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

There exists a (5,8,24)-Steiner system, and it is unique up to permutation of the elements of $\Omega$. 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 $\Omega^{\prime}$ to be a dodecad (element of weight 12) of $\mathcal{G}_{24}$. Then the subsets of size 6 in $\Omega^{\prime}$ 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 $244\,823\,040=2^{10}\cdot 3^{3}\cdot 5\cdot 7\cdot 11\cdot 23$.

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

The group $M_{22}$ is the subgroup of $M_{24}$ fixing two given of $\Omega$. 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 $|M_{23}|/23=443\,250=2^{7}\cdot 3^{2}\cdot 5\cdot 7\cdot 11$.

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

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

For further on the Mathieu groups, consult the ([1], Chapters 10 and 11) and ([2], Chapters 5-7).

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 MathieuGroups 2013-03-22 18:43:33 2013-03-22 18:43:33 monster (22721) monster (22721) 7 monster (22721) Definition msc 20B20 msc 20D08 $M_{24}$ $M_{23}$ $M_{22}$ $M_{12}$ $M_{11}$