|
Combinatorial Group Theory
In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a natural and geometric way such a presentation. A very closely related topic is geometric group theory, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides. It also comprises a number of algorithmically insoluble problems, most notably the word problem for groups; and the classical Burnside problem. History See the book by Chandler and Magnus for a detailed history of combinatorial group theory. A proto-form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. The foundations of combinatorial group theory were laid by Walther von Dyck, student of Felix Klein Felix Christ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Word Problem For Groups
A word is a basic element of language that carries meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no consensus among linguists on its definition and numerous attempts to find specific criteria of the concept remain controversial. Different standards have been proposed, depending on the theoretical background and descriptive context; these do not converge on a single definition. Some specific definitions of the term "word" are employed to convey its different meanings at different levels of description, for example based on phonological, grammatical or orthographic basis. Others suggest that the concept is simply a convention used in everyday situations. The concept of "word" is distinguished from that of a morpheme, which is the smallest unit of language that has a meaning, even if it cannot stand on its own. Words are made out of at least one morpheme. Morphemes can ... [...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 Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ... 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 T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Symmetry Group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object ''X'' is ''G'' = Sym(''X''). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure. Introduction We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a scalar field, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Icosahedral Symmetry
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 polyhedron, dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 Rotational symmetry, rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a Reflection symmetry, 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. As point group Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
William Rowan Hamilton
Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathematical equations are considered fundamental to modern theoretical physics, particularly Hamiltonian mechanics, his reformulation of Lagrangian mechanics. His career included the analysis of geometrical optics, Fourier analysis, and quaternions, the last of which made him one of the founders of modern linear algebra. Hamilton was Andrews Professor of Astronomy at Trinity College Dublin. He was also the third director of Dunsink Observatory from 1827 to 1865. The Hamilton Institute at Maynooth University is named after him. Early life Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819), who lived in Dublin at 29 Dominick Street, Dublin, Dominick Street, later renumbered to 36. Ham ... [...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 group, icosahedral rotation group by Generating set of a group, generators and relations. Hamilton's discovery derived from his attempts to find an algebra of tuple, "triplets" or 3-tuples that he believed would reflect the three Cartesian coordinate system#Cartesian coordinates in three dimensions, Cartesian axes. The symbols of the icosian calculus correspond to moves between vertices on a dodecahedron. (Hamilton originally thought in terms of moves between the faces of an icosahedron, which is equivalent by Platonic solid#Dual polyhedra, duality. This is the origin of the name "icosian".) 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 hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Burnside Problem
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory, and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants that differ in the additional conditions imposed on the orders of the group elements (see bounded and restricted below). Some of these variants are still open questions. Brief history Initial work pointed towards the affirmative answer. For example, if a group ''G'' is finitely generated and the order of each element of ''G'' is a divisor of 4, then ''G'' is finite. Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Algorithmically Insoluble
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 an arbitrary program eventually halts when run. Background A decision problem is a question which, for every input in some infinite set of inputs, requires a "yes" or "no" answer. Those inputs can be numbers (for example, the decision problem "is the input a prime number?") or values of some other kind, such as strings of a formal language. The formal representation of a decision problem is a subset of the natural numbers. For decision problems on natural numbers, the set consists of those numbers that the decision problem answers "yes" to. For example, the decision problem "is the input even?" is formalized as the set of even numbers. A decision probl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Free Group
In mathematics, the free group ''F''''S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1''t'' but ''s'' ≠ ''t''−1 for ''s'',''t'',''u'' ∈ ''S''). The members of ''S'' are called generators of ''F''''S'', and the number of generators is the rank of the free group. An arbitrary group ''G'' is called free if it is isomorphic to ''F''''S'' for some subset ''S'' of ''G'', that is, if there is a subset ''S'' of ''G'' such that every element of ''G'' can be written in exactly one way as a product of finitely many elements of ''S'' and their inverses (disregarding trivial variations such as ''st'' = ''suu''−1''t''). A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property. History ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Geometric Group Theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups can act non-trivially (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Simplicial Complex
In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersections of the elements are also included in the set (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely Combinatorics, combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex., Section 4.3 Definitions A simplicial complex \mathcal is a set of Simplex, simplices that satisfies the following conditions: # Every Simplex#Elements, face of a simplex from \mathcal is also in \mathcal. # The non-empty Set intersection, intersection of any two simplices \sigma_1, \sigma_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |