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Computational Group Theory
In mathematics, computational group theory is the study of groups by means of computers. It is concerned with designing and analysing algorithms and data structures to compute information about groups. The subject has attracted interest because for many interesting groups (including most of the sporadic groups) it is impractical to perform calculations by hand. Important algorithms in computational group theory include: * the Schreier–Sims algorithm for finding the order of a permutation group * the Todd–Coxeter algorithm and Knuth–Bendix algorithm for coset enumeration * the product-replacement algorithm for finding random elements of a group Two important computer algebra systems (CAS) used for group theory are GAP and Magma. Historically, other systems such as CAS (for character theory) and Cayley (a predecessor of Magma) were important. Some achievements of the field include: * complete enumeration of all finite groups of order less than 2000 * computation of ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character Theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations. Applications Characters of irreducible representations encode many important p ... [...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 King Henry VIII in 1534, it is the oldest university press 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 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 university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles C
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was "free man". The Old English descendant of this word was '' Ċearl'' or ''Ċeorl'', as the name of King Cearl of Mercia, that disappeared after the Norman conquest of England. The name was notably borne by Charlemagne (Charles the Great), and was at the time Latinized as ''Karolus'' (as in ''Vita Karoli Magni''), later also as '' Carolus''. Some Germanic languages, for example Dutch and German, have retained the word in two separate senses. In the particular case of Dutch, ''Karel'' refers to the given name, whereas the noun ''kerel'' means "a bloke, fellow, man". Etymology The name's etymology is a Common Germanic noun ''*karilaz'' meaning "free man", which survives in English as churl (< Old English ''ċeorl''), which developed its dep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RWTH Aachen
RWTH Aachen University (), also known as North Rhine-Westphalia Technical University of Aachen, Rhine-Westphalia Technical University of Aachen, Technical University of Aachen, University of Aachen, or ''Rheinisch-Westfälische Technische Hochschule Aachen'', is a German public research university located in Aachen, North Rhine-Westphalia, Germany. With more than 47,000 students enrolled in 144 study programs, it is the largest technical university in Germany. In 2018, the university was ranked 31st in the world university rankings in the field of engineering and technology, and 36th world-wide in the category of natural sciences.Daten & Fakten – RWTH AACHEN UNIVERSITY – Deutsch Rwth-aachen.de (12 December 2011). Retrieved on 2013-09-18. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rutgers University
Rutgers University (; RU), officially Rutgers, The State University of New Jersey, is a public land-grant research university consisting of four campuses in New Jersey. Chartered in 1766, Rutgers was originally called Queen's College, and was affiliated with the Dutch Reformed Church. It is the eighth-oldest college in the United States, the second-oldest in New Jersey (after Princeton University), and one of the nine U.S. colonial colleges that were chartered before the American Revolution.Stoeckel, Althea"Presidents, professors, and politics: the colonial colleges and the American revolution", ''Conspectus of History'' (1976) 1(3):45–56. In 1825, Queen's College was renamed Rutgers College in honor of Colonel Henry Rutgers, whose substantial gift to the school had stabilized its finances during a period of uncertainty. For most of its existence, Rutgers was a private liberal arts college but it has evolved into a coeducational public research university after being des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Sims (mathematician)
Charles Coffin Sims (April 14, 1937 – October 23, 2017J. J. O'Connor and E. F. Robertson''Charles Sims biography'' MacTutor History of Mathematics archive. Accessed 2018-12-20.) was an American mathematician best known for his work in group theory. Together with Donald G. Higman he discovered the Higman–Sims group, one of the sporadic groups. The permutation group software developed by Sims also led to the proof of existence of the Lyons group (also known as the Lyons–Sims group) and the O'Nan group (also known as the O'Nan–Sims group). Sims was born and raised in Elkhart, Indiana, and received his B.S. from the University of Michigan. He did his graduate studies at Harvard University, where he was a student of John G. Thompson and received his Ph.D. degree in 1963. In his thesis, he enumerated ''p''-groups, giving sharp asymptotic upper and lower bounds. Sims is one of the founders of computational group theory and is the eponym of the Schreier–Sims algorith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Notices Of The American Mathematical Society
''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since January 1995 is available in its entirety on the journal web site. Articles are peer-reviewed by an editorial board of mathematical experts. Since 2019, the editor-in-chief is Erica Flapan. The cover regularly features mathematical visualization Mathematical phenomena can be understood and explored via visualization. Classically this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century), while today it ...s. The ''Notices'' is self-described to be the world's most widely read mathematical journal. As the membership journal of the American Mathematical Society, the ''Notices'' is sent to the approximately 30,000 AMS members worldwide, one-third of whom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ohio State University
The Ohio State University, commonly called Ohio State or OSU, is a public land-grant research university in Columbus, Ohio. A member of the University System of Ohio, it has been ranked by major institutional rankings among the best public universities in the United States. Founded in 1870 as the state's land-grant university and the ninth university in Ohio with the Morrill Act of 1862, Ohio State was originally known as the Ohio Agricultural and Mechanical College and focused on various agricultural and mechanical disciplines, but it developed into a comprehensive university under the direction of then-Governor and later U.S. president Rutherford B. Hayes, and in 1878, the Ohio General Assembly passed a law changing the name to "the Ohio State University" and broadening the scope of the university. Admission standards tightened and became greatly more selective throughout the 2000s and 2010s. Ohio State's political science department and faculty have greatly contri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Black Box Group
In computational group theory, a black box group (black-box group) is a group ''G'' whose elements are encoded by bit strings of length ''N'', and group operations are performed by an oracle (the "black box"). These operations include: • taking a product ''g''·''h'' of elements ''g'' and ''h'', • taking an inverse ''g''−1 of element ''g'', • deciding whether ''g'' = 1. This class is defined to include both the permutation groups and the matrix groups. The upper bound on the order of ''G'' given by , ''G'', ≤ 2''N'' shows that ''G'' is finite. Applications The black box groups were introduced by Babai and Szemerédi in 1984. They were used as a formalism for (constructive) ''group recognition'' and '' property testing''. Notable algorithms include the ''Babai's algorithm'' for finding random group elements, the ''Product Replacement Algorithm'', and '' testing group commutativity''. Many early algorithms in CGT, such as the Schreier–Sims ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term ''representation of a group'' is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Small Groups
The following list in mathematics contains the finite groups of small order up to group isomorphism. Counts For ''n'' = 1, 2, … the number of nonisomorphic groups of order ''n'' is : 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, ... For labeled groups, see . Glossary Each group is named by their Small Groups library as G''o''''i'', where ''o'' is the order of the group, and ''i'' is the index of the group within that order. Common group names: * Z''n'': the cyclic group of order ''n'' (the notation C''n'' is also used; it is isomorphic to the additive group of Z/''n''Z). * Dih''n'': the dihedral group of order 2''n'' (often the notation D''n'' or D2''n'' is used ) ** K4: the Klein four-group of order 4, same as and Dih2. * S''n'': the symmetric group of degree ''n'', containing the ''n''! permutations of ''n'' elements. * A''n'': the alternating group of degree ''n'', containing the even permutations of ''n'' elements, of order 1 for , and order ''n''!/ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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