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).

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 
• 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, doi:10.1080/10586458.2006.10128958, ISSN 1058-6458, http://nrs.harvard.edu/urn-3:HUL.InstRepos:2794826 
• 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 
• Gill, Nick; Gillespie, Neil; Nixon, Anthony; Semeraro, Jason (2014). "Puzzle groups". arXiv:1405.1701v2 [math.GR].

External links

• The Mathieu groupoid