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Freiheitssatz
In mathematics, the Freiheitssatz (German: "freedom/independence theorem": '' Freiheit'' + '' Satz'') is a result in the presentation theory of groups, stating that certain subgroups of a one-relator group are free groups. Statement Consider a group presentation :G = \langle x_, \dots, x_ , r = 1 \rangle given by generators and a single cyclically reduced relator . If appears in , then (according to the freiheitssatz) the subgroup of generated by is a free group, freely generated by . In other words, the only relations involving are the trivial ones. History The result was proposed by the German mathematician Max Dehn and proved by his student, Wilhelm Magnus, in his doctoral thesis. Although Dehn expected Magnus to find a topological proof, Magnus instead found a proof based on mathematical induction and amalgamated products of groups. Different induction-based proofs were given later by and . Significance The freiheitssatz has become "the cornerstone of one-relato ... [...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 lette ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilhelm Magnus
Hans Heinrich Wilhelm Magnus known as Wilhelm Magnus (February 5, 1907 in Berlin, Germany – October 15, 1990 in New Rochelle, New York) was a German-American mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations. Biography In 1931, Magnus received his PhD from the University of Frankfurt, in Germany. His thesis, written under the direction of Max Dehn, was entitled ''Über unendlich diskontinuierliche Gruppen von einer definierenden Relation (der Freiheitssatz)''. Magnus was a faculty member in Frankfurt from 1933 until 1938. He refused to join the Nazi Party and, as a consequence, was not allowed to hold an academic post during World War II. In 1947 he became a professor at the University of Göttingen. In 1948 he emigrated to the United States to collaborate on the Bateman Manuscript Project as a co-editor, while a visiting professor at the California Institute o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Max Dehn
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. However, he was forced to retire in 1935 and eventually fled Germany in 1939 and emigrated to the United States. Dehn was a student of David Hilbert, and in his habilitation in 1900 Dehn resolved Hilbert's third problem, making him the first to resolve one of Hilbert's well-known 23 problems. Dehn's students include Ott-Heinrich Keller, Ruth Moufang, Wilhelm Magnus, and the artists Dorothea Rockburne and Ruth Asawa. Biography Dehn was born to a family of Jewish origin in Hamburg, Imperial Germany. He studied the foundations of geometry with Hilbert at Göttingen in 1899, and obtained a proof of the Jordan curve theorem for polygons. In 1900 he wrote his dissertation on the role of the Legendre angle sum theorem in axiomatic geome ... [...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] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amalgamated Product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from ''G'' and ''H'' into a group ''K'' factor uniquely through a homomorphism from to ''K''. Unless one of the groups ''G'' and ''H'' is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group (the universal group with a given set of generators). The free product is the coproduct in the category of groups. That is, the free product plays the same role in group theory that disjoint union plays in set theory, or that the direct sum plays in module theory. Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup of ''G'' if the restriction of ∗ to is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ). If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. Subgroup tests Suppose th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated between the Baltic and North seas to the north, and the Alps to the south; it covers an area of , with a population of almost 84 million within its 16 constituent states. Germany borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, and France, Luxembourg, Belgium, and the Netherlands to the west. The nation's capital and most populous city is Berlin and its financial centre is Frankfurt; the largest urban area is the Ruhr. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity. A region named Germania was documented before AD 100. In 962, the Kingdom of Germany formed the bulk of the Holy Roman Empire. During the 16th ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
German Language
German ( ) is a West Germanic languages, West Germanic language mainly spoken in Central Europe. It is the most widely spoken and Official language, official or co-official language in Germany, Austria, Switzerland, Liechtenstein, and the Italy, Italian province of South Tyrol. It is also a co-official language of Luxembourg and German-speaking Community of Belgium, Belgium, as well as a national language in Namibia. Outside Germany, it is also spoken by German communities in France (Bas-Rhin), Czech Republic (North Bohemia), Poland (Upper Silesia), Slovakia (Bratislava Region), and Hungary (Sopron). German is most similar to other languages within the West Germanic language branch, including Afrikaans, Dutch language, Dutch, English language, English, the Frisian languages, Low German, Luxembourgish, Scots language, Scots, and Yiddish. It also contains close similarities in vocabulary to some languages in the North Germanic languages, North Germanic group, such as Danish lan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
