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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. It was introduced by and studied in detail by . 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 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' ... [...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 \circ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Group M12
In the area of modern algebra known as group theory, the Mathieu group ''M12'' is a sporadic simple group of order : 12111098 = 2633511 = 95040. History and properties ''M12'' is one of the 26 sporadic groups and was introduced by . It is a sharply 5-transitive permutation group on 12 objects. showed that the Schur multiplier of M12 has order 2 (correcting a mistake in where they incorrectly claimed it has order 1). The double cover had been implicitly found earlier by , who showed that M12 is a subgroup of the projective linear group of dimension 6 over the finite field with 3 elements. The outer automorphism group has order 2, and the full automorphism group M12.2 is contained in M24 as the stabilizer of a pair of complementary dodecads of 24 points, with outer automorphisms of M12 swapping the two dodecads. Representations calculated the complex character table of M12. M12 has a strictly 5-transitive permutation representation on 12 points, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 be moved may consist of simple shapes, or they may be imprinted with colours, patterns, sections of a larger picture (like a jigsaw puzzle), numbers, or letters. Sliding puzzles are essentially two-dimensional in nature, even if the sliding is facilitated by mechanically interlinked pieces (like partially encaged marbles) or three-dimensional tokens. In manufactured wood and plastic products, the linking and encaging is often achieved in combination, through mortise-and-tenon key channels along the edges of the pieces. In at least one vintage case of the popular Chinese cognate game Huarong Road, a wire screen prevents lifting of the pieces, which remain loose. As the illustration shows, some sliding puzzles are mechanical puzzles. Howev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and are composed to yield a function that maps in domain to in codomain . Intuitively, if is a function of , and is a function of , then is a function of . The resulting ''composite'' function is denoted , defined by for all in . The notation is read as " of ", " after ", " circle ", " round ", " about ", " composed with ", " following ", " then ", or " on ", or "the composition of and ". Intuitively, composing functions is a chaining process in which the output of function feeds the input of function . The composition of functions is a special case of the composition of relations, sometimes also denoted by \circ. As a result, all properties of composition of relations are true of composition of functions, such as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sporadic Groups
In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group, in which case there would be 27 sporadic groups. The monster group is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it. Names Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
