Mathieu group

In mathematics, the Mathieu groups are five finite simple groupss 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 M_n.

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, ... a_k and b₁, ... b_k with the property that all the a_i are distinct and all the b_i are distinct, there is a group element g in G which maps a_i to b_i for each i between 1 and k.

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.

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.

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.

References

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