The Mathieu Group M₁₂ and Its Pseudogroup Extension M₁₃

Abstract

We study a construction of the Mathieu group M ₁₂ using a game reminiscent of Loyd's "15-puzzle." The elements of M ₁₂ are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There is a natural extension to a "pseudogroup" M ₁₃ acting on all 13 points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, akin to that induced by a Cayley graph, on both M ₁₂ and M ₁₃. We develop these results, and extend them to the double covers and automorphism groups of M ₁₂ and M ₁₃, using the ternary Golay code and 12 × 12 Hadamard matrices. In addition, we use experimental data on the quasi-Cayley metric to gain some insight into the structure of these groups and pseudogroups.

Keywords:

• Primary 20B25
• Secondary 05B25, 51E20, 20B20
• Mathieu group
• finite projective plane
• Golay code
• Hadamard matrix