13 - Natural Generators of the Mathieu Groups

Summary

In this chapter we describe how M₂₄ can be generated by seven involutions that are normalized as a set by a subgroup of M₂₄ isomorphic to the linear group L₃(2). These seven elements are described combinatorially as acting on a conjugacy class of 7-cycles in L₃(2). They are also described as acting on the 24 heptagonal faces of the Klein map κ, whose 84 edges fall into seven blocks of imprimitivity of size 12 under the action of L₃(2); thus there is one permutation for each block. So, remarkably, generators for M₂₄ may be read directly off κ. Analogously we describe how M₁₂ can be generated by five elements of order 3 that are normalized within M₁₂ by a subgroup isomorphic to the alternating group A₅. These five elements are described combinatorially as acting on a conjugacy class of 5-cycles in A₅. They are also described as acting on the 12 faces of the regular dodecahedron whose 20 vertices fall into five blocks of imprimitivity of size 4 (each of which forms a regular tetrahedron); thus there is one generator for each block. So generators for M₁₂ can be read directly off the faces of the dodecahedron. Galois showed that the only simple groups L₂(p) that can act faithfully with degree p are for p = 5, 7 and 11. These constructions for M₁₂ and M₂₄ owe their existence to the cases p = 5 and p = 7.