M₁₃

Summary

Summary The group M₁₂ has no transitive extension, but the object of the title is the next best thing: a set of permutations which is an extension of M₁₂. We give an elementary construction, based on a moving-counter puzzle on the projective plane of order 3, and provide easy proofs of some of its properties.

Introduction

Long ago I was intrigued by the fact that M₁₂, É. Mathieu's celebrated quintuply transitive group on 12 letters, shares some structure with L₃(3), which acts doubly transitively on the 13 points of the projective plane PG(2,3), of which it is the automorphism group.

To be more precise, the point-stabilizer in L₃(3) is a group of structure 3²: 2S₄ that permutes the 12 remaining points imprimitively in four blocks of 3, and there is an isomorphic subgroup of M₁₂ that permutes the 12 letters in precisely the same fashion. Again, the line-stabilizer in L₃(3) is a group of this same structure that permutes the 9 points not on that line in a doubly transitive manner, while the stabilizer of a triple in M₁₂ is an isomorphic group that permutes the 9 letters not in that triple in just the same manner.

In the heady days when new simple groups were being discovered right and left, this common structure inevitably suggested that there should be a new group that contained both M₁₂ and L₃(3), various copies of which would intersect in the subgroups mentioned above.