Mathieu group

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In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873. They are usually denoted by the symbols M₁₁, M₁₂, M₂₂, M₂₃, M₂₄, and can be thought of respectively as permutation groups on sets of 11, 12, 22, 23 or 24 objects (or points).

M₂₄, the largest of the groups, is the symmetry group of the binary Golay code, which has practical uses. Moreover, the Mathieu groups are fascinating to many group theorists as mathematical anomalies.

Simple groups are defined as having no nontrivial proper normal subgroups. Intuitively this means they cannot be broken down into products of smaller groups. For many years group theorists struggled to classify the simple groups and had found all of them by about 1980. Simple groups belong to a number of infinite families except for 26 groups including the Mathieu groups, called sporadic simple groups. After the Mathieu groups no new sporadic groups were found until 1965, when the group J₁ was discovered.

Contents 1 Multiply transitive groups 1.1 Order and transitivity table 2 Three constructions of the Mathieu groups 2.1 Permutation groups 2.2 Automorphism groups of Steiner systems 2.3 Automorphism group of the Golay code 3 Properties 4 Matrix representations in GL(11,2) 5 Sextet subgroup of M24 6 Subgroup structure 6.1 Maximal subgroups of M24 6.2 Maximal subgroups of M23 6.3 Maximal subgroups of M22 6.4 Maximal subgroups of M21 6.5 Maximal subgroups of M12 6.6 Maximal subgroups of M11 7 External links 8 Notes and references

[edit] Multiply transitive groups

Mathieu was interested in finding multiply transitive permutation groups, which will now be defined. For a natural number k, a permutation group G acting on n points is k-transitive if, given two sets of points a₁, ... a_k and b₁, ... b_k with the property that all the a_i are distinct and all the b_i are distinct, there is a group element g in G which maps a_i to b_i for each i between 1 and k. Such a group is called sharply k-transitive if the element g is unique (i.e. the action on k-tuples is regular, rather than just transitive).

The only 4-transitive groups are the symmetric groups S_k for k at least 4, the alternating groups A_k for k at least 6, and the Mathieu groups M₂₄, M₂₃, M₁₂ and M₁₁. The full proof requires the classification of finite simple groups, but some special cases have been known for much longer. It is a classical result of Jordan that the symmetric and alternating groups (of degree k and k − 2 respectively), and M₁₂ and M₁₁ are the only sharply k-transitive permutation groups for k at least 4.

[edit] Order and transitivity table

Group	Order	Factorised order	Transitivity
M24	244823040	210·33·5·7·11·23	5-transitive
M23	10200960	27·32·5·7·11·23	4-transitive
M22	443520	27·32·5·7·11	3-transitive
M12	95040	26·33·5·11	sharply 5-transitive
M11	7920	24·32·5·11	sharply 4-transitive

[edit] Three constructions of the Mathieu groups

[edit] Permutation groups

M₁₂ has a simple subgroup of order 660, a maximal subgroup. That subgroup can be represented as a linear fractional group on the field F₁₁ of 11 elements. With -1 written as a and infinity as b , two standard generators are (0123456789a) and (0b)(1a)(25)(37)(48)(69). A third generator giving M₁₂ sends an element x of F₁₁ to 4x²-3x⁷; as a permutation that is (26a7)(3945). The stabilizer of 4 points is a quaternion group.

Likewise M₂₄ has a maximal simple subgroup of order 6072 and this can be represented as a linear fractional group on the field F₂₃. One generator adds 1 to each element, i. e. (0123456789ABCDEFGHIJKLM), and the other is (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI). A third generator giving M₂₄ sends an element x of F₂₃ to 4x⁴-3x¹⁵; unexciting computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF).

These constructions were cited by Carmichael ^[1]; Dixon and Mortimer ascribe the permutations to Mathieu. ^[2]

[edit] Automorphism groups of Steiner systems

There exists up to equivalence a unique S(5,8,24) Steiner system W₂₄ (Witt geometry). The group M₂₄ is the automorphism group of this Steiner system; that is, the set of permutations which map every block to some other block. The subgroups M₂₃ and M₂₂ are defined to be the stabilizers of a single point and two points respectively.

Similarly, there exists up to equivalence a unique S(5,6,12) Steiner system W₁₂, and the group M₁₂ is its automorphism group. The subgroup M₁₁ is the stabilizer of a point.

A good nest egg for M₂₄ is PSL(3,4) ^[3], 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 group PGL(3,4) both sets of subgroups are 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. PGL(3,4) can therefore be extended to a group PΓL(3,4), which is a split extension of M₂₁ by the symmetric group S₃. PΓL(3,4) turns out to have an embedding as a maximal subgroup of M₂₄^[4].

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. The expansion of PSL(3,4) to M₂₄ is one of the remarkable phenomena of mathematics.

W₁₂ can be constructed from the affine geometry on the vector space F₃xF₃, an S(2,3,9) system.

There have been notable computer programs written to generate Steiner systems. For an introduction to a construction of W₂₄ via the Miracle Octad Generator of R. T. Curtis, see Geometry of the 4x4 Square. Another good account of this and Conway's analog for W₁₂, the miniMOG, may be found in the book by Conway and Sloane.

An alternative construction of W₁₂ is the 'Kitten' of R.T. Curtis^[5].

[edit] Automorphism group of the Golay code

The group M₂₄ also is the 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. 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.

M₁₂ has index 2 in its automorphism group. As a subgroup of M₂₄, M₁₂ acts on the second dodecad as an outer automorphic image of its action on the first dodecad. M₁₁ is a subgroup of M₂₃ but not of M₂₂. This representation of M₁₁ has orbits of 11 and 12. The automorphism group of M₁₂ is a maximal subgroup of M₂₄ of index 1288.

There is a very 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.

[edit] Properties

The Mathieu groups have fascinating properties; these groups happen because of a confluence of several anomalies of group theory.

For example, M₁₂ contains a copy of the exceptional outer automorphism of S₆. M₁₂ contains a subgroup isomorphic to S₆ acting differently on 2 sets of 6. In turn M₁₂ has an outer automorphism of index 2 and, as a subgroup of M₂₄, acts differently on 2 sets of 12.

The linear group GL(4,2) has an exceptional isomorphism to the alternating group A₈; this isomorphism is important to the structure of M₂₄. 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^[6]. There are 759 octads. Hence the order of M₂₄ is 759*16*20160.

[edit] Matrix representations in GL(11,2)

The binary Golay code is a vector space of dimension 12 over F₂. The fixed points under M₂₄ form a subspace of 2 vectors, those with coordinates all 0 or all 1. The quotient space, of dimension 11, order 2¹¹, can be constructed as a set of partitions of 24 bits into pairs of Golay codewords. It is intriguing that the number of non-zero vectors, 2¹¹-1 = 2047, is the smallest Mersenne number with prime exponent that is not prime, equal to 23*89. Then |M₂₄| divides |GL(11,2)| = 2⁵⁵*3⁶*5²*7³*11*17*23*73*89.

M₂₃ also requires dimension 11.

The groups M₂₂, M₁₂, and M₁₁ are represented in GL(10,2).

[edit] Sextet subgroup of M₂₄

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

B F J N R V

C G K O S W

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.

[edit] Subgroup structure

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.

[edit] Maximal subgroups of M₂₄

Robert T. Curtis (1977) completed the search for maximal subgroups of M₂₄.

• M₂₃, order 10200960
• M₂₂:2, order 887040, orbits of 2 and 22
• 2⁴:A₈, order 322560, orbits of 8 and 16: octad group
• M₁₂:2, order 190080, transitive and imprimitive: dodecad group

Copy of M₁₂ acting differently on 2 sets of 12, reflecting outer automorphism of M₁₂

• 2⁶:(3.S₆), order 138240: sextet group (vide supra)
• PSL(3,4):S₃, order 120960, orbits of 3 and 21
• 2⁶:(PSL(3,2) x S₃), order 64512, transitive and imprimitive: trio group

Stabilizer of partition into 3 octads

• PSL(2,23), order 6072: doubly transitive
• Octern group, order 168, simple, transitive and imprimitive, 8 blocks of 3

Last maximal subgroup of M₂₄ to be found.

This group's 7-elements fall into 2 conjugacy classes of 24.

[edit] Maximal subgroups of M₂₃

• 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

[edit] Maximal subgroups of M₂₂

There are no proper subgroups transitive on all 22 points.

• 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
• A₇, orbits of 7 and 15
• 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 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)

[edit] Maximal subgroups of M₂₁

There are no proper subgroups transitive on all 21 points.

• 2⁴:A₅ or M₂₀, order 960: one-point stabilizer

Imprimitive on 5 blocks of 4

• 2⁴:A₅, transpose of M₂₀, orbits of 5 and 16
• A₆, order 360, orbits of 6 and 15: hyperoval group
• A₆, orbits of 6 and 15
• A₆, orbits of 6 and 15
• PSL(3,2), order 168, orbits of 7 and 14: Fano subplane group
• PSL(3,2), orbits of 7 and 14
• PSL(3,2), orbits of 7 and 14
• 3²:Q or M₉, order 72, orbits of 9 and 12

[edit] Maximal subgroups of M₁₂

There are 11 conjugacy classes of maximal subgroups, 6 occurring in automorphic pairs.

• M₁₁, order 7920, degree 11
• M₁₁, degree 12

Outer automorphic image of preceding type

• S₆:2, order 1440, imprimitive and transitive, 2 blocks of 6

Example of the exceptional outer automorphism of S₆

• M₁₀.2, order 1440, orbits of 2 and 10

Outer automorphic image of preceding type

• PSL(2,11), order 660, doubly transitive on the 12 points
• 3²:(2.S₄), order 432, orbits of 3 and 9

Isomorphic to the affine group on the space C₃ x C₃.

• 3²:(2.S₄), imprimitive on 4 sets of 3

Outer automorphic image of preceding type

• S₅ x 2, order 240, doubly imprimitive, 6 by 2

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 by 3

[edit] Maximal subgroups of M₁₁

There are 5 conjugacy classes of maximal subgroups.

• M₁₀, order 720, one-point stabilizer in representation of degree 11
• PSL(2,11), order 660, one-point stabilizer in representation of degree 12
• M₉:2, order 144, stabilizer of a 9 and 2 partition.
• S₅, order 120, orbits of 5 and 6

Stabilizer of block in the S(4,5,11) Steiner system

• Q:S₃, order 48, orbits of 8 and 3

Centralizer of a quadruple transposition

Isomorphic to GL(2,3).

[edit] External links

• Moggie Java applet for studying the Curtis MOG construction

[edit] Notes and references

1. ^ Carmichael (1937): pp.151, 164, 263.
2. ^ Dixon and Mortimer (1996): p. 209.
3. ^ Dixon and Mortimer (1996): pp.192-205
4. ^ Griess (1998): p. 55
5. ^ Curtis, R.T. (1984), see below.
6. ^ Thomas Thompson (1983), pp. 197-208.

• Mathieu E., Mémoire sur l'étude des fonctions de plusieurs quantités, sur la manière de les former et sur les substitutions qui les laissent invariables J. Math. Pures Appl. (Liouville) (2) VI, 1861, pp. 241-323.

• Mathieu E., Sur la fonction cinq fois transitive de 24 quantités, Liouville Journ., (2) XVIII., 1873, pp. 25-47.

• Carmichael, Robert D. Groups of Finite Order, Dover (1937, reprint 1956).

• Conway, J.H.; Sloane N.J.A. Sphere Packings, Lattices and Groups: v. 290 (Grundlehren Der Mathematischen Wissenschaften.) Springer Verlag. ISBN 0-387-98585-9

• Curtis, R. T. A new combinatorial approach to M₂₄. Math. Proc. Camb. Phil. Soc. 79 (1976) 25-42.

• Curtis, R. T. The maximal subgroups of M₂₄. Math. Proc. Camb. Phil. Soc. 81 (1977) 185-192.

• Thompson, Thomas M.: From Error Correcting Codes through Sphere Packings to Simple Groups, Carus Mathematical Monographs, Mathematical Association of America, 1983.

• Curtis, R. T. The Steiner System S(5,6,12), the Mathieu Group M12 and the 'Kitten' , Computational Group Theory, Academic Press, London, 1984

• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985). Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Eynsham: Oxford University Press. ISBN 0-19-853199-0

• Dixon, John D. & Mortimer, Brian Permutation Groups, Springer-Verlag (1996).

• Griess, Robert L.: Twelve Sporadic Groups, Springer-Verlag, 1998.