# 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 *M*_{24}, *M*_{23}, *M*_{22}, *M*_{12} and *M*_{11}, 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 |
---|---|---|

M_{24} |
5-fold | 24.23.22.21.20.48 = 244,823,040 |

M_{23} |
4-fold | 23.22.21.20.48 = 10,200,960 |

M_{22} |
3-fold | 22.21.20.48 = 443,520 |

M_{12} |
5-fold | 12.11.10.9.8 = 95,040 |

M_{11} |
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 *M*_{24} the subgroup fixing one symbol is *M*_{23}. In *M*_{23} it is *M*_{22}, and in *M*_{12} it is *M*_{11}. Each Mathieu group is ‘simple’, meaning it cannot be deconstructed into anything simpler. In *M*_{22}, the subgroup fixing one symbol — which we may denote *M*_{21} — 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, *M*_{11} and *M*_{12} are subgroups of *M*_{23} and *M*_{24} respectively, but note that *M*_{11} is not a subgroup of *M*_{22}.

The easiest way to approach the Mathieu groups is to start with the largest one, namely *M*_{24}, and work downwards. *M*_{24} is the symmetry group of the Witt design, and can also be viewed as the symmetry group of the Golay code.