Mathieu group
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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 Mn.
The Mathieu groups were the first of the sporadic groups to be discovered.
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[edit] 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 a1, ... ak and b1, ... bk with the property that all the ai are distinct and all the bi are distinct, there is a group element g in G which maps ai to bi for each i between 1 and k. Such a group is called sharply k-transitive if the element g is unique (i.e. the action on k-tuples is regular, rather than just transitive).
The groups M24 and M12 are 5-transitive, the groups M23 and M11 are 4-transitive, and M22 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 M24, M23, M12 and M11.
It is a classical result of Jordan that the symmetric and alternating groups (of degree k and k − 2 respectively), and M12 and M11 are the only sharply k-transitive permutation groups for k at least 4.
[edit] Orders
| Group | Order | Factorised order |
|---|---|---|
| M24 | 244823040 | 210·33·5·7·11·23 |
| M23 | 10200960 | 27·32·5·7·11·23 |
| M22 | 443520 | 27·32·5·7·11 |
| M12 | 95040 | 26·33·5·11 |
| M11 | 7920 | 24·32·5·11 |
[edit] Two constructions of the Mathieu groups
[edit] Automorphism group of Steiner systems
There exists up to equivalence a unique Steiner system S(5,8,24). The group M24 is the automorphism group of this Steiner system; that is, the set of permutations which maps every block to some other block. The subgroups M23 and M22 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 M12 is its automorphism group. The subgroup M11 is the stabilizer of a point.
For an introduction to a construction of M24 as the automorphism group of S(5,8,24) via the Miracle Octad Generator of R. T. Curtis, see Geometry of the 4x4 Square. Another good account of this and Conway's analog for S(5,6,12), the miniMOG, may be found in the book by Conway and Sloane.
An alternative construction of S(5,6,12) is R.T. Curtis' Kitten[citation needed].
[edit] Automorphism group of the Golay code
The group M24 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 S24 and Stab(W) in Aut(V). Codewords correspond in a natural way to subsets of a set of 24 objects. Those subsets corresponding to codewords with 8 or 12 coordinates equal to 1 are called octads or dodecads respectively. The octads are blocks of an S(5,8,24) Steiner system.
The simple subgroups M23, M22, M12, and M11 can be defined as subgroups of M24, stabilizers respectively of a single coordinate, an ordered pair of coordinates, a pair of complementary dodecads, and a dodecad pair together with a single coordinate.
M12 has index 2 in its automorphism group. As a subgroup of M24, M12 acts on the second dodecad as an outer automorphic image of its action on the first dodecad. M11 is a subgroup of M23 but not of M22. This representation of M11 has orbits of 11 and 12. The automorphism group of M12 is a maximal subgroup of M24 of index 1288.
There is a very natural connection between the Mathieu groups and the larger Conway groups, because the binary Golay code and the Leech lattice both lie in spaces of dimension 24. The Conway groups in turn are found in the Monster group. Robert Griess refers to the 20 sporadic groups found in the Monster as the Happy Family, and to the Mathieu groups as the first generation.
[edit] 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
[edit] 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
- Conway, J.H.; Sloane N.J.A. Sphere Packings, Lattices and Groups: v. 290 (Grundlehren Der Mathematischen Wissenschaften.) Springer Verlag. ISBN 0-387-98585-9
- Curtis, R. T. A new combinatorial approach to M24. Math. Proc. Camb. Phil. Soc. 79 (1976) 25-42.
- Thompson, Thomas M.: "From Error Correcting Codes through Sphere Packings to Simple Groups", Carus Mathematical Monographs, Mathematical Association of America, 1983.
- Curtis, R. T. "The Steiner System S(5,6,12), the Mathieu Group M12 and the 'Kitten' ," Computational Group Theory, Academic Press, London, 1984
- Griess, Robert L.: "Twelve Sporadic Groups", Springer-Verlag, 1998.
