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Mathieu group M12

January 01, 1970

For general background and history of the Mathieu sporadic groups, see Mathieu group.

In the area of modern algebra known as group theory, the Mathieu group M12 is a sporadic simple group of order

   12 · 11 · 10 · 9 · 8 = 26 · 33 · 5 · 11 = 95040.

History and properties edit

M12 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a sharply 5-transitive permutation group on 12 objects. Burgoyne & Fong (1968) showed that the Schur multiplier of M12 has order 2 (correcting a mistake in (Burgoyne & Fong 1966) where they incorrectly claimed it has order 1).

The double cover had been implicitly found earlier by Coxeter (1958), who showed that M12 is a subgroup of the projective linear group of dimension 6 over the finite field with 3 elements.

The outer automorphism group has order 2, and the full automorphism group M12.2 is contained in M24 as the stabilizer of a pair of complementary dodecads of 24 points, with outer automorphisms of M12 swapping the two dodecads.

Representations edit

Frobenius (1904) calculated the complex character table of M12.

M12 has a strictly 5-transitive permutation representation on 12 points, whose point stabilizer is the Mathieu group M11. Identifying the 12 points with the projective line over the field of 11 elements, M12 is generated by the permutations of PSL2(11) together with the permutation (2,10)(3,4)(5,9)(6,7). This permutation representation preserves a Steiner system S(5,6,12) of 132 special hexads, such that each pentad is contained in exactly 1 special hexad, and the hexads are the supports of the weight 6 codewords of the extended ternary Golay code. In fact M12 has two inequivalent actions on 12 points, exchanged by an outer automorphism; these are analogous to the two inequivalent actions of the symmetric group S6 on 6 points.

The double cover 2.M12 is the automorphism group of the extended ternary Golay code, a dimension 6 length 12 code over the field of order 3 of minimum weight 6. In particular the double cover has an irreducible 6-dimensional representation over the field of 3 elements.

The double cover 2.M12 is the automorphism group of any 12×12 Hadamard matrix.

M12 centralizes an element of order 11 in the monster group, as a result of which it acts naturally on a vertex algebra over the field with 11 elements, given as the Tate cohomology of the monster vertex algebra.

Maximal subgroups edit

There are 11 conjugacy classes of maximal subgroups of M12, 6 occurring in automorphic pairs, as follows:

  • M11, order 7920, index 12. There are two classes of maximal subgroups, exchanged by an outer automorphism. One is the subgroup fixing a point with orbits of size 1 and 11, while the other acts transitively on 12 points.
  • S6:2 = M10.2, the outer automorphism group of the symmetric group S6 of order 1440, index 66. There are two classes of maximal subgroups, exchanged by an outer automorphism. One is imprimitive and transitive, acting with 2 blocks of 6, while the other is the subgroup fixing a pair of points and has orbits of size 2 and 10.
  • PSL(2,11), order 660, index 144, doubly transitive on the 12 points
  • 32:(2.S4), order 432. There are two classes of maximal subgroups, exchanged by an outer automorphism. One acts with orbits of 3 and 9, and the other is imprimitive on 4 sets of 3.
Isomorphic to the affine group on the space C3 x C3.
  • S5 x 2, order 240, doubly imprimitive on 6 sets of 2 points
Centralizer of a sextuple transposition
  • Q:S4, order 192, orbits of 4 and 8.
Centralizer of a quadruple transposition
  • 42:(2 x S3), order 192, imprimitive on 3 sets of 4
  • A4 x S3, order 72, doubly imprimitive, 4 sets of 3 points.

Conjugacy classes edit

The cycle shape of an element and its conjugate under an outer automorphism are related in the following way: the union of the two cycle shapes is balanced, in other words invariant under changing each n-cycle to an N/n cycle for some integer N.

Order Number Centralizer Cycles Fusion
1 1 95040 112
2 396 240 26
2 495 192 1424
3 1760 54 1333
3 2640 36 34
4 2970 32 2242 Fused under an outer automorphism
4 2970 32 1442
5 9504 10 1252
6 7920 12 62
6 15840 6 1 2 3 6
8 11880 8 122 8 Fused under an outer automorphism
8 11880 8 4 8
10 9504 10 2 10
11 8640 11 1 11 Fused under an outer automorphism
11 8640 11 1 11

References edit

  • Adem, Alejandro; Maginnis, John; Milgram, R. James (1991), "The geometry and cohomology of the Mathieu group M₁₂", Journal of Algebra, 139 (1): 90–133, doi:10.1016/0021-8693(91)90285-G, hdl:2027.42/29344, ISSN 0021-8693, MR 1106342
  • Burgoyne, N.; Fong, Paul (1966), "The Schur multipliers of the Mathieu groups", Nagoya Mathematical Journal, 27 (2): 733–745, doi:10.1017/S0027763000026519, 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, doi:10.1017/S0027763000012782, ISSN 0027-7630, MR 0219626
  • 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
  • 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
  • Coxeter, Harold Scott MacDonald (1958), "Twelve points in PG(5,3) with 95040 self-transformations", Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 247 (1250): 279–293, doi:10.1098/rspa.1958.0184, ISSN 0962-8444, JSTOR 100667, MR 0120289, S2CID 121676627
  • 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", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (in German), 16, Königliche Akademie der Wissenschaften, Berlin: 558–571, Reprinted in volume III of his collected works.
  • 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
  • Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, vol. 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, S2CID 123106337
  • 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, S2CID 123658601

External links edit

  • MathWorld: Mathieu Groups
  • Atlas of Finite Group Representations: M12

January 01, 1970
mathieu, group, general, background, history, mathieu, sporadic, groups, mathieu, group, area, modern, algebra, known, group, theory, mathieu, group, sporadic, simple, group, order, 95040, contents, history, properties, representations, maximal, subgroups, con. For general background and history of the Mathieu sporadic groups see Mathieu group In the area of modern algebra known as group theory the Mathieu group M12 is a sporadic simple group of order 12 11 10 9 8 26 33 5 11 95040 Contents 1 History and properties 2 Representations 3 Maximal subgroups 4 Conjugacy classes 5 References 6 External linksHistory and properties editM12 is one of the 26 sporadic groups and was introduced by Mathieu 1861 1873 It is a sharply 5 transitive permutation group on 12 objects Burgoyne amp Fong 1968 showed that the Schur multiplier of M12 has order 2 correcting a mistake in Burgoyne amp Fong 1966 where they incorrectly claimed it has order 1 The double cover had been implicitly found earlier by Coxeter 1958 who showed that M12 is a subgroup of the projective linear group of dimension 6 over the finite field with 3 elements The outer automorphism group has order 2 and the full automorphism group M12 2 is contained in M24 as the stabilizer of a pair of complementary dodecads of 24 points with outer automorphisms of M12 swapping the two dodecads Representations editFrobenius 1904 calculated the complex character table of M12 M12 has a strictly 5 transitive permutation representation on 12 points whose point stabilizer is the Mathieu group M11 Identifying the 12 points with the projective line over the field of 11 elements M12 is generated by the permutations of PSL2 11 together with the permutation 2 10 3 4 5 9 6 7 This permutation representation preserves a Steiner system S 5 6 12 of 132 special hexads such that each pentad is contained in exactly 1 special hexad and the hexads are the supports of the weight 6 codewords of the extended ternary Golay code In fact M12 has two inequivalent actions on 12 points exchanged by an outer automorphism these are analogous to the two inequivalent actions of the symmetric group S6 on 6 points The double cover 2 M12 is the automorphism group of the extended ternary Golay code a dimension 6 length 12 code over the field of order 3 of minimum weight 6 In particular the double cover has an irreducible 6 dimensional representation over the field of 3 elements The double cover 2 M12 is the automorphism group of any 12 12 Hadamard matrix M12 centralizes an element of order 11 in the monster group as a result of which it acts naturally on a vertex algebra over the field with 11 elements given as the Tate cohomology of the monster vertex algebra Maximal subgroups editThere are 11 conjugacy classes of maximal subgroups of M12 6 occurring in automorphic pairs as follows M11 order 7920 index 12 There are two classes of maximal subgroups exchanged by an outer automorphism One is the subgroup fixing a point with orbits of size 1 and 11 while the other acts transitively on 12 points S6 2 M10 2 the outer automorphism group of the symmetric group S6 of order 1440 index 66 There are two classes of maximal subgroups exchanged by an outer automorphism One is imprimitive and transitive acting with 2 blocks of 6 while the other is the subgroup fixing a pair of points and has orbits of size 2 and 10 PSL 2 11 order 660 index 144 doubly transitive on the 12 points 32 2 S4 order 432 There are two classes of maximal subgroups exchanged by an outer automorphism One acts with orbits of 3 and 9 and the other is imprimitive on 4 sets of 3 Isomorphic to the affine group on the space C3 x C3 S5 x 2 order 240 doubly imprimitive on 6 sets of 2 points Centralizer of a sextuple transposition Q S4 order 192 orbits of 4 and 8 Centralizer of a quadruple transposition 42 2 x S3 order 192 imprimitive on 3 sets of 4 A4 x S3 order 72 doubly imprimitive 4 sets of 3 points Conjugacy classes editThe cycle shape of an element and its conjugate under an outer automorphism are related in the following way the union of the two cycle shapes is balanced in other words invariant under changing each n cycle to an N n cycle for some integer N Order Number Centralizer Cycles Fusion 1 1 95040 112 2 396 240 26 2 495 192 1424 3 1760 54 1333 3 2640 36 34 4 2970 32 2242 Fused under an outer automorphism 4 2970 32 1442 5 9504 10 1252 6 7920 12 62 6 15840 6 1 2 3 6 8 11880 8 122 8 Fused under an outer automorphism 8 11880 8 4 8 10 9504 10 2 10 11 8640 11 1 11 Fused under an outer automorphism 11 8640 11 1 11References editAdem Alejandro Maginnis John Milgram R James 1991 The geometry and cohomology of the Mathieu group M Journal of Algebra 139 1 90 133 doi 10 1016 0021 8693 91 90285 G hdl 2027 42 29344 ISSN 0021 8693 MR 1106342 Burgoyne N Fong Paul 1966 The Schur multipliers of the Mathieu groups Nagoya Mathematical Journal 27 2 733 745 doi 10 1017 S0027763000026519 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 doi 10 1017 S0027763000012782 ISSN 0027 7630 MR 0219626 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 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 amp 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 Coxeter Harold Scott MacDonald 1958 Twelve points in PG 5 3 with 95040 self transformations Proceedings of the Royal Society of London Series A Mathematical Physical and Engineering Sciences 247 1250 279 293 doi 10 1098 rspa 1958 0184 ISSN 0962 8444 JSTOR 100667 MR 0120289 S2CID 121676627 Curtis R T 1984 The Steiner system S 5 6 12 the Mathieu group M 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 Uber die Charaktere der mehrfach transitiven Gruppen Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften in German 16 Konigliche Akademie der Wissenschaften Berlin 558 571 Reprinted in volume III of his collected works 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 Emile 1861 Memoire sur l etude des fonctions de plusieurs quantites sur la maniere de les former et sur les substitutions qui les laissent invariables Journal de Mathematiques Pures et Appliquees 6 241 323 Mathieu Emile 1873 Sur la fonction cinq fois transitive de 24 quantites Journal de Mathematiques Pures et Appliquees 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 vol 21 Mathematical Association of America ISBN 978 0 88385 023 7 MR 0749038 Witt Ernst 1938a uber Steinersche Systeme Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg 12 265 275 doi 10 1007 BF02948948 ISSN 0025 5858 S2CID 123106337 Witt Ernst 1938b Die 5 fach transitiven Gruppen von Mathieu Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg 12 256 264 doi 10 1007 BF02948947 S2CID 123658601External links editMathWorld Mathieu Groups Atlas of Finite Group Representations M12 Retrieved from https en wikipedia org w index php title Mathieu group M12 amp oldid 1223972326, wikipedia, wiki, book, books, library,

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