The Mathieu group $M_{12}$ and its pseudogroup extension $M_{13}$

TL;DR

This paper presents a novel construction of the Mathieu group M12 and its extension M13 using a puzzle analogy, explores their properties, and develops metrics on these groups through combinatorial and coding theory methods.

Contribution

It introduces a new puzzle-based construction of M12 and M13, extending their analysis with metrics and automorphism groups using Golay code and Hadamard matrices.

Findings

Construction of M12 via a puzzle analogy

Extension to a pseudogroup M13 with partial transitivity

Development of a Cayley-like metric on M12 and 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 $12 \x 12$ 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.