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Mathieu Group M23
In the area of modern algebra known as group theory, the Mathieu group ''M''23 is a sporadic simple group of order : 2732571123 = 10200960 : ≈ 1 × 107. History and properties ''M''23 is one of the 26 sporadic groups and was introduced by . It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial. calculated the integral cohomology, and showed in particular that M23 has the unusual property that the first 4 integral homology groups all vanish. The inverse Galois problem seems to be unsolved for M23. In other words, no polynomial in Z 'x''seems to be known to have M23 as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups. Construction using finite fields Let be the finite field with 211 elements. Its group of units has order − 1 = 2047 = 23 · 89, so it has a cyclic subgroup of order 23. The Mathieu group M23 can be identified with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modern Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form category (mathematics), mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism Group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the group of invertible linear transformations from ''X'' to itself (the general linear group of ''X''). If instead ''X'' is a group, then its automorphism group \operatorname(X) is the group consisting of all group automorphisms of ''X''. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. Examples If ''X'' is a set with no additional structure, then any bijection from ''X'' to itself is an automorphism, and hence the automorphism group of ''X'' in this case is precisely the symmetric group of ''X''. If the set ''X'' has additional struct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books by decree in 1586, it is the second oldest university press after Cambridge University Press. It is a department of the University of Oxford and is governed by a group of 15 academics known as the Delegates of the Press, who are appointed by the vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, opposite Somerville College, in the inner suburb of Jericho. For the last 500 years, OUP has primarily focused on the publication of pedagogical texts and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Well-known products include the ''Methods in Enzymology'' series and encyclopedias such as ''The International Encyclopedia of Public Health'' and the ''Encyclopedia of Neuroscience''. See also * Akademische Verlagsgesellschaft (AVG) — the German predecessor, founded in 1906 by Leo Jolowicz (1868–1940), the father of Walter Jolowicz Walter may refer to: People * Walter (name), both a surname and a given name * Little Walter, American blues harmonica player Marion Wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and public-domain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original pagination and ... [...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] |
Mathieu Group M11
In the area of modern algebra known as group theory, the Mathieu group ''M11'' is a sporadic simple group of order : 2432511 = 111098 = 7920. History and properties ''M11'' is one of the 26 sporadic groups and was introduced by . It is the smallest sporadic group and, along with the other four Mathieu groups, the first to be discovered. The Schur multiplier and the outer automorphism group are both trivial. ''M11'' is a sharply 4-transitive permutation group on 11 objects. It admits many generating sets of permutations, such as the pair (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) of permutations used by the GAP computer algebra system. Representations M11 has a sharply 4-transitive permutation representation on 11 points. The point stabilizer is sometimes denoted by M10, and is a non-split extension of the form A6.2 (an extension of the group of order 2 by the alternating group A6). This action is the automorphism group of a Steiner system S(4,5,11) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Group M22
In the area of modern algebra known as group theory, the Mathieu group ''M22'' is a sporadic simple group of Order (group theory), order : 27325711 = 443520 : ≈ 4. History and properties ''M22'' is one of the 26 sporadic groups and was introduced by . It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M22 is cyclic of order 12, and the outer automorphism group has order 2. There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature. incorrectly claimed that the Schur multiplier of M22 has order 3, and in a correction incorrectly claimed that it has order 6. This caused an error in the title of the paper announcing the discovery of the Janko group J4. showed that the Schur multiplier is in fact cyclic of order 12. calculated the 2-part of all the cohomology of M22. Representations M22 has a 3-transitive permutation representation on 22 points, with point stabilizer th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring (mathematics), ring , often denoted , is defined to be the smallest number of times one must use the ring's identity element, multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent (group theory), exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank 3 Action
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * Hierarchy of the Catholic Church * Military rank * Police ranks of the United States * Ranking member, S politicsthe most senior member of a committee from the minority party, and thus second-most senior member of a committee * Imperial, royal and noble ranks Level or position in society *Social class *Social position *Social status Places * Rank, Iran, a village * Rank, Nepal, a village development committee People * Rank (surname), a list of people with the name Arts, entertainment, and media Music * ''Rank'' (album), a live album by the Smiths * "Rank", a song by Artwork from '' A Bugged Out Mix'' Other arts, entertainment, and media * Rank (chess), a row of the chessboard * ''Rank'' (film), a short film directed by David Yates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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