Editing Mathieu group (section)

{{Short description|Five sporadic simple groups}}
{{Group theory sidebar |Finite}}

In [[group theory]], a topic in [[abstract algebra]], the '''Mathieu groups''' are the five [[sporadic simple group]]s [[Mathieu group M11|''M''<sub>11</sub>]], [[Mathieu group M12|''M''<sub>12</sub>]], [[Mathieu group M22|''M''<sub>22</sub>]], [[Mathieu group M23|''M''<sub>23</sub>]] and [[Mathieu group M24|''M''<sub>24</sub>]] introduced by {{harvs|txt |authorlink=Émile Léonard Mathieu |last=Mathieu |year1=1861 |year2=1873}}. They are multiply transitive [[permutation group]]s on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to be discovered.

Sometimes the notation ''M''<sub>8</sub>, ''M''<sub>9</sub>, ''M''<sub>10</sub>, ''M''<sub>20</sub>, and ''M''<sub>21</sub> is used for related groups (which act on sets of 8, 9,  10,  20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones.  [[John Horton Conway|John Conway]] has shown that one can also extend this sequence up, obtaining the [[Mathieu groupoid |Mathieu groupoid ''M''<sub>13</sub>]] acting on 13 points. ''M''<sub>21</sub> is simple, but is not a sporadic group, being isomorphic to [[Projective special linear group|PSL]](3,4).