Mathieu group M22

In the area of modern algebra known as group theory, the Mathieu group M₂₂ is a sporadic simple group of order

   2⁷ · 3² · 5 · 7 · 11 = 443520

≈ 4×10⁵.

History and properties

M₂₂ is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M₂₂ is cyclic of order 12, and the outer automorphism group has order 2.

There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature. Burgoyne & Fong (1966) incorrectly claimed that the Schur multiplier of M₂₂ has order 3, and in a correction Burgoyne & Fong (1968) incorrectly claimed that it has order 6. This caused an error in the title of the paper Janko (1976) announcing the discovery of the Janko group J4. Mazet (1979) showed that the Schur multiplier is in fact cyclic of order 12.

Adem & Milgram (1995) calculated the 2-part of all the cohomology of M₂₂.

Representations

M₂₂ has a 3-transitive permutation representation on 22 points, with point stabilizer the group PSL₃(4), sometimes called M₂₁. This action fixes a Steiner system S(3,6,22) with 77 hexads, whose full automorphism group is the automorphism group M₂₂.2 of M₂₂.

M₂₂ has three rank 3 permutation representations: one on the 77 hexads with point stabilizer 2⁴:A₆, and two rank 3 actions on 176 heptads that are conjugate under an outer automorphism and have point stabilizer A₇.

M₂₂ is the point stabilizer of the action of M₂₃ on 23 points, and also the point stabilizer of the rank 3 action of the Higman–Sims group on 100 = 1+22+77 points.

The triple cover 3.M₂₂ has a 6-dimensional faithful representation over the field with 4 elements.

The 6-fold cover of M₂₂ appears in the centralizer 2¹⁺¹².3.(M₂₂:2) of an involution of the Janko group J4.

Maximal subgroups

There are no proper subgroups transitive on all 22 points. There are 8 conjugacy classes of maximal subgroups of M₂₂ as follows:

• PSL(3,4) or M₂₁, order 20160: one-point stabilizer
• 2⁴:A₆, order 5760, orbits of 6 and 16

Stabilizer of W₂₂ block

• A₇, order 2520, orbits of 7 and 15

There are 2 sets, of 15 each, of simple subgroups of order 168. Those of one type have orbits of 1, 7 and 14; the others have orbits of 7, 8, and 7.

• A₇, orbits of 7 and 15

Conjugate to preceding type in M₂₂:2.

• 2⁴:S₅, order 1920, orbits of 2 and 20 (5 blocks of 4)

A 2-point stabilizer in the sextet group

• 2³:PSL(3,2), order 1344, orbits of 8 and 14
• M₁₀, order 720, orbits of 10 and 12 (2 blocks of 6)

A one-point stabilizer of M₁₁ (point in orbit of 11)

A non-split group extension of form A₆.2

• PSL(2,11), order 660, orbits of 11 and 11

Another one-point stabilizer of M₁₁ (point in orbit of 12)

Conjugacy classes

There are 12 conjugacy classes, though the two classes of elements of order 11 are fused under an outer automorphism.

Order	No. elements	Cycle structure
1 = 1	1	1^22
2 = 2	1155 = 3 · 5 · 7 · 11	1^62^8
3 = 3	12320 = 2^5 · 5 · 7 · 11	1^43^6
4 = 2^2	13860 = 2^2 · 3^2 · 5 · 7 · 11	1^22^24^4
	27720 = 2^3 · 3^2 · 5 · 7 · 11	1^22^24^4
5 = 5	88704 = 2^7 · 3^2 · 7 · 11	1^25^4
6 = 2 · 3	36960 = 2^5 · 3 · 5 · 7 · 11	2^23^26^2
7 = 7	63360= 2^7 · 3^2 · 5 · 11	1 7^3
7 = 7	63360= 2^7 · 3^2 · 5 · 11	1 7^3
8 = 2^3	55440 = 2^4 · 3^2 · 5 · 7 · 11	2·4·8^2
11 = 11	40320 = 2^7 · 3^2 · 5 · 7	11^2
11 = 11	40320 = 2^7 · 3^2 · 5 · 7	11^2

References

• Adem, Alejandro; Milgram, R. James (1995), "The cohomology of the Mathieu group M₂₂", Topology. An International Journal of Mathematics, 34 (2): 389–410, doi:10.1016/0040-9383(94)00029-K, ISSN 0040-9383, MR 1318884
• Burgoyne, N.; Fong, Paul (1966), "The Schur multipliers of the Mathieu groups", Nagoya Mathematical Journal, 27: 733–745, ISSN 0027-7630, MR 0197542
• Burgoyne, N.; Fong, Paul (1968), "A correction to: "The Schur multipliers of the Mathieu groups"", Nagoya Mathematical Journal, 31: 297–304, ISSN 0027-7630, MR 0219626
• 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
• 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: 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
• Harada, Koichiro; Solomon, Ronald (2008), "Finite groups having a standard component L of type M₁₂ or M₂₂", Journal of Algebra, 319 (2): 621–628, doi:10.1016/j.jalgebra.2006.09.034, ISSN 0021-8693, MR 2381799
• Z. Janko, A new finite simple group of order 86,775,570,046,077,562,880 which possesses M₂₄ and the full covering group of M₂₂ as subgroups, J. Algebra 42 (1976) 564-596.doi:10.1016/0021-8693(76)90115-0 (The title of this paper is incorrect, as the full covering group of M₂₂ was later discovered to be larger: center of order 12, not 6.)
• 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
• Mazet, Pierre (1979), "Sur le multiplicateur de Schur du groupe de Mathieu M₂₂", Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B, 289 (14): A659––A661, ISSN 0151-0509, MR 560327
• 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

• MathWorld: Mathieu Groups
• Atlas of Finite Group Representations: M₂₂