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Hopfian Group
In mathematics, a Hopfian group is a group ''G'' for which every epimorphism :''G'' → ''G'' is an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. A group ''G'' is co-Hopfian if every monomorphism :''G'' → ''G'' is an isomorphism. Equivalently, ''G'' is not isomorphic to any of its proper subgroups. Examples of Hopfian groups * Every finite group, by an elementary counting argument. * More generally, every polycyclic-by-finite group. * Any finitely generated free group. * The group Q of rationals. * Any finitely generated residually finite group. * Any word-hyperbolic group. Examples of non-Hopfian groups * Quasicyclic groups. * The group R of real numbers. * The Baumslag–Solitar group ''B''(2,3). Properties It was shown by that it is an undecidable problem to determine, given a finite presentation of a group, whether the group is Hopfian. Unlike the undecidability of many properties of groups th ... [...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] |
Word-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 finitely g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Encyclopedia Of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, and the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer. The CD-ROM contains animations and three-dimensional objects. The encyclopedia has been translated from the Soviet ''Matematicheskaya entsiklopediya'' (1977) originally edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles. Until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge online. This URL now redirects to the new wiki incarnation of the EOM. ''Encyclopedia of Mathematics'' wiki A new dynamic version of the encyclopedia is now ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Algebra
''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1984. From 1985 until 2000, Walter Feit served as its editor-in-chief. In 2004, ''Journal of Algebra'' announced (vol. 276, no. 1 and 2) the creation of a new section on computational algebra, with a separate editorial board. The first issue completely devoted to computational algebra was vol. 292, no. 1 (October 2005). The Editor-in-Chief of the ''Journal of Algebra'' is Michel Broué, Université Paris Diderot, and Gerhard Hiß, Rheinisch-Westfälische Technische Hochschule Aachen ( RWTH) is Editor of the computational algebra section. See also *Susan Montgomery M. Susan Montgomery (born 2 April 1943 in Lansing, MI) is a distinguished American mathematician whose current research interests concern noncommutative algebras: in parti ... [...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] |
Archiv Der Mathematik
'' Archiv der Mathematik'' is a peer-reviewed mathematics journal published by Springer, established in 1948. Abstracting and indexing The journal is abstracted and indexed in: Springer. 2022 * * * * According to the '' |
Adian–Rabin Theorem
In the mathematical subject of group theory, the Adian–Rabin theorem is a result that states that most "reasonable" properties of finitely presentable groups are algorithmically undecidable. The theorem is due to Sergei Adian (1955) and, independently, Michael O. Rabin (1958).Michael O. Rabin''Recursive unsolvability of group theoretic problems'' Annals of Mathematics (2), vol. 67, 1958, pp. 172–194 Markov property A ''Markov property'' ''P'' of finitely presentable groups is one for which: #''P'' is an abstract property, that is, ''P'' is preserved under group isomorphism. #There exists a finitely presentable group A_+ with property ''P''. #There exists a finitely presentable group A_- that cannot be embedded as a subgroup in any finitely presentable group with property ''P''. For example, being a finite group is a Markov property: We can take A_+ to be the trivial group and we can take A_- to be the infinite cyclic group \mathbb. Precise statement of the Adian–Rabin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presentation Of A Group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set ''R'' of relations among those generators. We then say ''G'' has presentation :\langle S \mid R\rangle. Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is isomorphic to the quotient of a free group on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation :\langle a \mid a^n = 1\rangle, where 1 is the group identity. This may be written equivalently as :\langle a \mid a^n\rangle, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undecidable Problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run. Background A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns ''yes''. These inputs can be natural numbers, but also other values of some other kind, such as strings of a formal language. Using some encoding, such as a Gödel numbering, the strings can be encoded as natural numbers. Thus, a decision problem informally phrased in terms of a formal language is also equivalent to a set of natural numbers. To keep the formal definition simple, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baumslag–Solitar Group
In the mathematical field of group theory, the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group theory and geometric group theory as (counter)examples and test-cases. They are given by the group presentation : \left \langle a, b \ : \ b a^m b^ = a^n \right \rangle. For each integer and , the Baumslag–Solitar group is denoted . The relation in the presentation is called the Baumslag–Solitar relation. Some of the various are well-known groups. is the free abelian group on two generators, and is the fundamental group of the Klein bottle. The groups were defined by Gilbert Baumslag and Donald Solitar in 1962 to provide examples of non- Hopfian groups. The groups contain residually finite groups, Hopfian groups that are not residually finite, and non-Hopfian groups. Linear representation Define :A= \begin1&1\\0&1\end, \qquad B= \begin\frac&0\\0&1\end. The matrix group generated by and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
