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Relator
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] |
Conjugate Closure
In group theory, the normal closure of a subset S of a group G is the smallest normal subgroup of G containing S. Properties and description Formally, if G is a group and S is a subset of G, the normal closure \operatorname_G(S) of S is the intersection of all normal subgroups of G containing S: \operatorname_G(S) = \bigcap_ N. The normal closure \operatorname_G(S) is the smallest normal subgroup of G containing S, in the sense that \operatorname_G(S) is a subset of every normal subgroup of G that contains S. The subgroup \operatorname_G(S) is generated by the set S^G=\ = \ of all conjugates of elements of S in G. Therefore one can also write \operatorname_G(S) = \. Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set \varnothing is the trivial subgroup. A variety of other notations are used for the normal closure in the literature, including \langle S^G\rangle, \langle S\rangle^G, \langle \langle S\rangle\rangle_G, and \langle\langl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Closure (group Theory)
In group theory, the normal closure of a subset S of a Group (mathematics), group G is the smallest normal subgroup of G containing S. Properties and description Formally, if G is a group and S is a subset of G, the normal closure \operatorname_G(S) of S is the intersection of all normal subgroups of G containing S: \operatorname_G(S) = \bigcap_ N. The normal closure \operatorname_G(S) is the smallest normal subgroup of G containing S, in the sense that \operatorname_G(S) is a subset of every normal subgroup of G that contains S. The subgroup \operatorname_G(S) is Generating set of a group, generated by the set S^G=\ = \ of all Conjugacy class, conjugates of elements of S in G. Therefore one can also write \operatorname_G(S) = \. Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set \varnothing is the trivial subgroup. A variety of other notations are used for the normal closure in the literature, including \langle S^G\rangle, \langle S ... [...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] |
Recursively Enumerable
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the set of input numbers for which the algorithm halts is exactly ''S''. Or, equivalently, *There is an algorithm that enumerates the members of ''S''. That means that its output is simply a list of all the members of ''S'': ''s''1, ''s''2, ''s''3, ... . If ''S'' is infinite, this algorithm will run forever. The first condition suggests why the term ''semidecidable'' is sometimes used. More precisely, if a number is in the set, one can ''decide'' this by running the algorithm, but if the number is not in the set, the algorithm runs forever, and no information is returned. A set that is "completely decidable" is a computable set. The second condition suggests why ''computably enumerable'' is used. The abbreviations c.e. and r.e. are oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dicyclic Group
In group theory, a dicyclic group (notation Dic''n'' or Q4''n'', Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST) is a particular kind of non-abelian group of order 4''n'' (''n'' > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2''n'', giving the name ''di-cyclic''. In the notation of exact sequences of groups, this extension can be expressed as: :1 \to C_ \to \mbox_n \to C_2 \to 1. \, More generally, given any finite abelian group with an order-2 element, one can define a dicyclic group. Definition For each integer ''n'' > 1, the dicyclic group Dic''n'' can be defined as the subgroup of the unit quaternions generated by :\begin a & = e^\frac = \cos\frac + i\sin\frac \\ x & = j \end More abstractly, one can define the dicyclic group Dic''n'' as the group with the following presentation :\operatorname_n = \left\langle a, x \mid a^ = 1,\ x^2 = a^n,\ x^ax = a^\right\rangle.\,\! Some things to note which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Dihedral Group
In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, ''p1m1'', seen as an infinite set of parallel reflections along an axis. Definition Every dihedral group is generated by a rotation ''r'' and a reflection; if the rotation is a rational multiple of a full rotation, then there is some integer ''n'' such that ''rn'' is the identity, and we have a finite dihedral group of order 2''n''. If the rotation is ''not'' a rational multiple of a full rotation, then there is no such ''n'' and the resulting group has infinitely many elements and is called Dih∞. It has presentations :\langle r, s \mid s^2 = 1, srs = r^ \rangle \,\! :\langle x, y \mid x^2 = y^2 = 1 \rangle \,\! and is isomorphic to a semidirect product of Z and Z/2, and to the free product Z/2 * Z/2. It is the automorphism group of the graph con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time. Life Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walther Von Dyck
Walther Franz Anton von Dyck (6 December 1856 – 5 November 1934), born Dyck () and later ennobled, was a German mathematician. He is credited with being the first to define a mathematical group, in the modern sense in . He laid the foundations of combinatorial group theory, being the first to systematically study a group by generators and relations. Biography Von Dyck was a student of Felix Klein, and served as chairman of the commission publishing Klein's encyclopedia. Von Dyck was also the editor of Kepler's works. He promoted technological education as rector of the Technische Hochschule of Munich. He was a Plenary Speaker of the ICM in 1908 at Rome. Von Dyck is the son of the Bavarian painter Hermann Dyck. Legacy The Dyck language in formal language theory is named after him, as are Dyck's theorem and Dyck's surface in the theory of surfaces, together with the von Dyck groups, the Dyck tessellations, Dyck paths, and the Dyck graph In the mathematical field of graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophical Magazine
The ''Philosophical Magazine'' is one of the oldest scientific journals published in English. It was established by Alexander Tilloch in 1798;John Burnett"Tilloch, Alexander (1759–1825)" Oxford Dictionary of National Biography, Oxford University Press, Sept 2004; online edn, May 2006, accessed 17 Feb 2010 in 1822 Richard Taylor became joint editor and it has been published continuously by Taylor & Francis ever since. Early history The name of the journal dates from a period when "natural philosophy" embraced all aspects of science. The very first paper published in the journal carried the title "Account of Mr Cartwright's Patent Steam Engine". Other articles in the first volume include "Methods of discovering whether Wine has been adulterated with any Metals prejudicial to Health" and "Description of the Apparatus used by Lavoisier to produce Water from its component Parts, Oxygen and Hydrogen". 19th century Early in the nineteenth century, classic papers by Humphry Davy, M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedral Group
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type . It may be represented by Coxeter notation and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters. Description Icosahedral symmetry is a mathematical property of objects indicating that an object has the same symmetries as a regular icosahedron. As point group Apart from the two infinite series of prismatic and antiprismatic symmetry, rotation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosian Calculus
The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856. In modern terms, he gave a group presentation of the icosahedral rotation group by generators and relations. Hamilton's discovery derived from his attempts to find an algebra of "triplets" or 3-tuples that he believed would reflect the three Cartesian axes. The symbols of the icosian calculus can be equated to moves between vertices on a dodecahedron. Hamilton's work in this area resulted indirectly in the terms Hamiltonian circuit and Hamiltonian path in graph theory. He also invented the icosian game as a means of illustrating and popularising his discovery. Informal definition The algebra is based on three symbols that are each roots of unity, in that repeated application of any of them yields the value 1 after a particular number of steps. They are: : \begin \iota^2 & = 1, \\ \kappa^3 & = 1, \\ \lambda^5 & = 1. \end Hamilton also gives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ireland, living at Dunsink Observatory. Hamilton's scientific career included the study of geometrical optics, ideas from Fourier analysis, and his work on quaternions which made him one of the founders of modern linear algebra. He made major contributions in optics, classical mechanics and abstract algebra. His work was fundamental to modern theoretical physics, particularly his reformulation of Newtonian mechanics, now called Hamiltonian mechanics. It is now central both to electromagnetism and to quantum mechanics. Early life Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819),Bruno (2003) who ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
