Editing Mathieu group (section)

== Constructions of the Mathieu groups ==

The Mathieu groups can be constructed in various ways.

===Permutation groups===

''M''<sub>12</sub> has a simple subgroup of order 660, a maximal subgroup. That subgroup is isomorphic to the [[projective special linear group]] PSL<sub>2</sub>('''F'''<sub>11</sub>) over the [[finite field|field of 11 elements]]. With &minus;1 written as '''a''' and infinity as '''b''', two standard generators are (0123456789a) and (0b)(1a)(25)(37)(48)(69). A third generator giving ''M''<sub>12</sub> sends an element ''x'' of '''F'''<sub>11</sub> to 4''x''<sup>2</sup>&nbsp;&minus;&nbsp;3''x''<sup>7</sup>; as a permutation that is (26a7)(3945).

This group turns out not to be isomorphic to any member of the infinite families of finite simple groups and is called sporadic. ''M''<sub>11</sub> is the stabilizer of a point in ''M''<sub>12</sub>, and turns out also to be a sporadic simple group. ''M''<sub>10</sub>, the stabilizer of two points, is not sporadic, but is an [[almost simple group]] whose [[commutator subgroup]] is the [[alternating group]] A<sub>6</sub>. It is thus related to the [[Automorphisms of the symmetric and alternating groups#exceptional outer automorphism|exceptional outer automorphism]] of A<sub>6</sub>. The stabilizer of 3 points is the [[projective special unitary group]] PSU(3,2<sup>2</sup>), which is solvable. The stabilizer of 4 points is the [[quaternion group]].

Likewise, ''M''<sub>24</sub> has a maximal simple subgroup of order 6072 isomorphic to PSL<sub>2</sub>('''F'''<sub>23</sub>). One generator adds 1 to each element of the field (leaving the point ''N'' at infinity fixed), i.e. (0123456789ABCDEFGHIJKLM)(''N''), and the other sends ''x'' to &minus;1/''x'', i.e. (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI). A third generator giving ''M''<sub>24</sub> sends an element ''x'' of '''F'''<sub>23</sub> to 4''x''<sup>4</sup>&nbsp;&minus;&nbsp;3''x''<sup>15</sup> (which sends perfect squares via ''x''<sup>4</sup> and non-perfect squares via 7''x''<sup>4</sup>);  computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF).

The stabilizers of 1 and 2 points, ''M''<sub>23</sub> and ''M''<sub>22</sub>, also turn out to  be sporadic simple groups. The stabilizer of 3 points is simple and isomorphic to the [[projective special linear group]] PSL<sub>3</sub>(4).

These constructions were cited by {{harvtxt|Carmichael|1956|loc= pp. 151, 164, 263}}. {{harvtxt|Dixon|Mortimer|1996|loc=p.209}} ascribe the permutations to Mathieu.

=== Automorphism groups of Steiner systems ===

There exists [[up to]] [[Equivalence relation|equivalence]] a unique ''S''(5,8,24) [[Steiner system]] '''W<sub>24</sub>''' (the [[Witt design]]). The group ''M''<sub>24</sub> is the automorphism group of this Steiner system; that is, the set of permutations which map every block to some other block. The subgroups ''M''<sub>23</sub> and ''M''<sub>22</sub> are defined to be the stabilizers of a single point and two points respectively.

Similarly, there exists up to equivalence a unique ''S''(5,6,12) Steiner system '''W<sub>12</sub>''', and the group ''M''<sub>12</sub> is its automorphism group. The subgroup ''M''<sub>11</sub> is the stabilizer of a point.

''W''<sub>12</sub> can be constructed from the [[affine geometry]] on the [[vector space]] {{nowrap|'''F'''<sub>3</sub> × '''F'''<sub>3</sub>}}, an ''S''(2,3,9) system.

An alternative construction of ''W''<sub>12</sub> is the "Kitten" of {{Harvtxt|Curtis|1984}}.

An introduction to a construction of ''W''<sub>24</sub> via the [[Miracle Octad Generator]] of R. T. Curtis and Conway's analog for ''W''<sub>12</sub>, the miniMOG, can be found in the book by Conway and [[Neil Sloane|Sloane]].

=== Automorphism groups on the Golay code ===
The group ''M''<sub>24</sub> is the [[Mathieu group M24#Automorphism group of the Golay code|permutation automorphism group]] of the [[binary Golay code|extended binary Golay code]] ''W'', i.e., the group of permutations on the 24 coordinates that map ''W'' to itself. All the Mathieu groups can be constructed as groups of permutations on the binary Golay code.

''M''<sub>12</sub> has index 2 in its automorphism group, and ''M''<sub>12</sub>:2 happens to be isomorphic to a subgroup of ''M''<sub>24</sub>. ''M''<sub>12</sub> is the stabilizer of a '''dodecad''', a codeword of 12 1's; ''M''<sub>12</sub>:2 stabilizes a partition into 2 complementary dodecads.

There is a natural connection between the Mathieu groups and the larger [[Conway groups]], because the [[Leech lattice]] was constructed on the binary Golay code and in fact both lie in spaces of dimension 24. The Conway groups in turn are found in the [[Monster group]].  [[Robert Griess]] refers to the 20 sporadic groups found in the Monster as the '''Happy Family''', and to the Mathieu groups as the '''first generation'''.

===Dessins d'enfants===

The Mathieu groups can be constructed via [[dessins d'enfants]], with the dessin associated to ''M''<sub>12</sub> suggestively called "Monsieur Mathieu" by {{harvtxt|le Bruyn|2007}}.