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Absolute Presentation Of A Group
In mathematics, an absolute presentation is one method of defining a group.B. Neumann, ''The isomorphism problem for algebraically closed groups,'' in: Word Problems, Decision Problems, and the Burnside Problem in Group Theory, Amsterdam-London (1973), pp. 553–562. Recall that to define a group G by means of a presentation, one specifies a set S of generators so that every element of the group can be written as a product of some of these generators, and a set R of relations among those generators. In symbols: :G \simeq \langle S \mid R \rangle. Informally G is the group generated by the set S such that r = 1 for all r \in R. But here there is a tacit assumption that G is the "freest" such group as clearly the relations are satisfied in any homomorphic image of G. One way of being able to eliminate this tacit assumption is by specifying that certain words in S should not be equal to 1. That is we specify a set I, called the set of irrelations, such that i \ne 1 for all i \in I ... [...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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Group
In group theory, a group A\ is algebraically closed if any finite set of equations and inequations that are applicable to A\ have a solution in A\ without needing a group extension. This notion will be made precise later in the article in . Informal discussion Suppose we wished to find an element x\ of a group G\ satisfying the conditions (equations and inequations): ::x^2=1\ ::x^3=1\ ::x\ne 1\ Then it is easy to see that this is impossible because the first two equations imply x=1\ . In this case we say the set of conditions are inconsistent with G\ . (In fact this set of conditions are inconsistent with any group whatsoever.) Now suppose G\ is the group with the multiplication table to the right. Then the conditions: ::x^2=1\ ::x\ne 1\ have a solution in G\ , namely x=a\ . However the conditions: ::x^4=1\ ::x^2a^ = 1\ Do not have a solution in G\ , as can easily be checked. However if we extend the group G \ to the group H \ with the adjacent multiplic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of such elements. By definition, every finite group is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated. Examples * Every quotient of a finitely generated group ''G'' is finitely generated; the quotient group is generated by the images of the generators of ''G'' under the canonical projection. * A subgroup of a finitely generated group need not be finitely generated. * A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Isomorphism Problem
In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given Presentation of a group, finite group presentations refer to Isomorphism, isomorphic Group (mathematics), groups. The isomorphism problem was formulated by Max Dehn, and together with the Word problem for groups, word problem and conjugacy problem, is one of three fundamental decision problems in group theory he identified in 1911. All three problems are Decidability (logic), undecidable: there does not exist a computer algorithm that correctly solves every instance of the isomorphism problem, or of the other two problems, regardless of how much time is allowed for the algorithm to run. In fact the problem of deciding whether a group is trivial is undecidable, (See Corollary 3.4) a consequence of the Adian–Rabin theorem due to Sergei Adian and Michael O. Rabin. References * Group theory Undecidable problems {{Abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernhard Neumann
Bernhard Hermann Neumann (15 October 1909 – 21 October 2002) was a German-born British-Australian mathematician, who was a leader in the study of group theory. Early life and education After gaining a D.Phil. from Friedrich-Wilhelms Universität in Berlin in 1932 he earned a Ph.D. at the University of Cambridge in 1935 and a Doctor of Science at the University of Manchester in 1954. His doctoral students included Gilbert Baumslag, László Kovács, Michael Newman, and James Wiegold. After war service with the British Army, he became a lecturer at University College, Hull, before moving in 1948 to the University of Manchester, where he spent the next 14 years. In 1954 he received a DSc from the University of Cambridge. In 1962 he migrated to Australia to take up the Foundation Chair of the Department of Mathematics within the Institute of Advanced Studies of the Australian National University (ANU), where he served as head of the department until retiring in 1975. In addition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Of A Group
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element of a group, is thus the smallest positive integer such that , where denotes the identity element of the group, and denotes the product of copies of . If no such exists, the order of is infinite. The order of a group is denoted by or , and the order of an element is denoted by or , instead of \operatorname(\langle a\rangle), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup of a finite group , the order of the subgroup divides the order of the group; that is, is a divisor of . In particular, the order of any element is a divisor of . Example The symmetric group S3 has th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a ''generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Retronym
A retronym is a newer name for an existing thing that helps differentiate the original form/version from a more recent one. It is thus a word or phrase created to avoid confusion between older and newer types, whereas previously (before there were more than one type) no clarification was required. Advances in technology are often responsible for the coinage of retronyms. For example, the term "acoustic guitar" was coined with the advent of electric guitars; analog watches were renamed to distinguish them from digital watches once the latter were invented; and "push bike" was created to distinguish from motorbikes and motorized bicycles; finally "feature phones" were also coined behind smartphones. Etymology The term ''retronym'', a neologism composed of the classical compound, combining forms '' retro-'' (from Latin ''retro'', "before") + '' -nym'' (from Greek '' ónoma'', "name"), was coined by Frank Mankiewicz in 1980 and popularized by William Safire in ''The New York Time ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigorchuk Topology
Hryhorchuk is a Ukrainian surname derived from the given name "Hryhor", or Gregory. A variant Ryhorchuk is derived form the simplified variant "Ryhor" of the given name "Hryhor". *The Russified form is Grigorchuk. *Belarusian forms: Hryharchuk (from Ukrainian), Grygarchuk (via Russian), or Ryharchuk (native or from "Ryhorchuk"). *Polonized form: Gregorczuk. *Lithuanized: Grigorčukas (via Russian), Gregorčukas (via Polish) Notable people with the surname include: * Lidiia Hryhorchuk * Roman Hryhorchuk * Rostislav Grigorchuk See also * * *Grigorchuk group In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate (that is, f ... {{surname Ukrainian-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng^ \in N for all g \in G and n \in N. The usual notation for this relation is N \triangleleft G. Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G are precisely the kernels of group homomorphisms with domain G, which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups. Definitions A subgroup N of a group G is called a normal subgroup of G if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is always in N. The usual notation for this re ... [...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] |
