Mathieu group

In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups M₁₁, M₁₂, M₂₂, M₂₃ and M₂₄ introduced by Mathieu (1861, 1873). They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to be discovered.

Sometimes the notation M₈, M₉, M₁₀, M₂₀, and M₂₁ is used for related groups (which act on sets of 8, 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 M₁₃ acting on 13 points. M₂₁ is simple, but is not a sporadic group, being isomorphic to PSL(3,4).

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 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 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.). These are the only two 5-transitive groups that are neither symmetric groups nor alternating groups (Cameron 1992, p. 139).

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 $q+1$ elements.

Order and transitivity table

Group	Order	Order (product)	Factorised order	Transitivity	Simple	Sporadic
M₂₄	244823040	3·16·20·21·22·23·24	2¹⁰·3³·5·7·11·23	5-transitive	yes	sporadic
M₂₃	10200960	3·16·20·21·22·23	2⁷·3²·5·7·11·23	4-transitive	yes	sporadic
M₂₂	443520	3·16·20·21·22	2⁷·3²·5·7·11	3-transitive	yes	sporadic
M₂₁	20160	3·16·20·21	2⁶·3²·5·7	2-transitive	yes	≈ PSL₃(4)
M₂₀	960	3·16·20	2⁶·3·5	1-transitive	no	≈2⁴:A₅

M₁₂	95040	8·9·10·11·12	2⁶·3³·5·11	sharply 5-transitive	yes	sporadic
M₁₁	7920	8·9·10·11	2⁴·3²·5·11	sharply 4-transitive	yes	sporadic
M₁₀	720	8·9·10	2⁴·3²·5	sharply 3-transitive	almost	M₁₀' ≈ Alt₆
M₉	72	8·9	2³·3²	sharply 2-transitive	no	≈ PSU₃(2)
M₈	8	8	2³	sharply 1-transitive (regular)	no	≈ Q

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 is isomorphic to the projective special linear group PSL₂(F₁₁) 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 M₁₂ sends an element x of F₁₁ to 4x² − 3x⁷; 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. M₁₁ is the stabilizer of a point in M₁₂, and turns out also to be a sporadic simple group. M₁₀, the stabilizer of two points, is not sporadic, but is an almost simple group whose commutator subgroup is the alternating group A₆. It is thus related to the exceptional outer automorphism of A₆. The stabilizer of 3 points is the projective special unitary group PSU(3,2²), which is solvable. The stabilizer of 4 points is the quaternion group.

Likewise, M₂₄ has a maximal simple subgroup of order 6072 isomorphic to PSL₂(F₂₃). One generator adds 1 to each element of the field (leaving the point N at infinity fixed), i.e. (0123456789ABCDEFGHIJKLM)(N), and the other sends x to −1/x, i.e. (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 x⁴ and non-perfect squares via 7x⁴); computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF).

The stabilizers of 1 and 2 points, M₂₃ and M₂₂, also turn out to be sporadic simple groups. The stabilizer of 3 points is simple and isomorphic to the projective special linear group PSL₃(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 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 F₃ × F₃, 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 groups on the Golay code

The group M₂₄ 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.

M₁₂ has index 2 in its automorphism group, and M₁₂:2 happens to be isomorphic to a subgroup of M₂₄. M₁₂ is the stabilizer of a dodecad, a codeword of 12 1's; M₁₂:2 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 M₁₂ suggestively called "Monsieur Mathieu" by le Bruyn (2007).

References

Cameron, Peter J. (1992), Projective and Polar Spaces (PDF), University of London, Queen Mary and Westfield College, ISBN 978-0-902-48012-4, S2CID 115302359
Cameron, Peter J. (1999), Permutation Groups, London Mathematical Society Student Texts, vol. 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 1972a), "On Subgroups of M₂₄. I: Stabilizers of Subsets", Transactions of the American Mathematical Society, 167: 1–27, doi:10.2307/1996123, JSTOR 1996123
Choi, C. (May 1972b). "On Subgroups of M₂₄. II: the Maximal Subgroups of M₂₄". Transactions of the 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 (eds.), 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 0827219
Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, vol. 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, 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, Bibcode:1976MPCPS..79...25C, 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, Bibcode:1977MPCPS..81..185C, 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. (ed.), 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 0760669
Cuypers, Hans, The Mathieu groups and their geometries (PDF)
Dixon, John D.; Mortimer, Brian (1996), Permutation groups, Graduate Texts in Mathematics, vol. 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
Gill, Nick; Hughes, Sam (2019), "The character table of a sharply 5-transitive subgroup of the alternating group of degree 12", International Journal of Group Theory, doi:10.22108/IJGT.2019.115366.1531, S2CID 119151614
Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03516-0, ISBN 978-3-540-62778-4, MR 1707296
Hughes, Sam (2018), Representation and Character Theory of the Small Mathieu Groups (PDF)
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
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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)
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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, archived from the original on 2010-05-01
Richter, David A., How to Make the Mathieu Group M₂₄, retrieved 2010-04-15
Mathieu group M₉ 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 M₁₂, presented as one "spin" permutation and a selectable "swap" permutation