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Group Isomorphism Problem
In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations refer to isomorphic groups. The isomorphism problem was formulated by Max Dehn, and together with the word problem and conjugacy problem, is one of three fundamental decision problems in group theory he identified in 1911. All three problems, formulated as ranging over all finitely presented groups, are undecidable. In the case of the isomorphism problem, this means that there does not exist a computer algorithm that takes two finite group presentations and decides whether or not the groups are isomorphic, regardless of how (finitely) much time is allowed for the algorithm to run and how (finitely) much memory is available. In fact the problem of deciding whether a finitely presented group is trivial is undecidable, a consequence of the Adian–Rabin theorem due to Sergei Adian and Michael O. Rabin. However, there are some classes of finitely p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. 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 it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by . The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory. In a very influential (over 1000 citations ) chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others. Definition Let G be a fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published 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. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-group
In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a group in which the order of every element is a power of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a nonnegative integer ''n'' such that the product of ''pn'' copies of ''g'', and not fewer, is equal to the identity element. The orders of different elements may be different powers of ''p''. Abelian ''p''-groups are also called ''p''-primary or simply primary. A finite group is a ''p''-group if and only if its order (the number of its elements) is a power of ''p''. Given a finite group ''G'', the Sylow theorems guarantee the existence of a subgroup of ''G'' of order ''pn'' for every prime power ''pn'' that divides the order of ''G''. Every finite ''p''-group is nilpotent. The remainder of this article deals with finite ''p''-groups. For an example of an infinite abelian ''p''-group, see Prüfer group, and for an example of an infinite simple ''p' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
László Babai
László "Laci" Babai (born July 20, 1950, in Budapest) a fellow of the American Academy of Arts and Sciences, and won the Knuth Prize. Babai was an invited speaker at the International Congresses of Mathematicians in Kyoto (1990), Zürich (1994, plenary talk), and Rio de Janeiro Rio de Janeiro, or simply Rio, is the capital of the Rio de Janeiro (state), state of Rio de Janeiro. It is the List of cities in Brazil by population, second-most-populous city in Brazil (after São Paulo) and the Largest cities in the America ... (2018). Sources Professor László Babai's algorithm is next big step in conquering isomorphism in graphs// Published on Nov 20, 2015 Division of the Physical Sciences / The University of Chicago Mathematician claims breakthrough in complexity theory by Adrian Cho 10 November 2015 17:45 // Posted iMath Science AAAS News A Quasipolynomial Time Algorithm for Graph Isomorphism: The Details+ Background on Graph Isomorphism + The Main Result // Math ∩ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Tarjan
Robert Endre Tarjan (born April 30, 1948) is an American computer scientist and mathematician. He is the discoverer of several graph theory algorithms, including his strongly connected components algorithm, and co-inventor of both splay trees and Fibonacci heaps. Tarjan is currently the James S. McDonnell Distinguished University Professor of Computer Science at Princeton University. Personal life and education He was born in Pomona, California. His father, George Tarjan (1912–1991), raised in Hungary, was a child psychiatrist, specializing in mental retardation, and ran a state hospital. Robert Tarjan's younger brother James became a chess grandmaster. As a child, Robert Tarjan read a lot of science fiction, and wanted to be an astronomer. He became interested in mathematics after reading Martin Gardner's mathematical games column in Scientific American. He became seriously interested in math in the eighth grade, thanks to a "very stimulating" teacher. While he was in hig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-polynomial Time
In computational complexity theory and the analysis of algorithms, an algorithm is said to take quasi-polynomial time if its time complexity is quasi-polynomially bounded. That is, there should exist a constant c such that the worst-case running time of the algorithm, on inputs of has an upper bound of the form 2^. The decision problems with quasi-polynomial time algorithms are natural candidates for being NP-intermediate, neither having polynomial time nor likely to be NP-hard. Complexity class The complexity class QP consists of all problems that have quasi-polynomial time algorithms. It can be defined in terms of DTIME as follows. :\mathsf = \bigcup_ \mathsf \left(2^\right) Examples An early example of a quasi-polynomial time algorithm was the Adleman–Pomerance–Rumely primality test. However, the problem of testing whether a number is a prime number has subsequently been shown to have a polynomial time algorithm, the AKS primality test. In some cases, quasi-polynomi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quanta Magazine
''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. History ''Quanta Magazine'' was initially launched as ''Simons Science News'' in October 2012, but it was renamed to its current title in July 2013. It was founded by the former ''New York Times'' journalist Thomas Lin, who was the magazine's editor-in-chief until 2024. The two deputy editors are John Rennie and Michael Moyer, formerly of ''Scientific American'', and the art director is Samuel Velasco. In 2024, Samir Patel became the magazine's second editor in chief. Content The articles in the magazine are freely available to read online. ''Scientific American'', ''Wired'', ''The Atlantic'', and ''The Washington Post'', as well as international science publications like '' Spektrum der Wissenschaft'', have reprinted articles from the magazine. In November 2018, MIT Press The MIT Press is the uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Isomorphism Problem
The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational complexity class NP-intermediate. It is known that the graph isomorphism problem is in the low hierarchy of class NP, which implies that it is not NP-complete unless the polynomial time hierarchy collapses to its second level. At the same time, isomorphism for many special classes of graphs can be solved in polynomial time, and in practice graph isomorphism can often be solved efficiently. This problem is a special case of the subgraph isomorphism problem, which asks whether a given graph ''G'' contains a subgraph that is isomorphic to another given graph ''H''; this problem is known to be NP-complete. It is also known to be a special case of the non-abelian hidden subgroup problem over the symmetric group. In the area of image r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center (group Theory)
In abstract algebra, the center of a group (mathematics), group is the set (mathematics), set of elements that commutative, commute with every element of . It is denoted , from German ''wikt:Zentrum, Zentrum,'' meaning ''center''. In set-builder notation, :. The center is a normal subgroup, Z(G)\triangleleft G, and also a characteristic subgroup, characteristic subgroup, but is not necessarily fully characteristic subgroup, fully characteristic. The quotient group, , is group isomorphism, isomorphic to the inner automorphism group, . A group is abelian if and only if . At the other extreme, a group is said to be centerless if is trivial group, trivial; i.e., consists only of the identity element. The elements of the center are central elements. As a subgroup The center of ''G'' is always a subgroup (mathematics), subgroup of . In particular: # contains the identity element of , because it commutes with every element of , by definition: , where is the identity; # If an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-relator Group
In the mathematical subject of group theory, a one-relator group is a group given by a group presentation with a single defining relation. One-relator groups play an important role in geometric group theory by providing many explicit examples of finitely presented groups. Formal definition A one-relator group is a group ''G'' that admits a group presentation of the form where ''X'' is a set (in general possibly infinite), and where r\in F(X) is a freely and cyclically reduced word. If ''Y'' is the set of all letters x\in X that appear in ''r'' and X'=X\setminus Y then :G=\langle Y\mid r=1\, \rangle \ast F(X'). For that reason ''X'' in () is usually assumed to be finite where one-relator groups are discussed, in which case () can be rewritten more explicitly as where X=\ for some integer n\ge 1. Freiheitssatz Let ''G'' be a one-relator group given by presentation () above. Recall that ''r'' is a freely and cyclically reduced word in ''F''(''X''). Let y\in X be a lett ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent Group
In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, it has a central series of finite length or its lower central series terminates with . Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups. Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series. Definition The definition uses the idea of a central series for a gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
