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The Mathieu groups are simple, meaning that they have no nontrivial Normal Subgroup s (intuitively this means that they cannot be broken down into simpler pieces). However they do not belong to any of the infinite main families of simple groups, and so they are said to be '''''sporadic''' simple groups'' (or simply '' Sporadic Group s''). The Mathieu groups were the first of the sporadic simple groups to be discovered. The Classification Of Finite Simple Groups asserts that there are only 21 other sporadic simple groups, and several of these are related to the Mathieu groups. The Mathieu groups happen because of a confluence of several anomalies of group theory. For example, S6 is the only symmetric group with an outer automorphism; the extension of S6 of order 1440 is found both in M12 and in M24. The linear group GL(4,2) is isomorphic to the alternating group A8; this isomorphism is quite fundamental to the structure of M24. MULTIPLY TRANSITIVE GROUPS The Mathieu groups are examples of multiply transitive groups. For a natural number ''k'', a permutation group ''G'' acting on ''n'' points is ''' ''k''-transitive''' if, given two sets of points ''a''1, ... ''a''''k'' and ''b''1, ... ''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 groups M24 and M12 are 5-transitive, the groups M23 and M11 are 4-transitive, and M22 is 3-transitive. It follows from the Classification Of Finite Simple Groups that the only groups which are ''k''-transitive for ''k'' at least 4 are the Symmetric and Alternating Group s (of degree ''k'' and ''k''-2 respectively) and the Mathieu groups M24, M23, M12 and M11. It is a classical result of Jordan that the Symmetric and Alternating Group s (of degree ''k'' and ''k'' − 2 respectively), and M12 and M11 are the only ''sharply'' ''k''-transitive permutation groups for ''k'' at least 4. Order and transitivity table
THREE CONSTRUCTIONS OF THE MATHIEU GROUPS 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 for M12 sends an element x of F11 to 4x2-3x7; as a permutatation 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 of 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 Carmichael (1937): pp.151, 164, 263.; Dixon and Mortimer ascribe the permutations to Mathieu. Dixon and Mortimer (1996): p. 209. Automorphism group 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) Dixon and 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 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 M24Griess (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 the lines, a different new point to the Fano subplanes in each of the sets of 120, and append appropriate pairs of new points to all of the hyperovals. That accounts for all but 210 of the octads. Those remaining octads are subsets of W21 and are Symmetric Difference s 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. CurtisCurtis, R.T. (1984), see below.. Automorphism group of the Golay code The group M24 can also be thought of as the Automorphism Group of the Binary Golay Code ''W'', i.e., the group of permutations of coordinates mapping ''W'' to itself. We can also regard it as the intersection of S24 and Stab(''W'') in Aut(''V''). 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 blocks of an S(5,8,24) 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'''. EXTERNAL LINKS
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