Mathieu groups, M24, M23, M22, M12, M11

In the early 1860s the French mathematician, …mile Mathieu discovered some remarkable groups of permutations, now called the Mathieu groups. They are denoted M₂₄, M₂₃, M₂₂, M₁₂ and M₁₁, where the notation M_s indicates a permutation group on s symbols. 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 transitivity	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.

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.