Mathieu group M24

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In the area of modern algebra known as group theory, the Mathieu group M₂₄ is a sporadic simple group of order

   2¹⁰ · 3³ · 5 · 7 · 11 · 23 = 244823040

≈ 2×10⁸.

History and properties

M₂₄ is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 5-transitive permutation group on 24 objects. The Schur multiplier and the outer automorphism group are both trivial.

The Mathieu groups can be constructed in various ways. Initially, Mathieu and others constructed them as permutation groups. It was difficult to see that M₂₄ actually existed, that its generators did not just generate the alternating group A₂₄. The matter was clarified when Ernst Witt constructed M₂₄ as the automorphism (symmetry) group of an S(5,8,24) Steiner system W₂₄ (the Witt design). M₂₄ is the group of permutations that map every block in this design to some other block. The subgroups M₂₃ and M₂₂ then are easily defined to be the stabilizers of a single point and a pair of points respectively.

M₂₄ from PSL(3,4)

M₂₄ can be built starting from PSL(3,4), the projective special linear group of 3-dimensional space over the finite field with 4 elements (Dixon & Mortimer 1996, pp. 192–205), also called M₂₁ which acts on the projective plane over the field F₄, an S(2,5,21) system called W₂₁. Its 21 blocks are called lines. Any 2 lines intersect at one point.

M₂₁ has 168 simple subgroups of order 360 and 360 simple subgroups of order 168. In the larger projective general linear group PGL(3,4) both sets of subgroups form single conjugacy classes, but in M₂₁ both sets split into 3 conjugacy classes. The subgroups respectively have orbits of 6, called hyperovals, and orbits of 7, called Fano subplanes. These sets allow creation of new blocks for larger Steiner systems. M₂₁ is normal in PGL(3,4), of index 3. PGL(3,4) has an outer automorphism induced by transposing conjugate elements in F₄ (the field automorphism). PGL(3,4) can therefore be extended to the group PΓL(3,4) of projective semilinear transformations, which is a split extension of M₂₁ by the symmetric group S₃. PΓL(3,4) has an embedding as a maximal subgroup of M₂₄.(Griess 1998, p. 55])

A hyperoval has no 3 points that are colinear. A Fano subplane likewise satisfies suitable uniqueness conditions.

To W₂₁ append 3 new points and let the automorphisms in PΓL(3,4) but not in M₂₁ permute these new points. An S(3,6,22) system W₂₂ is formed by appending just one new point to each of the 21 lines and new blocks are 56 hyperovals conjugate under M₂₁.

An S(5,8,24) system would have 759 blocks, or octads. Append all 3 new points to each line of W₂₁, a different new point to the Fano subplanes in each of the sets of 120, and append appropriate pairs of new points to all the hyperovals. That accounts for all but 210 of the octads. Those remaining octads are subsets of W₂₁ and are symmetric differences of pairs of lines. There are many possible ways to expand the group PΓL(3,4) to M₂₄.

Automorphism group of the Golay code

The group M₂₄ also is the permutation automorphism group of the binary Golay code W, i.e., the group of permutations of coordinates mapping W to itself. Codewords correspond in a natural way to subsets of a set of 24 objects. (In coding theory the term "binary Golay code" often refers to a shorter related length 23 code, and the length 24 code used here is called the "extended binary Golay code".) Those subsets corresponding to codewords with 8 or 12 coordinates equal to 1 are called octads or dodecads respectively. The octads are the blocks of an S(5,8,24) Steiner system and the binary Golay code is the vector space over field F₂ spanned by the octads of the Steiner system.

The simple subgroups M₂₃, M₂₂, M₁₂, and M₁₁ can be defined as subgroups of M₂₄, stabilizers respectively of a single coordinate, an ordered pair of coordinates, a dodecad, and a dodecad together with a single coordinate.

There is a natural connection between the Mathieu groups and the larger Conway groups, because the binary Golay code and the Leech lattice both lie in spaces of dimension 24. The Conway groups in turn are found in the Monster group. Robert Griess refers to the 20 sporadic groups found in the Monster as the Happy Family, and to the Mathieu groups as the first generation.

Polyhedral symmetries

M₂₄ can be constructed from symmetries of the Klein quartic, augmented by a (non-geometric) symmetry of its immersion as the small cubicuboctahedron.

M₂₄ can be constructed starting from the symmetries of the Klein quartic (the symmetries of a tessellation of the genus three surface), which is PSL(2,7), which can be augmented by an additional permutation. This permutation can be described by starting with the tiling of the Klein quartic by 20 triangles (with 24 vertices – the 24 points on which the group acts), then forming squares of out some of the 2 triangles, and octagons out of 6 triangles, with the added permutation being "interchange the two endpoints of the lines bisecting the squares and octagons". This can be visualized by coloring the triangles – the corresponding tiling is topologically but not geometrically the t_0,1{4, 3, 3} tiling, and can be (polyhedrally) immersed in Euclidean 3-space as the small cubicuboctahedron (which also has 24 vertices).(Richter)

Applications

The theory of umbral moonshine is a partly conjectural relationship between K3 surfaces and M₂₄.

The Conway group Co1, the Fischer group Fi24, and the Janko group J4 all have maximal subgroups that are an extension of the Mathieu group M₂₄ by a group 2¹¹. (These extensions are not all the same.)

Representations

Frobenius (1904) calculated the complex character table of M₂₄.

The Mathieu group M₂₄ has a 5-fold transitive permutation representation on 24 points. The corresponding linear representation over the complex numbers is the sum of the trivial representation and a 23-dimensional irreducible representation.

M₂₄ has two rank 3 permutation representations: one on the 276 = 1+44+231 pairs of points (or duads) with stabilizer M₂₂.2, and one on the 1288 = 1+495+792 duads, with stabilizer M₁₂.2.

The quotient of the 24-dimensional linear representation of the permutation representation by its 1-dimensional fixed subspace gives a 23-dimensional representation, which is irreducible over any field of characteristic not 2 or 3, and gives the smallest faithful representation over such fields.

Reducing the 24-dimensional representation mod 2 gives an action on F24
2. This has invariant subspaces of dimension 1, 12 (the Golay code), and 23. The subquotients give two irreducible representations of dimension 11 over the field with 2 elements.

Maximal subgroups

Choi (1972b) found the 9 conjugacy classes of maximal subgroups of M₂₄. Curtis (1977) gave a short proof of the result, describing the 9 classes in terms of combinatorial data on the 24 points: the subgroups fix a point, duad, octad, duum, sextet, triad, trio, projective line, or octern, as described below. Todd (1966) gave the character tables of M₂₄ (originally calculated by Frobenius (1904)) and the 8 maximal subgroups that were known at the time.

M₂₄ contains non-abelian simple subgroups of 13 isomorphism types: five classes of A₅, four classes of PSL(3,2), two classes of A₆, two classes of PSL(2,11), one class each of A₇, PSL(2,23), M₁₁, PSL(3,4), A₈, M₁₂, M₂₂, M₂₃, and M₂₄. A₆ is also noted below as a subquotient in the sextet subgroup.

The Mathieu group acts on the 2048 = 1+759+1288 points of the Golay code modulo the fixed space with 3 orbits, and on the 4096 = 1+24+276+2024+1771 points of the cocode with 5 orbits, and the subgroups fixing a non-trivial point of the code or cocode give 6 of the 9 classes of maximal subgroups.

The 9 classes of maximal subgroups are as follows:

Point subgroup

M₂₃, order 10200960

Duad subgroup

A duad is a pair of points. The subgroup fixing a duad is M₂₂:2, order 887040, with orbits of 2 and 22.

Octad subgroup

The subgroup fixing one of the 759 (= 3·11·23) octads of the Golay code or Steiner system is the octad group 2⁴:A₈, order 322560, with orbits of size 8 and 16. The linear group GL(4,2) has an exceptional isomorphism to the alternating group A₈. The pointwise stabilizer O of an octad is an abelian group of order 16, exponent 2, each of whose involutions moves all 16 points outside the octad. The stabilizer of the octad is a split extension of O by A₈. (Thompson 1983, pp. 197-208)

Duum subgroup

A duum is a pair of complementary dodecads (12 point sets) in the Golay code. The subgroup fixing a duad is M₁₂:2, order 190080, transitive and imprimitive. This subgroup was discovered by Frobenius. The subgroup M₁₂ acts differently on 2 sets of 12, reflecting the outer automorphism of M₁₂.

Sextet subgroup

2⁶:(3.S₆), order 138240: sextet group

Consider a tetrad, any set of 4 points in the Steiner system W₂₄. An octad is determined by choice of a fifth point from the remaining 20. There are 5 octads possible. Hence any tetrad determines a partition into 6 tetrads, called a sextet, whose stabilizer in M₂₄ is called a sextet group.

The total number of tetrads is 24*23*22*21/4! = 23*22*21. Dividing that by 6 gives the number of sextets, 23*11*7 = 1771. Furthermore, a sextet group is a subgroup of a wreath product of order 6!*(4!)⁶, whose only prime divisors are 2, 3, and 5. Now we know the prime divisors of |M₂₄|. Further analysis would determine the order of the sextet group and hence |M₂₄|.

It is convenient to arrange the 24 points into a 6-by-4 array:

A E I M Q U

D H L P T X

Moreover, it is convenient to use the elements of the field F₄ to number the rows: 0, 1, u, u².

The sextet group has a normal abelian subgroup H of order 64, isomorphic to the hexacode, a vector space of length 6 and dimension 3 over F₄. A non-zero element in H does double transpositions within 4 or 6 of the columns. Its action can be thought of as addition of vector co-ordinates to row numbers.

The sextet group is a split extension of H by a group 3.S₆ (a stem extension). Here is an instance within the Mathieu groups where a simple group (A₆) is a subquotient, not a subgroup. 3.S₆ is the normalizer in M₂₄ of the subgroup generated by r=(BCD)(FGH)(JKL)(NOP)(RST)(VWX), which can be thought of as a multiplication of row numbers by u². The subgroup 3.A₆ is the centralizer of <r>. Generators of 3.A₆ are:

(AEI)(BFJ)(CGK)(DHL)(RTS)(VWX) (rotating first 3 columns)

(AQ)(BS)(CT)(DR)(EU)(FX)(GV)(HW)

(AUEIQ)(BXGKT)(CVHLR)(DWFJS) (product of preceding two)

(FGH)(JLK)(MQU)(NRV)(OSW)(PTX) (rotating last 3 columns).

An odd permutation of columns, say (CD)(GH)(KL)(OP)(QU)(RV)(SX)(TW), then generates 3.S₆.

The group 3.A₆ is isomorphic to a subgroup of SL(3,4) whose image in PSL(3,4) has been noted above as the hyperoval group.

The applet Moggie has a function that displays sextets in color.

Triad subgroup

A triad is a set of 3 points. The subgroup fixing a triad is PSL(3,4):S₃, order 120960, with orbits of size 3 and 21.

Trio subgroup

A trio is a set of 3 disjoint octads of the Golay code. The subgroup fixing a trio is the trio group 2⁶:(PSL(2,7) x S₃), order 64512, transitive and imprimitive.

Projective line subgroup

The subgroup fixing a projective line structure on the 24 points is PSL(2,23), order 6072, whose action is doubly transitive. This subgroup was observed by Mathieu.

Octern subgroup

An octern is a certain partition of the 24 points into 8 blocks of 3. The subgroup fixing an octern is the octern group isomorphic to PSL₂(7), of order 168, simple, transitive and imprimitive. It was the last maximal subgroup of M₂₄ to be found.

Conjugacy classes

There are 26 conjugacy classes. The cycle shapes are all balanced in the sense that they remain invariant under changing length k cycles to length N/k cycles for some integer N depending on the conjugacy class.

References

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• Frobenius, Ferdinand Georg (1904), "Über die Charaktere der mehrfach transitiven Gruppen", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (in German), Königliche Akademie der Wissenschaften, Berlin, 16: 558–571, Reprinted in volume III of his collected works. 
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• Ronan, Mark (2006), Symmetry and the Monster, Oxford, ISBN 978-0-19-280722-9  (an introduction for the lay reader, describing the Mathieu groups in a historical context)
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• Todd, J. A. (1966), "A representation of the Mathieu group M₂₄ as a collineation group", Annali di Matematica Pura ed Applicata. Serie Quarta, 71: 199–238, doi:10.1007/BF02413742, ISSN 0003-4622, MR 0202854 
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External links

• MathWorld: Mathieu Groups
• Atlas of Finite Group Representations: M₂₄
• Richter, David A., How to Make the Mathieu Group M₂₄, retrieved 2010-04-15