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Artin Transfer (group Theory)
In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index. Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, a partial order on the kernels and targets of Artin transfers has recently turned out to be compatible with parent-descendant relations between finite ''p''-groups (with a prime number ''p''), which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite ''p''-groups and for searching and identifying particular groups in descendant trees by looking for patterns defined by the kernels and ... [...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] |
Artin Transfer (group Theory)
In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index. Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, a partial order on the kernels and targets of Artin transfers has recently turned out to be compatible with parent-descendant relations between finite ''p''-groups (with a prime number ''p''), which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite ''p''-groups and for searching and identifying particular groups in descendant trees by looking for patterns defined by the kernels and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Homomorphism (quotient Group)
In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space ''X'' to a topological space ''Y'' induces a group homomorphism from the fundamental group of ''X'' to the fundamental group of ''Y''. More generally, in category theory, any functor by definition provides an induced morphism in the target category for each morphism in the source category. For example, fundamental groups, higher homotopy groups, singular homology, and De Rham cohomology are algebraic structures that are ''functorial'', meaning that their definition provides a functor from (e.g.) the category of topological spaces to (e.g.) the category of groups or rings. This means that each space is associated with an algebraic structure, while each continuous map between spaces is associated with a structure-preserving map between structures, called an induced homomorphism. A homomorphism in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Coset
In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let be a group, and let and be subgroups. Let act on by left multiplication and let act on by right multiplication. For each in , the -double coset of is the set :HxK = \. When , this is called the -double coset of . Equivalently, is the equivalence class of under the equivalence relation : if and only if there exist in and in such that . The set of all double cosets is denoted by H \,\backslash G / K. Properties Suppose that is a group with subgroups and acting by left and right multiplication, respectively. The -double cosets of may be equivalently described as orbits for the product group acting on by . Many of the basic properties of double cosets follow immediately from the fact that they are orbits. However, because is a group and and are subgroups acting by multiplicati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology Group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfer (group Theory)
In the mathematical field of group theory, the transfer defines, given a group ''G'' and a subgroup of finite index ''H'', a group homomorphism from ''G'' to the abelianization of ''H''. It can be used in conjunction with the Sylow theorems to obtain certain numerical results on the existence of finite simple groups. The transfer was defined by and rediscovered by . Construction The construction of the map proceeds as follows:Following Scott 3.5 Let 'G'':''H''= ''n'' and select coset representatives, say :x_1, \dots, x_n,\, for ''H'' in ''G'', so ''G'' can be written as a disjoint union :G = \bigcup\ x_i H. Given ''y'' in ''G'', each ''yxi'' is in some coset ''xjH'' and so :yx_i = x_jh_i for some index ''j'' and some element ''h''''i'' of ''H''. The value of the transfer for ''y'' is defined to be the image of the product :\textstyle \prod_^n h_i in ''H''/''H''′, where ''H''′ is the commutator subgroup of ''H''. The order of the factors is irrelevant since ''H''/''H''†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x \mapsto -x), reciprocation (x \mapsto 1/x), and complex conjugation (z \mapsto \bar z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions ''f'' and ''g'' is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: :a_0 = a_1 = 1 and a_n = a_ + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
