Mathieu groups

automorphism group automorphism 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 ${𝒢}_{24}$. 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 ${𝒢}_{24}$. Then the subsets of size 6 in ${\mathrm{\Omega }}^{\prime }$ which are contained in an octad of ${𝒢}_{24}$ 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 $|M_{12}/12|=\mathrm{7\hspace{0.17em}920}=2^4\cdot 3^2\cdot 5\cdot 11$.

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, $M_{21}\cong PSL(3,{𝔽}_4)$, 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.

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	$M_{11}$