Editing Mathieu group (section)

== Multiply transitive groups ==

Mathieu was interested in finding '''multiply transitive''' permutation groups, which will now be defined. For a natural number ''k'', a permutation group ''G'' acting on ''n'' points is ''' ''k''-transitive''' if, given two sets of points ''a''<sub>1</sub>, ... ''a''<sub>''k''</sub> and ''b''<sub>1</sub>, ... ''b''<sub>''k''</sub> with the property that all the ''a''<sub>''i''</sub> are distinct and all the ''b''<sub>''i''</sub> are distinct, there is a group element ''g'' in ''G'' which maps ''a''<sub>''i''</sub> to ''b''<sub>''i''</sub> for each ''i'' between 1 and ''k''.  Such a group is called '''sharply ''k''-transitive''' if the element ''g'' is unique (i.e. the action on ''k''-tuples is [[Group action (mathematics)#Remarkable properties of actions|regular]], rather than just transitive).

''M''<sub>24</sub> is 5-transitive, and ''M''<sub>12</sub> is sharply 5-transitive, with the other Mathieu groups (simple or not) being the subgroups corresponding to stabilizers of ''m'' points, and accordingly of lower transitivity (''M''<sub>23</sub> is 4-transitive, etc.). These are the only two 5-transitive groups that are neither [[symmetric group]]s nor [[alternating group]]s {{harv|Cameron|1992|loc= p. 139}}.

The only 4-transitive groups are the [[symmetric group]]s ''S''<sub>''k''</sub> for ''k'' at least 4, the [[alternating group]]s ''A''<sub>''k''</sub> for ''k'' at least 6, and the Mathieu groups [[Mathieu group M24|''M''<sub>24</sub>]], [[Mathieu group M23|''M''<sub>23</sub>]], [[Mathieu group M12|''M''<sub>12</sub>]], and [[Mathieu group M11|''M''<sub>11</sub>]]. {{harv|Cameron|1999|loc= p. 110}} The full proof requires the [[classification of finite simple groups]], but some special cases have been known for much longer.

It is [[Jordan's theorem (symmetric group)|a classical result of Jordan]] that the [[symmetric group|symmetric]] and [[alternating group]]s (of degree ''k'' and ''k''&nbsp;+&nbsp;2 respectively), and ''M''<sub>12</sub> and ''M''<sub>11</sub> are the only ''sharply'' ''k''-transitive permutation groups for ''k'' at least&nbsp;4.

Important examples of multiply transitive groups are the [[2-transitive group]]s and the [[Zassenhaus group]]s. The Zassenhaus groups notably include the [[projective general linear group]] of a projective line over a finite field, PGL(2,'''F'''<sub>''q''</sub>), which is sharply 3-transitive (see [[cross ratio]]) on <math>q+1</math> elements.

=== Order and transitivity table ===
{| class="wikitable"
! Group
! Order
! Order (product)
! Factorised order
! Transitivity
! Simple
! Sporadic
|-
| ''M''<sub>24</sub>
| 244823040
| 3·16·20·21·22·23·24
| 2<sup>10</sup>·3<sup>3</sup>·5·7·11·23
| 5-transitive
| yes
| sporadic
|-
| ''M''<sub>23</sub>
| 10200960
| 3·16·20·21·22·23
| 2<sup>7</sup>·3<sup>2</sup>·5·7·11·23
| 4-transitive
| yes
| sporadic
|-
| ''M''<sub>22</sub>
| 443520
| 3·16·20·21·22
| 2<sup>7</sup>·3<sup>2</sup>·5·7·11
| 3-transitive
| yes
| sporadic
|-
| ''M''<sub>21</sub>
| 20160
| 3·16·20·21
| 2<sup>6</sup>·3<sup>2</sup>·5·7
| 2-transitive
| yes
| ≈ [[projective special linear group#Finite fields|PSL]]<sub>3</sub>(4)
|-
| ''M''<sub>20</sub>
| 960
| 3·16·20
| 2<sup>6</sup>·3·5
| 1-transitive
| no
| ≈2<sup>4</sup>:A<sub>5</sub>
|-
| colspan = "7" |
|-
| ''M''<sub>12</sub>
| 95040
| 8·9·10·11·12
| 2<sup>6</sup>·3<sup>3</sup>·5·11
| sharply 5-transitive
| yes
| sporadic
|- 
| ''M''<sub>11</sub>
| 7920
| 8·9·10·11
| 2<sup>4</sup>·3<sup>2</sup>·5·11
| sharply 4-transitive
| yes
| sporadic
|- 
| ''M''<sub>10</sub>
| 720
| 8·9·10
| 2<sup>4</sup>·3<sup>2</sup>·5
| sharply 3-transitive
| [[almost simple group|almost]]
| ''M''<sub>10</sub>' ≈ [[Alternating group|Alt]]<sub>6</sub>
|- 
| ''M''<sub>9</sub>
| 72
| 8·9
| 2<sup>3</sup>·3<sup>2</sup>
| sharply 2-transitive
| no
| ≈ [[projective special unitary group#Finite fields|PSU]]<sub>3</sub>(2)
|- 
| ''M''<sub>8</sub>
| 8
| 8
| 2<sup>3</sup>
| sharply 1-transitive (regular)
| no
| ≈ [[quaternion group|Q]]
|}