Papers › The Mathieu group M₁₂ and its pseudogroup extension M₁₃
The Mathieu group M₁₂ and its pseudogroup extension M₁₃
John H. Conway, Noam D. Elkies, Jeremy L. Martin
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We study a construction of the Mathieu group M₁₂ using a game reminiscent of Loyd's ``15-puzzle''. The elements of M₁₂ 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'' M₁₃ 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 M₁₂ and M₁₃. We develop these results, and extend them to the double covers and automorphism groups of M₁₂ and M₁₃, 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.
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