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.

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.

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

Two constructions of the Mathieu groups

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.

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).

The simple subgroups M23, M22, M12, and M11 can be defined as the stabilizers in M24 of a single coordinate, an ordered pair of coordinates, a 12-element subset of the coordinates corresponding to a code word, and a 12-element code word together with a single coordinate, respectively.

External links

  • Moggie Java applet for studying the Curtis MOG construction

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
  • Curtis, R. T. A new combinatorial approach to M24. Math. Proc. Camb. Phil. Soc. 79 (1976) 25-42.

it:Gruppo di Mathieu