# Mathieu group

In mathematics, the **Mathieu groups** are five finite simple groups discovered by the French mathematician Emile Léonard Mathieu. They are usually thought of as permutation groups on *n* points (where *n* can take the values 11, 12, 22, 23 or 24) and are named M_{n}.

The Mathieu groups were the first of the sporadic groups to be discovered.

## Contents

## Multiply transitive groups

The Mathieu groups are examples of **multiply transitive** groups. For a natural number *k*, a permutation group *G* acting on *n* points is ** k-transitive** if, given two sets of points

*a*

_{1}, ...

*a*

_{k}and

*b*

_{1}, ...

*b*

_{k}with the property that all the

*a*

_{i}are distinct and all the

*b*

_{i}are distinct, there is a group element

*g*in

*G*which maps

*a*

_{i}to

*b*

_{i}for each

*i*between 1 and

*k*. Such a group is called

**sharply**if the element

*k*-transitive*g*is unique (i.e. the action on

*k*-tuples is regular, rather than just transitive).

The groups M_{24} and M_{12} are 5-transitive, the groups M_{23} and M_{11} are 4-transitive, and M_{22} is 3-transitive.

It follows from the classification of finite simple groups that the only groups which are *k*-transitive for *k* at least 4 are the symmetric and alternating groups (of degree *k* and *k*-2 respectively) and the Mathieu groups M_{24}, M_{23}, M_{12} and M_{11}.

It is a classical result of Jordan that the symmetric and alternating groups (of degree *k* and *k*-2 respectively), and M_{12} and M_{11} are the only *sharply* *k*-transitive permutation groups for *k* at least 4.

## Orders

Group | Order | Factorised order |
---|---|---|

M_{24} |
244823040 | 2^{10}.3^{3}.5.7.11.23 |

M_{23} |
10200960 | 2^{7}.3^{2}.5.7.11.23 |

M_{22} |
443520 | 2^{7}.3^{2}.5.7.11 |

M_{12} |
95040 | 2^{6}.3^{3}.5.11 |

M_{11} |
7920 | 2^{4}.3^{2}.5.11 |

## Two constructions of the Mathieu groups

### Automorphism group of Steiner systems

There exists up to equivalence a unique Steiner system S(5,8,24). The group M_{24} is the automorphism group of
this Steiner system; that is, the set of permutations which maps every block to some other block. The subgroups M_{23} and M_{22} are defined to be the stabilizers of a single point and two points respectively.

Similarly, there exists up to equivalence a unique Steiner system S(5,6,12), and the group M_{12} is its
automorphism group. The subgroup M_{11} is the stabilizer of a point.

For an introduction to a construction of M_{24} as the automorphism group of S(5,8,24) via the Miracle Octad Generator of R. T. Curtis, see Geometry of the 4x4 Square.

### Automorphism group of the Golay code

The group M_{24} can also be thought of as the automorphism group of the binary Golay code *W*, i.e., the group of permutations of coordinates mapping *W* to itself. We can also regard it as the intersection of S_{24} and Stab(*W*) in Aut(*V*).

The simple subgroups M_{23}, M_{22}, M_{12}, and M_{11} can be defined as the stabilizers in M_{24} of a single coordinate, an ordered pair of coordinates, a 12-element subset of the coordinates corresponding to a code word, and a 12-element code word together with a single coordinate, respectively.

## External links

- Geometry of the 4x4 square Describes the Curtis "miracle octad generator" (MOG) construction of S(5,8,24).

- Moggie Java applet for studying the Curtis MOG construction

## References

- Mathieu E.,
*Sur la fonction cinq fois transitive de 24 quantités*, Liouville Journ., (2) XVIII., 1873, pp. 25-47.

- Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985).
*Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray.*Eynsham: Oxford University Press. ISBN 0-19-853199-0

- Curtis, R. T. A new combinatorial approach to M
_{24}. Math. Proc. Camb. Phil. Soc.**79**(1976) 25-42.