Mathieu group M11

In mathematics, the Mathieu group M₁₁, introduced by Mathieu (1861, 1873), is a sharply 4-transitive permutation group on 11 objects, of order

2⁴ · 3² · 5 · 11 (= 7920).

The Schur multiplier and the outer automorphism group are both trivial.

Construction

The Mathieu group can be defined as a permutation group on 11 points generated by some set of permutations, such as the pair (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) of permutations used by the GAP computer algebra system.

Representations

M₁₁ has a strictly 4-transitive permutation representation on 11 points, whose point stabilizer is sometimes denoted by M₁₀, and is a non-split extension of the form A₆.2 (an extension of the group of order 2 by the alternating group A₆). This action is the automorphism group of a Steiner system S(4,5,11). The induced action on unordered pairs of points gives a rank 3 action on 55 points.

M₁₁ has a 2-transitive permutation representation on 12 points with point stabilizer PSL₂(11). The permutation representations on 11 and 12 points can both be seen inside the Mathieu group M12 as two different embeddings of M₁₁ in M₁₂, exchanged by an outer automorphism.

The permutation representation on 11 points gives a complex irreducible representation in 10 dimensions. This is the smallest possible dimension of a faithful complex representation, though there are also two other such representations in 10 dimensions forming a complex conjugate pair.

M₁₁ has two 5-dimensional irreducible representations over the field with 5 elements, related to the restrictions of 6-dimensional representations of the double cover of M₁₂. These have the smallest dimension of any faithful linear representations of M₁₁ over any field.

Maximal subgroups

There are 5 conjugacy classes of maximal subgroups.

• M₁₀, order 720, one-point stabilizer in representation of degree 11
• PSL(2,11), order 660, one-point stabilizer in representation of degree 12
• M₉:2, order 144, stabilizer of a 9 and 2 partition.
• S₅, order 120, orbits of 5 and 6

Stabilizer of block in the S(4,5,11) Steiner system

• Q:S₃, order 48, orbits of 8 and 3

Centralizer of a quadruple transposition

Isomorphic to GL(2,3).

Number of elements of each order

The maximum order of any element in M₁₁ is 11. The conjugacy class orders and sizes are found in the ATLAS.ATLAS: Mathieu group M₁₁

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 
• 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. (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 
• 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 
• 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 M₁₁