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Universal Embedding Theorem
The universal embedding theorem, or Krasner–Kaloujnine universal embedding theorem, is a theorem from the mathematical discipline of 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 ... first published in 1951 by Marc Krasner and Lev Kaluznin. The theorem states that any group extension In 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 mod ... of a group by a group is isomorphic to a subgroup of the regular wreath product In group theory, the wreath product is a special combination of two Group (mathematics), groups based on the semidirect product. It is formed by the Actio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [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] [Bing] [Yahoo] [DuckDuckGo] [Baidu] |
Homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [Baidu] |
Simple Group
SIMPLE Group Limited is a list of conglomerates, conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The date of incorporation is listed as 1999 by Companies House of Gibraltar, who class it as a holding company; however it is understood that SIMPLE Group's business and trading activities date to the second part of the 90s, probably as an incorporated body. SIMPLE Group Limited is a Conglomerate (company), conglomerate that cultivate secrecy, they are not listed on any Stock Exchange and the group is owned by a complicated series of offshore trusts. The Sunday Times stated that SIMPLE Group's interests could be evaluated at £77 million; the shareholders are understood to be a group of Swiss Private Banks' owners. Each of the companies operating under the SIMPLE Group umbrella is independent, with most being majority ow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [Baidu] |
Subsemigroup
In mathematics, a semigroup is an algebraic structure In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ... consisting of a set together with an associative In 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 i ... internal binary operation In 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 ... on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [Baidu] |
Image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensional picture, that resembles a subject. In the context of signal processing, an image is a distributed amplitude of color(s). In optics, the term “image” may refer specifically to a 2D image. An image does not have to use the entire visual system to be a visual representation. A popular example of this is of a greyscale image, which uses the visual system's sensitivity to brightness across all wavelengths, without taking into account different colors. A black and white visual representation of something is still an image, even though it does not make full use of the visual system's capabilities. Images are typically still, but in some cases can be moving or animated. Characteristics Images may be two or three-dimension, dimensional, su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ... consisting of a set together with an associative internal binary operation In 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 ... on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [Baidu] |
Krohn–Rhodes Theory
In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata theory, automata that seeks to decompose them in terms of elementary components. These components correspond to finite aperiodic semigroups and finite simple groups that are combined in a feedback-free manner (called a "wreath product" or "cascade"). Krohn and John Rhodes (mathematician), Rhodes found a general decomposition for finite automata. In doing their research, though, the authors discovered and proved an unexpected major result in finite semigroup theory, revealing a deep connection between finite automata and semigroups. Definitions and description of the Krohn–Rhodes theorem Let ''T'' be a semigroup. A semigroup ''S'' that is a homomorphic image of a subsemigroup of ''T'' is said to be a divisor of ''T''. The Krohn–Rhodes theorem for finite semigroups states that every finite semigroup ''S'' is a divisor of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [Baidu] |
Surjective Function
In mathematics, a surjective function (also known as surjection, or onto function) is a Function (mathematics), function that every element can be mapped from element so that . In other words, every element of the function's codomain is the Image (mathematics), image of one element of its Domain of a function, domain. It is not required that be unique (mathematics), unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms ''injective function, injective'' and ''bijective function, bijective'' were introduced by Nicolas Bourbaki, a group of mainly France, French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word ''wikt:sur#French, sur'' means ''over'' or ''above'', and relates to the fact that the image (mathematics), image of the domain of a surjective function completely covers the functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [Baidu] |
Injective Function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function (mathematics), function that maps Distinct (mathematics), distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent Contraposition, contrapositive statement.) In other words, every element of the function's codomain is the Image (mathematics), image of one element of its Domain of a function, domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [Baidu] |
Marc Krasner
Marc Krasner (1912 – 13 May 1985, in Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), ma ...) was a Russian Empire-born French mathematician, who worked on algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob .... Krasner emigrated from the Soviet Union to France and received in 1935 his PhD from the University of Paris , image_name = Coat of arms of the University of Paris.svg , image_size = 150px , caption = Coat of Arms , latin_name = Universitas magistrorum et scholarium Parisiensis , motto = ''Hic et ubique terrarum'' (Latin Latin (, or , ) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [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 inner automorphism, conjugation by members of the Group (mathematics), 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 Kernel of a homomorphism, kernels of Group homomorphism, group homomorphisms with domain of a function, 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 inner automorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [Baidu] |
Group Action (mathematics)
In mathematics, a group action on a space (mathematics), space is a group homomorphism of a given group (mathematics), group into the group of transformation (geometry), transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometry, Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertex (geometry), vertices, the edge (geometry), edges, and the face (geometry), faces of the polyhedron. A group action on a vector space is called a Group representation, representation of the group. In the case of a finite-dimensional vector space, it allo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Bing] [Yahoo] [DuckDuckGo] [Baidu] |
