Mathieu group M₂₃

In the area of modern algebra known as group theory, the Mathieu group M₂₃ is a sporadic simple group of order

   2⁷ · 3² · 5 · 7 · 11 · 23 = 10200960

≈ 1×10⁷.

History and properties

M₂₃ is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial.

Milgram (2000) calculated the integral cohomology, and showed in particular that M₂₃ has the unusual property that the first 4 integral homology groups all vanish.

The inverse Galois problem seems to be unsolved for M₂₃. In other words, no polynomial in Z[x] seems to be known to have M₂₃ as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups.

Construction using finite fields

Let F_2¹¹ be the finite field with 2¹¹ elements. Its group of units has order 2¹¹ − 1 = 2047 = 23 · 89, so it has a cyclic subgroup C of order 23.

The Mathieu group M₂₃ can be identified with the group of F₂-linear automorphisms of F_2¹¹ that stabilize C. More precisely, the action of this automorphism group on C can be identified with the 4-fold transitive action of M₂₃ on 23 objects.

Representations

M₂₃ is the point stabilizer of the action of the Mathieu group M24 on 24 points, giving it a 4-transitive permutation representation on 23 points with point stabilizer the Mathieu group M22.

M₂₃ has 2 different rank 3 actions on 253 points. One is the action on unordered pairs with orbit sizes 1+42+210 and point stabilizer M₂₁.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 2⁴.A₇.

The integral representation corresponding to the permutation action on 23 points decomposes into the trivial representation and a 22-dimensional representation. The 22-dimensional representation is irreducible over any field of characteristic not 2 or 23.

Over the field of order 2, it has 2 11-dimensional representations, the restrictions of the corresponding representations of the Mathieu group M24.

Maximal subgroups

There are 7 conjugacy classes of maximal subgroups of M₂₃ as follows:

• M₂₂, order 443520
• PSL(3,4):2, order 40320, orbits of 21 and 2
• 2⁴:A₇, order 40320, orbits of 7 and 16

Stabilizer of W₂₃ block

• A₈, order 20160, orbits of 8 and 15
• M₁₁, order 7920, orbits of 11 and 12
• (2⁴:A₅):S₃ or M₂₀:S₃, order 5760, orbits of 3 and 20 (5 blocks of 4)

One-point stabilizer of the sextet group

• 23:11, order 253, simply transitive

Conjugacy classes

Order	No. elements	Cycle structure
1 = 1	1	1^23
2 = 2	3795 = 3 · 5 · 11 · 23	1^72^8
3 = 3	56672 = 2^5 · 7 · 11 · 23	1^53^6
4 = 2^2	318780 = 2^2 · 3^2 · 5 · 7 · 11 · 23	1^32^24^4
5 = 5	680064 = 2^7 · 3 · 7 · 11 · 23	1^35^4
6 = 2 · 3	850080 = 2^5 · 3 · 5 · 7 · 11 · 23	1·2^23^26^2
7 = 7	728640 = 2^6 · 3^2 · 5 · 11 · 23	1^27^3	power equivalent
	728640 = 2^6 · 3^2 · 5 · 11 · 23	1^27^3
8 = 2^3	1275120 = 2^4 · 3^2 · 5 · 7 · 11 · 23	1·2·4·8^2
11 = 11	927360= 2^7 · 3^2 · 5 · 7 · 23	1·11^2	power equivalent
	927360= 2^7 · 3^2 · 5 · 7 · 23	1·11^2
14 = 2 · 7	728640= 2^6 · 3^2 · 5 · 11 · 23	2·7·14	power equivalent
	728640= 2^6 · 3^2 · 5 · 11 · 23	2·7·14
15 = 3 · 5	680064= 2^7 · 3 · 7 · 11 · 23	3·5·15	power equivalent
	680064= 2^7 · 3 · 7 · 11 · 23	3·5·15
23 = 23	443520= 2^7 · 3^2 · 5 · 7 · 11	23	power equivalent
	443520= 2^7 · 3^2 · 5 · 7 · 11	23

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 (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, 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, 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, doi:10.1007/978-3-662-03516-0, 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
• Milgram, R. James (2000), "The cohomology of the Mathieu group M₂₃", Journal of Group Theory, 3 (1): 7–26, doi:10.1515/jgth.2000.008, ISSN 1433-5883, MR 1736514
• 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 0749038
• Witt, Ernst (1938a), "über Steinersche Systeme", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 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: M₂₃