Open Access
2006 The Mathieu Group $M_{12}$ and Its Pseudogroup Extension $M_{13}$
John H. Conway, Noam D. Elkies, Jeremy L. Martin
Experiment. Math. 15(2): 223-236 (2006).


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.


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John H. Conway. Noam D. Elkies. Jeremy L. Martin. "The Mathieu Group $M_{12}$ and Its Pseudogroup Extension $M_{13}$." Experiment. Math. 15 (2) 223 - 236, 2006.


Published: 2006
First available in Project Euclid: 5 April 2007

zbMATH: 1112.20003
MathSciNet: MR2253008

Primary: 20B25
Secondary: 05B25 , 20B20 , 51E20

Keywords: finite projective plane , Golay code , Hadamard matrix , Mathieu group

Rights: Copyright © 2006 A K Peters, Ltd.

Vol.15 • No. 2 • 2006
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