Mathieu group

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.

History

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

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, ... 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. 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 S_k for k at least 4, the alternating groups A_k 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,F_q), which is sharply 3-transitive (see cross ratio) on elements.

Order and transitivity table

Constructions of the Mathieu groups

The Mathieu groups can be constructed in various ways.

Permutation groups

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 and non-perfect squares via ); 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

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

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

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

• Cameron, Peter J. (1999), Permutation Groups, London Mathematical Society Student Texts 45, Cambridge University Press, ISBN 978-0-521-65378-7 
• Carmichael, Robert D. (1956) [1937], Introduction to the theory of groups of finite order, New York: Dover Publications, ISBN 978-0-486-60300-1, MR 0075938 
• Choi, C. (May 1972), "On Subgroups of M₂₄. 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 1972). "On Subgroups of M₂₄. II: the Maximal Subgroups of M₂₄". 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 978-0-12-563850-0, 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 978-0-19-853199-9, 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: Springer-Verlag, ISBN 978-0-387-98585-5, 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 0305-0041, 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 0305-0041, 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 978-0-12-066270-8, MR 760669 
• Cuypers, Hans, The Mathieu groups and their geometries 
• Dixon, John D.; Mortimer, Brian (1996), Permutation groups, Graduate Texts in Mathematics 163, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0731-3, ISBN 978-0-387-94599-6, MR 1409812 
• Frobenius, Ferdinand Georg (1904), Über die Charaktere der mehrfach transitiven Gruppen, Berline Berichte, Mouton De Gruyter, pp. 558–571, ISBN 978-3-11-109790-9 
• Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62778-4, 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 five-fold 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 978-0-19-280722-9  (an introduction for the lay reader, describing the Mathieu groups in a historical context)
• Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs 21, Mathematical Association of America, ISBN 978-0-88385-023-7, 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 0025-5858 
• Witt, Ernst (1938b), "Die 5-fach transitiven Gruppen von Mathieu", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 12: 256–264, doi:10.1007/BF02948947 

External links

• ATLAS: Mathieu group M₁₀
• ATLAS: Mathieu group M₁₁
• ATLAS: Mathieu group M₁₂
• ATLAS: Mathieu group M₂₀
• ATLAS: Mathieu group M₂₁
• ATLAS: Mathieu group M₂₂
• ATLAS: Mathieu group M₂₃
• ATLAS: Mathieu group M₂₄
• le Bruyn, Lieven (2007), Monsieur Mathieu 
• Richter, David A., How to Make the Mathieu Group M₂₄, retrieved 2010-04-15 

• 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