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.
Referencesedit
Conway, John Horton (1987), "Graphs and groups and M13", Graph Theory Notes of New York, XIV: 18–29
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, hdl:1808/6365, 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
mathieu, groupoid, mathematics, groupoid, acting, points, such, that, stabilizer, each, point, mathieu, group, introduced, conway, 1987, 1997, studied, detail, conway, elkies, martin, 2006, construction, editthe, projective, plane, order, points, lines, each, . 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 amp Martin 2006 Construction editThe 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 editConway John Horton 1987 Graphs and groups and M13 Graph Theory Notes of New York XIV 18 29 Conway John Horton 1997 M Surveys in combinatorics 1997 London London Math Soc Lecture Note Ser vol 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 hdl 1808 6365 ISSN 1058 6458 MR 2253008 Nakashima Yasuhiro 2008 The transitivity of Conway s M 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 External links editThe Mathieu groupoid Retrieved from https en wikipedia org w index php title Mathieu groupoid amp oldid 1222661879, wikipedia, wiki, book, books, library,
