Geometric interpretations of the ‘natural’ generators of the Mathieu groups

Abstract

In [1] we used the alternating group A₅ to produce a set of five permutations of order three acting on 12 letters; this set was normalized by A₅ and generated the Mathieu group M₁₂. We analogously used the linear group PSL₂(7) to produce a set of seven involutions acting on 24 letters which was normalized by our PSL₂(7) and generated M₂₄. In this paper we describe the combinatorial and geometric interpretations of our generators, and aim to justify our use of the word ‘natural’. The generators for M₁₂ are defined both as permutations of twelve 5-cycles in a conjugacy class of A₅, and as permutations of the faces of a dodecahedron. The outer automorphism of M¹² emerges as a bonus. The generators of M₂₄ are defined analogously both as permutations of twenty-four 7-cycles in the action of PSL₂(7) on seven letters, and as permutations of the 24 faces of a certain cubic graph drawn on a surface of genus three. In addition we mention the connection between the latter construction and the binary Golay code.