Editing Mathieu group (section)

===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.