In mathematics, the Mathieu groupoid M
13 is a
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
*'' Group'' with a partial func ...
acting on 13 points such that the stabilizer of each point is the
Mathieu group M12. It was introduced by and studied in detail by .
Construction
The
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
of order 3 has 13 points and 13 lines, each containing 4 points. The Mathieu groupoid can be visualized as a
sliding block puzzle
A sliding puzzle, sliding block puzzle, or sliding tile puzzle is a combination puzzle that challenges a player to slide (frequently flat) pieces along certain routes (usually on a board) to establish a certain end-configuration. The pieces to ...
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 M
12 with 12×11×10×9×8 elements.
References
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*{{cite arXiv , last1=Gill , first1=Nick , last2=Gillespie , first2=Neil , last3=Nixon , first3=Anthony , last4=Semeraro , first4=Jason , eprint=1405.1701v2 , class=math.GR , title=Puzzle groups , year=2014
External links
The Mathieu groupoid
Sporadic groups
John Horton Conway