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 two 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	123
2 = 2	3795 = 3 · 5 · 11 · 23	1728
3 = 3	56672 = 25 · 7 · 11 · 23	1536
4 = 22	318780 = 22 · 32 · 5 · 7 · 11 · 23	132244
5 = 5	680064 = 27 · 3 · 7 · 11 · 23	1354
6 = 2 · 3	850080 = 25 · 3 · 5 · 7 · 11 · 23	1·223262
7 = 7	728640 = 26 · 32 · 5 · 11 · 23	1273	power equivalent
	728640 = 26 · 32 · 5 · 11 · 23	1273
8 = 23	1275120 = 24 · 32 · 5 · 7 · 11 · 23	1·2·4·82
11 = 11	927360= 27 · 32 · 5 · 7 · 23	1·112	power equivalent
	927360= 27 · 32 · 5 · 7 · 23	1·112
14 = 2 · 7	728640= 26 · 32 · 5 · 11 · 23	2·7·14	power equivalent
	728640= 26 · 32 · 5 · 11 · 23	2·7·14
15 = 3 · 5	680064= 27 · 3 · 7 · 11 · 23	3·5·15	power equivalent
	680064= 27 · 3 · 7 · 11 · 23	3·5·15
23 = 23	443520= 27 · 32 · 5 · 7 · 11	23	power equivalent
	443520= 27 · 32 · 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, https://archive.org/details/permutationgroup0000came 
• Carmichael, Robert D. (1956), Introduction to the theory of groups of finite order, New York: Dover Publications, ISBN 978-0-486-60300-1, https://books.google.com/books?id=McMgAAAAMAAJ 
• 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, https://books.google.com/books?id=TPPkAAAAIAAJ  Reprinted in (Conway Sloane)
• 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, https://books.google.com/books?id=38fEMl2-Fp8C 
• 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, https://books.google.com/books?id=upYwZ6cQumoC 
• Cuypers, Hans, The Mathieu groups and their geometries, http://www.win.tue.nl/~hansc/mathieu.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, https://archive.org/details/permutationgroup0000dixo 
• 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 
• 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, http://gallica.bnf.fr/ark:/12148/bpt6k16405f/f249 
• 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 
• 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, https://books.google.com/books?id=ggqxuG31B3cC 
• 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