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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, Mathieu group M12. It was introduced by and studied in detail by . Construction The Projective plane#A finite example, 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 Function composition, 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 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 function replacing the binary operation; *''Category'' in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called ''inverse'' by analogy with group theory. A groupoid where there is only one object is a usual group. In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed g:A \rightarrow B, h:B \rightarrow C, say. Composition is then a total function: \circ : (B \rightarrow C) \rightarrow (A \rightarrow B) \rightarrow A \rightarrow C , so that h \ci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |