Mathieu group
In mathematics, the Mathieu groups M_{11}, M_{12}, M_{22}, M_{23}, M_{24}, introduced by Template:Harvs, are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They were the first sporadic simple groups discovered.
Sometimes the notation M_{10}, M_{20} and M_{21} 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 M_{13} acting on 13 points. M_{21} is simple, but is not a sporadic group, being isomorphic to PSL(3,4).
History
Template:Harvtxt introduced the group M_{12} as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) the group M_{24}, giving its order. In Template:Harvtxt 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. Template:Harvtxt even published a paper mistakenly claiming to prove that M_{24} does not exist, though shortly afterwards in Template:Harv he pointed out that his proof was wrong, and gave a proof that the Mathieu groups are simple. Template:Harvs 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 J_{1} was discovered.
Multiply transitive groups
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 a_{1}, ... a_{k} and b_{1}, ... 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. 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).
M_{24} is 5-transitive, and M_{12} 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 (M_{23} is 4-transitive, etc.).
The only 4-transitive groups are the symmetric groups S_{k} for k at least 4, the alternating groups A_{k} for k at least 6, and the Mathieu groups M_{24}, M_{23}, M_{12} and M_{11}. Template:Harv 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 M_{12} and M_{11} 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,F_{q}), which is sharply 3-transitive (see cross ratio) on elements.
Order and transitivity table
Group | Order | Order (product) | Factorised order | Transitivity | Simple |
---|---|---|---|---|---|
M_{24} | 244823040 | 3·16·20·21·22·23·24 | 2^{10}·3^{3}·5·7·11·23 | 5-transitive | simple |
M_{23} | 10200960 | 3·16·20·21·22·23 | 2^{7}·3^{2}·5·7·11·23 | 4-transitive | simple |
M_{22} | 443520 | 3·16·20·21·22 | 2^{7}·3^{2}·5·7·11 | 3-transitive | simple |
M_{21} | 20160 | 3·16·20·21 | 2^{6}·3^{2}·5·7 | 2-transitive | simple |
M_{20} | 960 | 3·16·20 | 2^{6}·3·5 | 1-transitive | not simple |
M_{12} | 95040 | 8·9·10·11·12 | 2^{6}·3^{3}·5·11 | sharply 5-transitive | simple |
M_{11} | 7920 | 8·9·10·11 | 2^{4}·3^{2}·5·11 | sharply 4-transitive | simple |
M_{10} | 720 | 8·9·10 | 2^{4}·3^{2}·5 | sharply 3-transitive | not simple |
Constructions of the Mathieu groups
The Mathieu groups can be constructed in various ways.
Permutation groups
M_{12} has a simple subgroup of order 660, a maximal subgroup. That subgroup can be represented as a linear fractional group on the field F_{11} 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 M_{12} sends an element x of F_{11} to 4x^{2} − 3x^{7}; as a permutation that is (26a7)(3945). The stabilizer of 4 points is a quaternion group.
Likewise M_{24} has a maximal simple subgroup of order 6072 and this can be represented as a linear fractional group on the field F_{23}. 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 M_{24} sends an element x of F_{23} to 4x^{4} − 3x^{15} (which sends perfect squares via and non-perfect squares via ); computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF).
These constructions were cited by Template:Harvtxt. Template:Harvtxt ascribe the permutations to Mathieu.
Automorphism groups of Steiner systems
There exists up to equivalence a unique S(5,8,24) Steiner system W_{24} (the Witt design). The group M_{24} is the automorphism group of this Steiner system; that is, the set of permutations which map every block to some other block. The subgroups M_{23} and M_{22} 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 W_{12}, and the group M_{12} is its automorphism group. The subgroup M_{11} is the stabilizer of a point.
W_{12} can be constructed from the affine geometry on the vector space F_{3}xF_{3}, an S(2,3,9) system.
An alternative construction of W_{12} is the 'Kitten' of Template:Harvtxt.
An introduction to a construction of W_{24} via the Miracle Octad Generator of R. T. Curtis and Conway's analog for W_{12}, the miniMOG, can be found in the book by Conway and Sloane.
Automorphism group of the Golay code
The group M_{24} 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 F_{2} spanned by the octads of the Steiner system.
The simple subgroups M_{23}, M_{22}, M_{12}, and M_{11} can be defined as subgroups of M_{24}, stabilizers respectively of a single coordinate, an ordered pair of coordinates, a dodecad, and a dodecad together with a single coordinate.
M_{12} has index 2 in its automorphism group. As a subgroup of M_{24}, M_{12} acts on the second dodecad as an outer automorphic image of its action on the first dodecad. M_{11} is a subgroup of M_{23} but not of M_{22}. This representation of M_{11} has orbits of 11 and 12. The automorphism group of M_{12} is a maximal subgroup of M_{24} 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
The Mathieu groups can be constructed via dessins d'enfants, with the dessin associated to M_{12} suggestively called "Monsieur Mathieu" by Template:Harvtxt.
References
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External links
- ATLAS: Mathieu group M_{10}
- ATLAS: Mathieu group M_{11}
- ATLAS: Mathieu group M_{12}
- ATLAS: Mathieu group M_{20}
- ATLAS: Mathieu group M_{21}
- ATLAS: Mathieu group M_{22}
- ATLAS: Mathieu group M_{23}
- ATLAS: Mathieu group M_{24}
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- Scientific American A set of puzzles based on the mathematics of the Mathieu groups
- Sporadic M12 An iPhone app that implements puzzles based on M_{12}, presented as one "spin" permutation and a selectable "swap" permutation