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The Mathieu groups are a series 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 $\gc$ . 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 $\Omega'$ to be a dodecad (element of weight 12) of $\gc$ . Then the subsets of size 6 in $\Omega'$ which are contained in an octad of $\gc$ form a (5,6,12)-Steiner system. We denote it by $S(5,6,12)$ below.
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 point 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 points of $\Omega$ . It is a subgroup of index 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 point of $\Omega'$ . 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 information on the Mathieu groups, consult the references ([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 index 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. The group $M_{20}$ is a solvable group. The group $M_{10}$ is not simple; it contains, with index 2, the alternating group $A_6$ (sometimes denoted $M_{10}'$ in this context since it is the derived subgroup of
$M_{10}$ ). The groups $M_9$ and $M_8$ are solvable.
- 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.
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