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Schreier Coset Graph
In the area of mathematics called combinatorial group theory, the Schreier coset graph is a graph associated with a group ''G'', a generating set of ''G'', and a subgroup ''H'' ≤ ''G''. The Schreier graph encodes the abstract structure of a group modulo an equivalence relation formed by the coset. The graph is named after Otto Schreier, who used the term “Nebengruppenbild”. An equivalent definition was made in an early paper of Todd and Coxeter. Description The vertices of the graph are the right cosets ''Hg'' = for ''g'' in ''G''. The edges of the graph are of the form (''Hg'', ''Hgxi''). The Cayley graph of the group ''G'' with is the Schreier coset graph for ''H'' = . A spanning tree of a Schreier coset graph corresponds to a Schreier transversal, as in Schreier's subgroup lemma . The book "Categories and Groupoids" listed below relates this to the theory of covering morphisms of groupoids. A subgroup ''H'' of a group ''G'' determines a covering morph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spanning Tree
In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree (see about spanning forests below). If all of the edges of ''G'' are also edges of a spanning tree ''T'' of ''G'', then ''G'' is a tree and is identical to ''T'' (that is, a tree has a unique spanning tree and it is itself). Applications Several pathfinding algorithms, including Dijkstra's algorithm and the A* search algorithm, internally build a spanning tree as an intermediate step in solving the problem. In order to minimize the cost of power networks, wiring connections, piping, automatic speech recognition, etc., people often use algorithms that gradually build a spanning tree (or many such trees) as intermediate steps in the process of finding the minimum spanning tree. The Interne ... [...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 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 university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Spo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex-transitive Graph
In the mathematical field of graph theory, a vertex-transitive graph is a graph in which, given any two vertices and of , there is some automorphism :f : G \to G\ such that :f(v_1) = v_2.\ In other words, a graph is vertex-transitive if its automorphism group acts transitively on its vertices.. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph and Tietze's graph). Finite examples Finite vertex-transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The finite Cayley graphs (such as cube-connected cycles) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurwitz Group
In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus ''g'' > 1, stating that the number of such automorphisms cannot exceed 84(''g'' − 1). A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve.Technically speaking, there is an equivalence of categories between the category of compact Riemann surfaces with the orientation-preserving conformal maps and the category of non-singular complex projective algebraic curves with the algebraic morphisms. The theorem is named after Adolf Hurwitz, who proved it in . Hurwitz's bound also holds for algebraic curves over a field of characteristic 0, and over fields of posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic properties For , the group A''n'' is the commutator subgroup of the symmetric group S''n'' with index 2 and has therefore ''n''!/2 elements. It is the kernel of the signature group homomorphism explained under symmetric group. The group A''n'' is abelian if and only if and simple if and only if or . A5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group. The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto . We have the exact sequence . In Galois theory, this map, or rather the corresponding map , corresponds to associating the Lagrange resolvent cubic to a quartic, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graham Higman
Graham Higman FRS (19 January 1917 – 8 April 2008) was a prominent English mathematician known for his contributions to group theory. Biography Higman was born in Louth, Lincolnshire, and attended Sutton High School, Plymouth, winning a scholarship to Balliol College, Oxford. In 1939 he co-founded The Invariant Society, the student mathematics society, and earned his DPhil from the University of Oxford in 1941. His thesis, ''The units of group-rings'', was written under the direction of J. H. C. Whitehead. From 1960 to 1984 he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. Higman was awarded the Senior Berwick Prize in 1962 and the De Morgan Medal of the London Mathematical Society in 1974. He was the founder of the Journal of Algebra and its editor from 1964 to 1984. Higman had 51 D.Phil. students, including Jonathan Lazare Alperin, Rosemary A. Bailey, Marston Conder, John Mackintosh Howie, and Peter M. Neumann. He was also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Representation
In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices. The term also refers to the combination of the two. Abstract permutation representation A permutation representation of a group G on a set X is a homomorphism from G to the symmetric group of X: : \rho\colon G \to \operatorname(X). The image \rho(G)\sub \operatorname(X) is a permutation group and the elements of G are represented as permutations of X. A permutation representation is equivalent to an action of G on the set X: :G\times X \to X. See the article on group action for further details. Linear permutation representation If G is a permutation group of degree n, then the permutation representation of G is the linear representation of G :\rho\colon G\to \operatorname_n(K) which maps g\in G to the corresponding permutation matrix (here K is an arb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coset Enumeration
In mathematics, coset enumeration is the problem of counting the cosets of a subgroup ''H'' of a group ''G'' given in terms of a presentation. As a by-product, one obtains a permutation representation for ''G'' on the cosets of ''H''. If ''H'' has a known finite order, coset enumeration gives the order of ''G'' as well. For small groups it is sometimes possible to perform a coset enumeration by hand. However, for large groups it is time-consuming and error-prone, so it is usually carried out by computer. Coset enumeration is usually considered to be one of the fundamental problems in computational group theory. The original algorithm for coset enumeration was invented by John Arthur Todd and H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t .... Various improvement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Image
In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B. Image and inverse image may also be defined for general binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a function from the set X to the set Y. Image of an element If x is a member of X, then the image of x under f, denoted f(x), is the value of f when applied to x. f(x) is alternatively known as the output of f for argument x. Given y, the function f is said to "" or "" if there exists some x in the function's domain such that f(x) = y. Similarly, given a set S, f is said to "" if the ... [...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 \c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schreier's Subgroup Lemma
In mathematics, Schreier's lemma is a theorem in group theory used in the Schreier–Sims algorithm and also for finding a presentation of a subgroup. Statement Suppose H is a subgroup of G, which is finitely generated with generating set S, that is, G = \langle S\rangle. Let R be a right transversal of H in G. In other words, R is (the image of) a section of the quotient map G \to H\backslash G, where H\backslash G denotes the set of right cosets of H in G. We make the definition that given g\in G, \overline is the chosen representative in the transversal R of the coset Hg, that is, :g\in H\overline. Then H is generated by the set :\. Hence, in particular, Schreier's lemma implies that every subgroup of finite index of a finitely generated group is again finitely generated. Example Let us establish the evident fact that the group Z3 = Z/3Z is indeed cyclic. Via Cayley's theorem, Z3 is a subgroup of the symmetric group In abstract algebra, the symmetric group defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
