Mathieu group
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Modular groups

Infinite dimensional Lie group

In the area of modern algebra known as group theory, the Mathieu groups are the five sporadic simple groups M_{11}, M_{12}, M_{22}, M_{23} and M_{24} introduced by Mathieu (1861, 1873). They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They were the first sporadic groups to be discovered.
Sometimes the notation M_{9}, M_{10}, M_{20} and M_{21} is used for related groups (which act on sets of 9, 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
Mathieu (1861, p.271) 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 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 M_{24} 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 successive transitive extensions of permutation groups, as well 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 ktransitive 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 ktransitive if the element g is unique (i.e. the action on ktuples is regular, rather than just transitive).
M_{24} is 5transitive, and M_{12} is sharply 5transitive, 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 4transitive, etc.).
The only 4transitive 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}. (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 M_{12} and M_{11} are the only sharply ktransitive permutation groups for k at least 4.
Important examples of multiply transitive groups are the 2transitive 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 3transitive (see cross ratio) on elements.
Order and transitivity table
Group  Order  Order (product)  Factorised order  Transitivity  Simple  Sporadic 

M_{24}  244823040  3·16·20·21·22·23·24  2^{10}·3^{3}·5·7·11·23  5transitive  yes  sporadic 
M_{23}  10200960  3·16·20·21·22·23  2^{7}·3^{2}·5·7·11·23  4transitive  yes  sporadic 
M_{22}  443520  3·16·20·21·22  2^{7}·3^{2}·5·7·11  3transitive  yes  sporadic 
M_{21}  20160  3·16·20·21  2^{6}·3^{2}·5·7  2transitive  yes  ≈ PSL_{3}(4) 
M_{20}  960  3·16·20  2^{6}·3·5  1transitive  no  
M_{12}  95040  8·9·10·11·12  2^{6}·3^{3}·5·11  sharply 5transitive  yes  sporadic 
M_{11}  7920  8·9·10·11  2^{4}·3^{2}·5·11  sharply 4transitive  yes  sporadic 
M_{10}  720  8·9·10  2^{4}·3^{2}·5  sharply 3transitive  almost  M_{10}' ≈ Alt_{6} 
M_{9}  72  8·9  2^{3}·3^{2}  sharply 2transitive  no  ≈ PSU_{3}(2^{2}) 
M_{8}  8  8  2^{3}  sharply 1transitive (regular)  no  ≈ Q 
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 is isomorphic to the projective special linear group PSL_{2}(F_{11}) over the field 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).
This group turns out not to be isomorphic to any member of the infinite families of finite simple groups and is called sporadic. M_{11} is the stabilizer of a point in M_{12}, and turns out also to be a sporadic simple group. M_{10}, the stabilizer of two points, is not sporadic, but is an almost simple group whose commutator subgroup is the alternating group A_{6}. It is thus related to the exceptional outer automorphism of A_{6}. The stabilizer of 3 points is the projective special unitary group PSU(3,2^{2}), which is solvable. The stabilizer of 4 points is the quaternion group.
Likewise, M_{24} has a maximal simple subgroup of order 6072 isomorphic to PSL_{2}(F_{23}). One generator adds 1 to each element of the field (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 nonperfect squares via ); computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF).
The stabilizers of 1 and 2 points, M_{23} and M_{22} also turn out ot be sporadic simple groups. The stabilizer of 3 points is simple and isomorphic to the projective special linear group PSL_{3}(4).
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
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}×F_{3}, an S(2,3,9) system.
An alternative construction of W_{12} is the 'Kitten' of Curtis (1984).
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 groups on the Golay code
The group M_{24} is the permutation automorphism group of the extended binary Golay code W, i.e., the group of permutations on the 24 coordinates that map W to itself. All the Mathieu groups can be constructed as groups of permutations on the binary Golay code.
M_{12} has index 2 in its automorphism group, and M_{12}:2 happens to be isomorphic to a subgroup of M_{24}. M_{12} is the stabilizer of a dodecad, a codeword of 12 1's; M_{12}:2 stabilizes a partition into 2 complementary dodecads.
There is a natural connection between the Mathieu groups and the larger Conway groups, because the Leech lattice was constructed on the binary Golay code and in fact 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 le Bruyn (2007).
References
 Cameron, Peter J. (1999), Permutation Groups, London Mathematical Society Student Texts, 45, Cambridge University Press, ISBN 9780521653787
 Carmichael, Robert D. (1956) [1937], Introduction to the theory of groups of finite order, New York: Dover Publications, ISBN 9780486603001, MR 0075938
 Choi, C. (May 1972a), "On Subgroups of M_{24}. I: Stabilizers of Subsets", Transactions of the American Mathematical Society, American Mathematical Society, 167: 1–27, doi:10.2307/1996123, JSTOR 1996123
 Choi, C. (May 1972b). "On Subgroups of M_{24}. II: the Maximal Subgroups of M_{24}". Transactions of the American Mathematical Society. American Mathematical Society. 167: 29–47. doi:10.2307/1996124. JSTOR 1996124.
 Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham, Finite simple groups, Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 215–247, ISBN 9780125638500, MR 0338152 Reprinted in Conway & Sloane (1999, 267–298)
 Conway, John Horton; Parker, Richard A.; Norton, Simon P.; Curtis, R. T.; Wilson, Robert A. (1985), Atlas of finite groups, Oxford University Press, ISBN 9780198531999, MR 827219
 Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, 290 (3rd ed.), Berlin, New York: SpringerVerlag, ISBN 9780387985855, MR 0920369
 Curtis, R. T. (1976), "A new combinatorial approach to M₂₄", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (1): 25–42, doi:10.1017/S0305004100052075, ISSN 03050041, MR 0399247
 Curtis, R. T. (1977), "The maximal subgroups of M₂₄", Mathematical Proceedings of the Cambridge Philosophical Society, 81 (2): 185–192, doi:10.1017/S0305004100053251, ISSN 03050041, MR 0439926
 Curtis, R. T. (1984), "The Steiner system S(5, 6, 12), the Mathieu group M₁₂ and the "kitten"", in Atkinson, Michael D., Computational group theory. Proceedings of the London Mathematical Society symposium held in Durham, July 30–August 9, 1982., Boston, MA: Academic Press, pp. 353–358, ISBN 9780120662708, MR 760669
 Cuypers, Hans, The Mathieu groups and their geometries (PDF)
 Dixon, John D.; Mortimer, Brian (1996), Permutation groups, Graduate Texts in Mathematics, 163, Berlin, New York: SpringerVerlag, doi:10.1007/9781461207313, ISBN 9780387945996, MR 1409812
 Frobenius, Ferdinand Georg (1904), Über die Charaktere der mehrfach transitiven Gruppen, Berline Berichte, Mouton De Gruyter, pp. 558–571, ISBN 9783111097909
 Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: SpringerVerlag, ISBN 9783540627784, MR 1707296
 Mathieu, Émile (1861), "Mémoire sur l'étude des fonctions de plusieurs quantités, sur la manière de les former et sur les substitutions qui les laissent invariables", Journal de Mathématiques Pures et Appliquées, 6: 241–323
 Mathieu, Émile (1873), "Sur la fonction cinq fois transitive de 24 quantités", Journal de Mathématiques Pures et Appliquées (in French), 18: 25–46, JFM 05.0088.01
 Miller, G. A. (1898), "On the supposed fivefold transitive function of 24 elements and 19!/48 values.", Messenger of Mathematics, 27: 187–190
 Miller, G. A. (1900), "Sur plusieurs groupes simples", Bulletin de la Société Mathématique de France, 28: 266–267
 Ronan, Mark (2006), Symmetry and the Monster, Oxford, ISBN 9780192807229 (an introduction for the lay reader, describing the Mathieu groups in a historical context)
 Thompson, Thomas M. (1983), From errorcorrecting codes through sphere packings to simple groups, Carus Mathematical Monographs, 21, Mathematical Association of America, ISBN 9780883850237, MR 749038
 Witt, Ernst (1938a), "über Steinersche Systeme", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Springer Berlin / Heidelberg, 12: 265–275, doi:10.1007/BF02948948, ISSN 00255858
 Witt, Ernst (1938b), "Die 5fach transitiven Gruppen von Mathieu", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 12: 256–264, doi:10.1007/BF02948947
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}
 le Bruyn, Lieven (2007), Monsieur Mathieu
 Richter, David A., How to Make the Mathieu Group M_{24}, retrieved 20100415
 Mathieu group M_{9} on GroupNames
 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