Mathieu groupoid

In mathematics, the Mathieu groupoid M13 is a groupoid acting on 13 points such that the stabilizer of each point is the Mathieu group M12. It was introduced by Conway (1987, 1997) and studied in detail by Conway, Elkies & Martin (2006).

Construction

The projective plane of order 3 has 13 points and 13 lines, each containing 4 points. The Mathieu groupoid can be visualized as a sliding block puzzle by placing 12 counters on 12 of the 13 points of the projective plane. A move consists of moving a counter from any point x to the empty point y, then exchanging the 2 other counters on the line containing x and y. The Mathieu groupoid consists of the permutations that can be obtained by composing several moves.

This is not a group because two operations A and B can only be composed if the empty point after carrying out A is the empty point at the beginning of B. It is in fact a groupoid (a category such that every morphism is invertible) whose 13 objects are the 13 points, and whose morphisms from x to y are the operations taking the empty point from x to y. The morphisms fixing the empty point form a group isomorphic to the Mathieu group M12 with 12×11×10×9×8 elements.

References
  • Conway, John Horton (1987), "Graphs and groups and M13", Graph Theory Notes of New York, XIV: 18–29
  • Conway, John Horton (1997), "M₁₃", Surveys in combinatorics, 1997 (London), London Math. Soc. Lecture Note Ser., 241, Cambridge University Press, pp. 1–11, doi:10.1017/CBO9780511662119.002, ISBN 9780511662119, MR 1477742
  • Conway, John Horton; Elkies, Noam D.; Martin, Jeremy L. (2006), "The Mathieu group M12 and its pseudogroup extension M13", Experimental Mathematics, 15 (2): 223–236, arXiv:math/0508630, doi:10.1080/10586458.2006.10128958, ISSN 1058-6458, MR 2253008
  • Nakashima, Yasuhiro (2008), "The transitivity of Conway's M₁₃", Discrete Mathematics, 308 (11): 2273–2276, doi:10.1016/j.disc.2007.04.053, ISSN 0012-365X, MR 2404553
  • Gill, Nick; Gillespie, Neil; Nixon, Anthony; Semeraro, Jason (2014). "Puzzle groups". arXiv:1405.1701v2 [math.GR].
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