John Conway has shown that one can also extend this sequence up, obtaining the Mathieu groupoid M13 acting on 13 points.
Miller (1898) even published a paper mistakenly claiming to prove that M24 does not exist, though shortly afterwards in (Miller 1900) he pointed out that his proof was wrong, and gave a proof that the Mathieu groups are simple.
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 a1, ... ak and b1, ... bk with the property that all the ai are distinct and all the bi are distinct, there is a group element g in G which maps ai to bi 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 regular, rather than just transitive).
M24 is 5-transitive, and M12 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 (M23 is 4-transitive, etc.).
(Cameron 1999, p. 110) The full proof requires the classification of finite simple groups, but some special cases have been known for much longer.
The Zassenhaus groups notably include the projective general linear group of a projective line over a finite field, PGL(2,Fq), which is sharply 3-transitive (see cross ratio) on
That subgroup is isomorphic to the projective special linear group PSL2(F11) over the field of 11 elements.
The stabilizer of 3 points is the projective special unitary group PSU(3,22), which is solvable.
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 −1/x, i.e. (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI).
The stabilizer of 3 points is simple and isomorphic to the projective special linear group PSL3(4).