# Mathieu groups

The Mathieu groups^{} are a of 5 sporadic simple groups discovered by the French mathematician Émile Léonard Mathieu. They are usually denoted by ${M}_{11}$, ${M}_{12}$, ${M}_{22}$, ${M}_{23}$, ${M}_{24}$. They are defined as automorphism groups of certain Steiner systems^{}, and the subscript denotes the size of the underlying set of the Steiner system.

If $\mathrm{\Omega}$ is a set of $n$ elements, then a $(t,k,n)$-Steiner system on $\mathrm{\Omega}$ is a set $S$ of subsets of $\mathrm{\Omega}$, each of size $k$, such that every subset of $\mathrm{\Omega}$ of size $t$ is contained in a unique element of $S$. The automorphism group of the Steiner system is defined as the permutations^{} of $\mathrm{\Omega}$ which map $S$ to itself.

There exists a (5,8,24)-Steiner system, and it is unique up to permutation of the elements of $\mathrm{\Omega}$. It can be constructed as the set of octads of the extended binary Golay Code ${\mathcal{G}}_{24}$. We denote it by $S(5,8,24)$ below.

There exists a (5,6,12)-Steiner system, and it is unique up to permutation of the elements. It can be constructed as follows. Take ${\mathrm{\Omega}}^{\prime}$ to be a dodecad (element of weight 12) of ${\mathcal{G}}_{24}$. Then the subsets of size 6 in ${\mathrm{\Omega}}^{\prime}$ which are contained in an octad of ${\mathcal{G}}_{24}$ form a (5,6,12)-Steiner system. We denote it by $S(5,6,12)$ below.

## 1 Definition of the Mathieu groups

The group ${M}_{24}$ is the automorphism group of $S(5,8,24)$. It has order $\mathrm{244\hspace{0.17em}823\hspace{0.17em}040}={2}^{10}\cdot {3}^{3}\cdot 5\cdot 7\cdot 11\cdot 23$.

The group ${M}_{23}$ is the subgroup^{} of ${M}_{24}$ fixing a given of $\mathrm{\Omega}$. It is the automorphism group of a (4,7,23)-Steiner system). It has order $|{M}_{24}|/24=\mathrm{10\hspace{0.17em}200\hspace{0.17em}960}={2}^{7}\cdot {3}^{2}\cdot 5\cdot 7\cdot 11\cdot 23$.

The group ${M}_{22}$ is the subgroup of ${M}_{24}$ fixing two given of $\mathrm{\Omega}$. It is a subgroup of index (http://planetmath.org/Coset) 2 in the automorphism group of a (3,6,22)-Steiner system. It has order $|{M}_{23}|/23=\mathrm{443\hspace{0.17em}250}={2}^{7}\cdot {3}^{2}\cdot 5\cdot 7\cdot 11$.

The group ${M}_{12}$ is the automorphism group of $S(5,6,12)$. It has order $\mathrm{95\hspace{0.17em}040}={2}^{6}\cdot {3}^{3}\cdot 5\cdot 11$.

The group ${M}_{11}$ is the subgroup of ${M}_{12}$ fixing a of ${\mathrm{\Omega}}^{\prime}$. It is the automorphism group of a (4,5,11)-Steiner system. It has order $|{M}_{12}/12|=\mathrm{7\hspace{0.17em}920}={2}^{4}\cdot {3}^{2}\cdot 5\cdot 11$.

*Note.* It is possible to continue the pattern above and define groups ${M}_{21}$, ${M}_{20}$, ${M}_{10}$, ${M}_{9}$, ${M}_{8}$. However, they are no longer sporadic simple groups. The group ${M}_{21}$ is a subgroup of 3! = 6 in the automorphism group of a (2,5,21)-Steiner system, which are the points and lines of the projective plane^{} over the field of 4 elements. In fact, ${M}_{21}\cong PSL(3,{\mathbb{F}}_{4})$, a simple group^{} of Lie type (http://planetmath.org/ProjectiveSpecialLinearGroup). The group ${M}_{20}$ is a solvable group^{}. The group ${M}_{10}$ is not simple; it contains, with 2, the alternating group^{} ${A}_{6}$ (sometimes denoted ${M}_{10}^{\prime}$ in this context since it is the derived subgroup of ${M}_{10}$). The groups ${M}_{9}$ and ${M}_{8}$ are solvable.

## References

- 1 J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1999.
- 2 Robert L. Griess, Jr. Twelve Sporadic Groups. Springer-Verlag, 1998.

Title | Mathieu groups |
---|---|

Canonical name | MathieuGroups |

Date of creation | 2013-03-22 18:43:33 |

Last modified on | 2013-03-22 18:43:33 |

Owner | monster (22721) |

Last modified by | monster (22721) |

Numerical id | 7 |

Author | monster (22721) |

Entry type | Definition |

Classification | msc 20B20 |

Classification | msc 20D08 |

Defines | ${M}_{24}$ |

Defines | ${M}_{23}$ |

Defines | ${M}_{22}$ |

Defines | ${M}_{12}$ |

Defines | ${M}_{11}$ |