Mathieu group

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In mathematics, the Mathieu groups M11, M12, M22, M23, M24, introduced by Mathieu (1861, 1873), are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They were the first sporadic simple groups discovered.

Sometimes the notation M10, M20 and M21 is used for related groups (which act on sets of 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones. John Conway has shown that one can also extend this sequence up, obtaining the Mathieu groupoid M13 acting on 13 points. M21 is simple, but is not a sporadic group, being isomorphic to PSL(3,4).

History[edit]

Mathieu (1861, p.271) introduced the group M12 as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) the group M24, giving its order. In Mathieu (1873) he gave further details, including explicit generating sets for his groups, but it was not easy to see from his arguments that the groups generated are not just alternating groups, and for several years the existence of his groups was controversial. Miller (1898) even published a paper mistakenly claiming to prove that M24 does not exist, though shortly afterwards in (Miller 1900) he pointed out that his proof was wrong, and gave a proof that the Mathieu groups are simple. Witt (1938a, 1938b) finally removed the doubts about the existence of these groups, by constructing them as automorphism groups of Steiner systems.

After the Mathieu groups no new sporadic groups were found until 1965, when the group J1 was discovered.

Multiply transitive groups[edit]

Mathieu was interested in finding multiply transitive permutation groups, which will now be defined. 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).

M24 is 5-transitive, and M12 is sharply 5-transitive, with the other Mathieu groups (simple or not) being the subgroups corresponding to stabilizers of m points, and accordingly of lower transitivity (M23 is 4-transitive, etc.).

The only 4-transitive groups are the symmetric groups Sk for k at least 4, the alternating groups Ak for k at least 6, and the Mathieu groups M24, M23, M12 and M11. (Cameron 1999, p. 110) The full proof requires the classification of finite simple groups, but some special cases have been known for much longer.

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.

Important examples of multiply transitive groups are the 2-transitive groups and the Zassenhaus groups. The Zassenhaus groups notably include the projective general linear group of a projective line over a finite field, PGL(2,Fq), which is sharply 3-transitive (see cross ratio) on q+1 elements.

Order and transitivity table[edit]

Group Order Order (product) Factorised order Transitivity Simple
M24 244823040 3·16·20·21·22·23·24 210·33·5·7·11·23 5-transitive simple
M23 10200960 3·16·20·21·22·23 27·32·5·7·11·23 4-transitive simple
M22 443520 3·16·20·21·22 27·32·5·7·11 3-transitive simple
M21 20160 3·16·20·21 26·32·5·7 2-transitive simple
M20 960 3·16·20 26·3·5 1-transitive not simple
M12 95040 8·9·10·11·12 26·33·5·11 sharply 5-transitive simple
M11 7920 8·9·10·11 24·32·5·11 sharply 4-transitive simple
M10 720 8·9·10 24·32·5 sharply 3-transitive not simple

Constructions of the Mathieu groups[edit]

The Mathieu groups can be constructed in various ways.

Permutation groups[edit]

M12 has a simple subgroup of order 660, a maximal subgroup. That subgroup can be represented as a linear fractional group on the field F11 of 11 elements. With −1 written as a and infinity as b, two standard generators are (0123456789a) and (0b)(1a)(25)(37)(48)(69). A third generator giving M12 sends an element x of F11 to 4x2 − 3x7; as a permutation that is (26a7)(3945). The stabilizer of 4 points is a quaternion group.

Likewise M24 has a maximal simple subgroup of order 6072 and this can be represented as a linear fractional group on the field F23. One generator adds 1 to each element (leaving the point N at infinity fixed), i. e. (0123456789ABCDEFGHIJKLM)(N), and the other is the order reversing permutation, (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI). A third generator giving M24 sends an element x of F23 to 4x4 − 3x15 (which sends perfect squares via  x^4 and non-perfect squares via  7 x^4); computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF).

These constructions were cited by Carmichael (1956, pp. 151, 164, 263). Dixon & Mortimer (1996, p.209) ascribe the permutations to Mathieu.

Automorphism groups of Steiner systems[edit]

There exists up to equivalence a unique S(5,8,24) Steiner system W24 (the Witt design). The group M24 is the automorphism group of this Steiner system; that is, the set of permutations which map 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 S(5,6,12) Steiner system W12, and the group M12 is its automorphism group. The subgroup M11 is the stabilizer of a point.

W12 can be constructed from the affine geometry on the vector space F3xF3, an S(2,3,9) system.

An alternative construction of W12 is the 'Kitten' of Curtis (1984).

An introduction to a construction of W24 via the Miracle Octad Generator of R. T. Curtis and Conway's analog for W12, the miniMOG, can be found in the book by Conway and Sloane.

Automorphism group of the Golay code[edit]

The group M24 also is the permutation automorphism group of the binary Golay code W, i.e., the group of permutations of coordinates mapping W to itself. (In coding theory the term "binary Golay code" often refers to a shorter related length 23 code, and the length 24 code used here is called the "extended binary Golay code".) 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 the blocks of an S(5,8,24) Steiner system and the binary Golay code is the vector space over field F2 spanned by the octads of the 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 dodecad, and a dodecad 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 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.

Dessins d'enfants[edit]

The Mathieu groups can be constructed via dessins d'enfants, with the dessin associated to M12 suggestively called "Monsieur Mathieu" by le Bruyn (2007).

References[edit]

External links[edit]

  • Scientific American A set of puzzles based on the mathematics of the Mathieu groups
  • Sporadic M12 An iPhone app that implements puzzles based on M12, presented as one "spin" permutation and a selectable "swap" permutation