Mathieu Groups
In the early 1860s the French mathematician, Émile Mathieu discovered some remarkable groups of permutations, now called the Mathieu groups. They are denoted M24, M23, M22, M12 and M11, where the subscript indicates the number of symbols being permuted. Each one is multiply transitive, or more precisely n-fold transitive, meaning that any n-tuple of symbols can be carried to any other by a suitable permutation in the group.
Here is the size and level of transitivity of each of these five groups.
| Group | Level of transivity | Size |
|---|---|---|
| M24 | 5-fold | 24.23.22.21.20.48 = 244,823,040 |
| M23 | 4-fold | 23.22.21.20.48 = 10,200,960 |
| M22 | 3-fold | 22.21.20.48 = 443,520 |
| M12 | 5-fold | 12.11.10.9.8 = 95,040 |
| M11 | 4-fold | 11.10.9.8 = 7,920 |
Apart from these Mathieu groups, there is no finite group that is 4- or 5-fold transitive unless it contains all even permutations (and is therefore a symmetric or alternating group), but this fact has never been proved directly. It is only known as a consequence of the classification of finite simple groups, described in my book Symmetry and the Monster.
In M24 the subgroup fixing one symbol is M23. In M23 it is M22, and in M12 it is M11. Each Mathieu group is ‘simple’, meaning it cannot be deconstructed into anything simpler. In M22, the subgroup fixing one symbol — which we may denote M21 — is also a simple group, but unlike the other five it is not a sporadic group. It is isomorphic to a group of Lie type. The two smaller Mathieu groups, M11 and M12 are subgroups of M23 and M24 respectively, but note that M11 is not a subgroup of M22.
The easiest way to approach the Mathieu groups is to start with the largest one, namely M24, and work downwards. M24 is the symmetry group of the Witt design, and can also be viewed as the symmetry group of the Golay code.