[Paper Review] The Mathieu group $M_{12}$ and its pseudogroup extension $M_{13}$

J. Conway, Noam D. Elkies+1 more|ArXiv.org|Aug 31, 2005

graph theory and CDMA systems3 references3 citations

TL;DR

This paper constructs the Mathieu group $M_{12}$ and its pseudogroup extension $M_{13}$ using a combinatorial game on the 13-point projective plane of order 3, analogous to the 15-puzzle. By defining moves as double transpositions along lines containing a hole, the authors show that closed move sequences generate $M_{12}$, while all move sequences form the pseudogroup $M_{13}$, which exhibits limited sextuple transitivity. A key contribution is the introduction of a quasi-Cayley metric on $M_{12}$ and $M_{13}$, with computational data revealing structural insights including the appearance of the 9-element tetracode at maximal distance.

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.

Motivation & Objective

• To construct the Mathieu group $M_{12}$ as the group of permutations generated by closed move sequences in a puzzle game on the 13-point projective plane $\mathbb{P}_3$.
• To define and study the pseudogroup $M_{13}$, formed by all move sequences (not necessarily closed), which acts on all 13 points and exhibits partial sextuple transitivity.
• To develop a quasi-Cayley metric on $M_{12}$ and $M_{13}$ based on minimal move sequences, enabling computational analysis of group structure.
• To extend the construction to the signed game, realizing the double cover $2M_{12}$ as the automorphism group of the ternary Golay code $\mathscr{C}_{12}$, and to study $2M_{13}$ as its pseudogroup extension.
• To provide a new geometric and computational interpretation of outer automorphisms of $M_{12}$ using dualized games and Hadamard matrices.

Proposed method

• Realizing $M_{12}$ as the group of permutations induced by closed move sequences in a 13-point game where a hole moves via double transpositions along lines containing it.
• Defining $M_{13}$ as the set of all permutations reachable by arbitrary move sequences, which is not a group due to concatenation restrictions based on hole positions.
• Using the ternary Golay code $\mathscr{C}_{12}$ to model the signed game, where counters are two-sided and moves flip signs, yielding the double cover $2M_{12}$.
• Employing $12 \times 12$ Hadamard matrices to provide an alternative construction of $M_{12}$ via the dualized game on lines of $\mathbb{P}_3$.
• Computing the quasi-Cayley metric by enumerating minimal move sequences to realize each group element, using computer-generated data on position distributions.
• Analyzing depth distributions and symmetries in the Cayley graph of $2M_{12}$ and $2M_{13}$, including the role of the central involution $-\mathbf{1}$ that flips all counters.

Experimental results

Research questions

• RQ1 Can the Mathieu group $M_{12}$ be realized as the group of permutations generated by a puzzle-like game on the 13-point projective plane $\mathbb{P}_3$?
• RQ2 Does the pseudogroup $M_{13}$, formed by all move sequences in this game, exhibit any form of sextuple transitivity?
• RQ3 What is the structure of the quasi-Cayley metric on $M_{12}$ and $M_{13}$, and how does it reflect the underlying group and code-theoretic properties?
• RQ4 How do the signed and dualized extensions of the game yield realizations of the double covers $2M_{12}$ and $2M_{13}$, and what is their metric structure?
• RQ5 Can the tetracode and outer automorphism of $M_{12}$ be geometrically interpreted through the game's configuration space?

Key findings

• The basic game group $G_{\rm bas}$ is isomorphic to the Mathieu group $M_{12}$, realized as the group of permutations from closed move sequences on $\mathbb{P}_3$.
• The pseudogroup $M_{13}$ acts on all 13 points and exhibits limited forms of sextuple transitivity, though full sextuple transitivity does not hold.
• The depth distribution for $M_{13}$ shows that the 9-element tetracode appears as the set of positions at maximal distance (depth 9) from the starting position.
• The depth distribution for $2M_{13}$ is symmetric near the poles (depths 0–2 and 10–12), but breaks down at depths 3–5 and 7–9, indicating non-uniformity in antipodal distances.
• The unique element at maximal depth 12 in $2M_{13}$ is $-\mathbf{1}$, the permutation that flips all counters in place, and it is central in $2M_{12}$.
• For $\sigma \in 2M_{13}$ with $d(\sigma) \in \{1,2\}$, the antipodal distance satisfies $d(-\sigma) = 12 - d(\sigma)$, confirming a partial symmetry in the metric structure.