'')}} acts non-trivially on ''p'' points if and only if {{nowrap|1=''p'' = 2}}, 3, 5, 7, or 11; for 2 and 3 the group is not simple, while for 5, 7, and 11, the group is simple – further, it does not act non-trivially on ''fewer'' than ''p'' points.<ref group="note">Since ''p'' divides the order of the group, the group does not embed in (or, since simple, map non-trivially to) ''S<sub>k</sub>'' for {{nowrap|''k'' < ''p''}}, as ''p'' does not divide the order of this latter group.</ref> This was first observed by [[Évariste Galois]] in his last letter to Chevalier, 1832.<ref>Letter, pp. 411–412</ref>

This can be analyzed as follows; note that for 2 and 3 the action is not faithful (it is a non-trivial quotient, and the PSL group is not simple), while for 5, 7, and 11 the action is faithful (as the group is simple and the action is non-trivial), and yields an embedding into S<sub>''p''</sub>. In all but the last case, {{nowrap|PSL(2, 11)}}, it corresponds to an exceptional isomorphism, where the right-most group has an obvious action on ''p'' points:
* ''L''<sub>2</sub>(2) ≅ S<sub>3</sub> <math>\twoheadrightarrow</math> S<sub>2</sub> via the sign map;
* ''L''<sub>2</sub>(3) ≅ A<sub>4</sub> <math>\twoheadrightarrow</math> A<sub>3</sub> ≅ C<sub>3</sub> via the quotient by the Klein 4-group;
* ''L''<sub>2</sub>(5) ≅ A<sub>5</sub>. To construct such an isomorphism, one needs to consider the group ''L''<sub>2</sub>(5) as a Galois group of a Galois cover ''a''<sub>5</sub>: {{nowrap|1=''X''(5) → ''X''(1) = '''P'''<sup>1</sup>}}, where ''X''(''N'') is  a [[modular curve]] of level ''N''. This cover is ramified at 12 points. The modular curve X(5) has genus 0 and is isomorphic to a sphere over the field of complex numbers, and then the action of ''L''<sub>2</sub>(5) on these 12 points becomes the [[Icosahedral symmetry|symmetry group of an icosahedron]]. One then needs to consider the action of the symmetry group of icosahedron on the [[Compound of five tetrahedra|five associated tetrahedra]].
* {{nowrap|''L''<sub>2</sub>(7) ≅ ''L''<sub>3</sub>(2)}} which acts on the {{nowrap|1=1 + 2 + 4 = 7}} points of the [[Fano plane]] (projective plane over '''F'''<sub>2</sub>); this can also be seen as the action on order 2 [[biplane geometry|biplane]], which is the ''complementary'' Fano plane.
* ''L''<sub>2</sub>(11) is subtler, and elaborated below; it acts on the order 3 biplane.<ref>{{Citation | id = see: The Embedding of PSl(2, 5) into PSl(2, 11) and Galois’ Letter to Chevalier. | title = The Graph of the Truncated Icosahedron and the Last Letter of Galois | first = Bertram | last = Kostant | journal = Notices Amer. Math. Soc. | volume = 42 | pages = 959–968 | year = 1995 | url = https://www.ams.org/notices/199509/kostant.pdf | issue = 4 }}</ref>
Further, ''L''<sub>2</sub>(7) and ''L''<sub>2</sub>(11) have two ''inequivalent'' actions on ''p'' points; geometrically this is realized by the action on a biplane, which has ''p'' points and ''p'' blocks – the action on the points and the action on the blocks are both actions on ''p'' points, but not conjugate (they have different point stabilizers); they are instead related by an outer automorphism of the group.<ref>[[Noam Elkies]], Math 155r, [http://www.math.harvard.edu/~elkies/M155.09/apr14 Lecture notes for April 14, 2010]</ref>

More recently, these last three exceptional actions have been interpreted as an example of the [[ADE classification]]:<ref>{{Harv|Kostant|1995|p=964}}</ref> these actions correspond to products (as sets, not as groups) of the groups as {{nowrap|A<sub>4</sub> × '''Z'''{{hsp}}/{{hsp}}5'''Z'''}}, {{nowrap|S<sub>4</sub> × '''Z'''{{hsp}}/{{hsp}}7'''Z'''}}, and {{nowrap|A<sub>5</sub> × '''Z'''{{hsp}}/{{hsp}}11'''Z'''}}, where the groups A<sub>4</sub>, S<sub>4</sub> and A<sub>5</sub> are the isometry groups of the [[Platonic solid]]s, and correspond to ''E''<sub>6</sub>, ''E''<sub>7</sub>, and ''E''<sub>8</sub> under the [[McKay correspondence]]. These three exceptional cases are also realized as the geometries of polyhedra (equivalently, tilings of [[Riemann surface]]s), respectively: the [[compound of five tetrahedra]] inside the icosahedron (sphere, genus 0), the order 2 biplane (complementary [[Fano plane]]) inside the Klein quartic (genus 3), and the order 3 biplane ([[Paley biplane]]) inside the [[buckyball surface]] (genus 70).<ref>[http://www.neverendingbooks.org/index.php/galois-last-letter.html Galois’ last letter] {{webarchive|url=https://web.archive.org/web/20100815034546/http://www.neverendingbooks.org/index.php/galois-last-letter.html |date=2010-08-15 }}, Never Ending Books</ref><ref name="martinsingerman">{{citation | title = From Biplanes to the Klein quartic and the Buckyball | first1 = Pablo | last1 = Martin | first2 = David | last2 = Singerman | date = April 17, 2008 | url = http://www.neverendingbooks.org/DATA/biplanesingerman.pdf}}</ref>

The action of ''L''<sub>2</sub>(11) can be seen algebraically as due to an exceptional inclusion {{nowrap|''L''<sub>2</sub>(5) <math>\hookrightarrow</math> ''L''<sub>2</sub>(11)}} – there are two conjugacy classes of subgroups of ''L''<sub>2</sub>(11) that are isomorphic to ''L''<sub>2</sub>(5), each with 11 elements: the action of ''L''<sub>2</sub>(11) by conjugation on these is an action on 11 points, and, further, the two conjugacy classes are related by an outer automorphism of ''L''<sub>2</sub>(11). (The same is true for subgroups of ''L''<sub>2</sub>(7) isomorphic to S<sub>4</sub>, and this also has a biplane geometry.)

Geometrically, this action can be understood via a ''[[biplane geometry]]'', which is defined as follows. A biplane geometry is a [[symmetric design]] (a set of points and an equal number of "lines", or rather blocks) such that any set of two points is contained in two lines, while any two lines intersect in two points; this is similar to a finite projective plane, except that rather than two points determining one line (and two lines determining one point), they determine two lines (respectively, points). In this case (the [[Paley biplane]], obtained from the [[Paley digraph]] of order 11), the points are the affine line (the finite field) '''F'''<sub>11</sub>, where the first line is defined to be the five non-zero [[quadratic residue]]s (points which are squares: 1, 3, 4, 5, 9), and the other lines are the affine translates of this (add a constant to all the points). ''L''<sub>2</sub>(11) is then isomorphic to the subgroup of S<sub>11</sub> that preserve this geometry (sends lines to lines), giving a set of 11 points on which it acts – in fact two: the points or the lines, which corresponds to the outer automorphism – while ''L''<sub>2</sub>(5) is the stabilizer of a given line, or dually of a given point.

More surprisingly, the coset space ''L''<sub>2</sub>(11){{hsp}}/{{hsp}}('''Z'''{{hsp}}/{{hsp}}11'''Z'''), which has order {{nowrap|1=660/11 = 60}} (and on which the icosahedral group acts) naturally has the structure of a [[buckeyball]], which is used in the construction of the [[buckyball surface]].