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3-transposition Group
In mathematical group theory, a 3-transposition group is a group (mathematics), group generated by a conjugacy class of involution (mathematics), involutions, called the 3-transpositions, such that the product of any two involutions from the conjugacy class has order (group theory), order at most 3. They were first studied by who discovered the three Fischer groups as examples of 3-transposition groups. History first studied 3-transposition groups in the special case when the product of any two distinct 3-transpositions has order 3. He showed that a finite group with this property is solvable, and has a (nilpotent) 3-group of index 2. used these groups to construct examples of non-abelian CH-quasigroups and to describe the structure of commutative Moufang loops of exponent 3. Fischer's theorem Suppose that ''G'' is a group that is generated by a conjugacy class ''D'' of 3-transpositions and such that the p-core, 2 and 3 cores ''O''2(''G'') and ''O''3(''G'') are both contained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Special Unitary Group
In mathematics, the projective unitary group is the quotient of the unitary group by the right multiplication of its center, , embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space. In terms of matrices, elements of are complex unitary matrices, and elements of the center are diagonal matrices equal to multiplied by the identity matrix. Thus, elements of correspond to equivalence classes of unitary matrices under multiplication by a constant phase . Abstractly, given a Hermitian space , the group is the image of the unitary group in the automorphism group of the projective space . Projective special unitary group The projective special unitary group PSU() is equal to the projective unitary group, in contrast to the orthogonal case. The connections between the U(), SU(), their centers, and the projective unitary groups is shown at right. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Zeitschrift .
''Mathematische Zeitschrift'' (German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag. It was founded in 1918 and edited by Leon Lichtenstein together with Konrad Knopp, Erhard Schmidt, and Issai Schur. Past editors include Erich Kamke, Friedrich Karl Schmidt, Rolf Nevanlinna, Helmut Wielandt, and Olivier Debarre Olivier Debarre (born 1959) is a French mathematician who specializes in complex algebraic geometry.Debarr ... External links * * Mathematics journals Publications established in 1918 {{math ...[...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] |
Petersen Graph
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring. Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by . Kempe observed that its vertices can represent the ten lines of the Desargues configuration, and its edges represent pairs of lines that do not meet at one of the ten points of the configuration. Donald Knuth states that the Petersen graph is "a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general." The Petersen graph also makes an appearance in tropical geometry. The cone over the Petersen graph is naturally identif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transposition (mathematics)
In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, mapping to themselves) all other elements of ''X''. If ''S'' has ''k'' elements, the cycle is called a ''k''-cycle. Cycles are often denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted. For example, given ''X'' = , the permutation (1, 3, 2, 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 (so ''S'' = ''X'') is a 4-cycle, and the permutation (1, 3, 2) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 (so ''S'' = and 4 is a fixed element) is a 3-cycle. On the other hand, the permutation that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs and . The set ''S'' is called the orbit of the cycle. Every permutation on finitely many eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by Function composition, composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrix, orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose invertible matrix, inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact group, compact. The orthogonal group in dimension has two connected component (topology), connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted . It consists of all orthogonal matrices of determinant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by \mathrm(n). Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name "symplectic group" is due to Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The metaplect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-core
In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the ''p''-core of a group. The normal core Definition For a group ''G'', the normal core or normal interiorRobinson (1996) p.16 of a subgroup ''H'' is the largest normal subgroup of ''G'' that is contained in ''H'' (or equivalently, the intersection of the conjugates of ''H''). More generally, the core of ''H'' with respect to a subset ''S'' ⊆ ''G'' is the intersection of the conjugates of ''H'' under ''S'', i.e. :\mathrm_S(H) := \bigcap_. Under this more general definition, the normal core is the core with respect to ''S'' = ''G''. The normal core of any normal subgroup is the subgroup itself. Significance Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup of any point acts as the identity on its entire orbit. Thus, in ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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