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Mathieu groupoid
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<h2 id="construction">Construction</h2> <p>The <a href="/facts/Projective_plane/p6ot8oNv">projective plane</a> of order 3 has 13 points and 13 lines, each containing 4 points. The Mathieu groupoid can be visualized as a <a href="/facts/Sliding_block_puzzle/Qu7RofMh">sliding block puzzle</a> by placing 12 counters on 12 of the 13 points of the projective plane. A move consists of moving a counter from any point <i>x</i> to the empty point <i>y</i>, then exchanging the 2 other counters on the line containing <i>x</i> and <i>y</i>. The Mathieu groupoid consists of the permutations that can be obtained by <a href="/facts/Function_composition/r5Pvnmth">composing</a> several moves. </p><p>This is not a group because two operations <i>A</i> and <i>B</i> can only be composed if the empty point after carrying out <i>A</i> is the empty point at the beginning of <i>B</i>. 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 <i>x</i> to <i>y</i> are the operations taking the empty point from <i>x</i> to <i>y</i>. The morphisms fixing the empty point form a group isomorphic to the Mathieu group M12 with 12×11×10×9×8 elements. </p> <ul><li><a href="/facts/John_Horton_Conway/g3PA1zoK">Conway, John Horton</a> (1987), "Graphs and groups and M13", <i>Graph Theory Notes of New York</i>, XIV: 18–29</li> <li><a href="/facts/John_Horton_Conway/g3PA1zoK">Conway, John Horton</a> (1997), "M₁₃", <i>Surveys in combinatorics, 1997 (London)</i>, London Math. Soc. Lecture Note Ser., vol. 241, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, pp. 1–11, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9780511662119.002">10.1017/CBO9780511662119.002</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780511662119, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1477742">1477742</a></li> <li><a href="/facts/John_Horton_Conway/g3PA1zoK">Conway, John Horton</a>; <a href="/facts/Noam_Elkies/t1Uerz9E">Elkies, Noam D.</a>; Martin, Jeremy L. (2006), <a href="http://nrs.harvard.edu/urn-3:HUL.InstRepos:2794826">"The Mathieu group M12 and its pseudogroup extension M13"</a>, <i>Experimental Mathematics</i>, 15 (2): 223–236, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0508630">math/0508630</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1080%2F10586458.2006.10128958">10.1080/10586458.2006.10128958</a>, <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/1808%2F6365">1808/6365</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1058-6458">1058-6458</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2253008">2253008</a></li> <li>Nakashima, Yasuhiro (2008), "The transitivity of Conway's M₁₃", <i>Discrete Mathematics</i>, 308 (11): 2273–2276, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.disc.2007.04.053">10.1016/j.disc.2007.04.053</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0012-365X">0012-365X</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2404553">2404553</a></li> <li>Gill, Nick; Gillespie, Neil; Nixon, Anthony; Semeraro, Jason (2014). "Puzzle groups". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1405.1701v2">1405.1701v2</a> [<a href="https://arxiv.org/archive/math.GR">math.GR</a>].</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.neverendingbooks.org/the-mathieu-groupoid-1">The Mathieu groupoid</a></li></ul>