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Transfer (group Theory)
In the mathematical field of group theory, the transfer defines, given a group ''G'' and a subgroup of finite index ''H'', a group homomorphism from ''G'' to the abelianization of ''H''. It can be used in conjunction with the Sylow theorems to obtain certain numerical results on the existence of finite simple groups. The transfer was defined by and rediscovered by . Construction The construction of the map proceeds as follows:Following Scott 3.5 Let 'G'':''H''= ''n'' and select coset representatives, say :x_1, \dots, x_n,\, for ''H'' in ''G'', so ''G'' can be written as a disjoint union :G = \bigcup\ x_i H. Given ''y'' in ''G'', each ''yxi'' is in some coset ''xjH'' and so :yx_i = x_jh_i for some index ''j'' and some element ''h''''i'' of ''H''. The value of the transfer for ''y'' is defined to be the image of the product :\textstyle \prod_^n h_i in ''H''/''H''′, where ''H''′ is the commutator subgroup of ''H''. The order of the factors is irrelevant since ''H''/''H'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. Homology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stamm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Fields
''Corps Locaux'' by Jean-Pierre Serre, originally published in 1962 and translated into English as ''Local Fields'' by Marvin Jay Greenberg in 1979, is a seminal graduate-level algebraic number theory text covering local fields, ramification, group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology ..., and local class field theory. The book's end goal is to present local class field theory from the cohomological point of view. This theory concerns extensions of "local" (i.e., complete for a discrete valuation) fields with finite residue field. Contents #''Part I, Local Fields (Basic Facts)'': Discrete valuation rings, Dedekind domains, and Completion. #''Part II, Ramification'': Discriminant & Different, Ramification Groups, The Norm, and Artin Representation. #''Par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Focal Subgroup Theorem
In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to . The focal subgroup theorem relates the ideas of transfer and fusion such as described in . Various applications of these ideas include local criteria for ''p''-nilpotence and various non-simplicity criteria focussing on showing that a finite group has a normal subgroup of index ''p''. Background The focal subgroup theorem relates several lines of investigation in finite group theory: normal subgroups of index a power of ''p'', the transfer homomorphism, and fusion of elements. Subgroups The following three normal subgroups of index a power of ''p'' are naturally defined, and arise as the smallest normal subgroups such that the quotient is (a certain kind of) ''p''-group. Formally, they are kernels of the reflection onto the reflective subcatego ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Tate (mathematician)
John Torrence Tate Jr. (March 13, 1925 – October 16, 2019) was an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He was awarded the Abel Prize in 2010. Biography Tate was born in Minneapolis, Minnesota. His father, John Tate Sr., was a professor of physics at the University of Minnesota, and a longtime editor of ''Physical Review''. His mother, Lois Beatrice Fossler, was a high school English teacher. Tate Jr. received his bachelor's degree in mathematics in 1946 from Harvard University, and entered the doctoral program in physics at Princeton University. He later transferred to the mathematics department and received his PhD in mathematics in 1950 after completing a doctoral dissertation, titled "Fourier analysis in number fields and Hecke's zeta functions", under the supervision of Emil Artin. Tate taught at Harvard for 36 years before joining the Univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields. Along with Emmy Noether, he is considered the founder of modern abstract algebra. Early life and education Parents Emil Artin was born in Vienna to parents Emma Maria, née Laura (stage name Clarus), a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of mixed Austrian and Armenian descent. His Armenian last name was Artinian which was shortened to Artin. Several documents, including Emil's birth certificate, list the father's occupation as “opera singer” though others list it as “art dealer.” It seems at least plausible that he and Emma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Field Theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem). One of the major results is: given a number field ''F'', and writing ''K'' for the maximal abelian unramified extension of ''F'', the Galois group of ''K'' over ''F'' is canonically isomorphic to the ideal class group of ''F''. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing ''CF' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Ideal Theorem
In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called ''principalization'', or sometimes ''capitulation''. Formal statement For any algebraic number field ''K'' and any ideal ''I'' of the ring of integers of ''K'', if ''L'' is the Hilbert class field of ''K'', then :IO_L\ is a principal ideal α''O''''L'', for ''O''''L'' the ring of integers of ''L'' and some element α in it. History The principal ideal theorem was conjectured by , and was the last remaining aspect of his program on class fields to be completed, in 1929. reduced the principal ideal theorem to a question about finite abelian groups: he showed that it would follow if the transfer from a finite group to its derived subg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutator Subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, G/N is abelian if and only if N contains the commutator subgroup of G. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is. Commutators For elements g and h of a group ''G'', the commutator of g and h is ,h= g^h^gh. The commutator ,h/math> is equal to the identity element ''e'' if and only if gh = hg , that is, if and only if g and h commute. In general, gh = hg ,h/math>. However, the notation is somewhat arbitrary and there is a non-equivalent variant definition for the commutator that has the inverses on the right hand side of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Hasse
Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and diophantine geometry (Hasse principle), and to local zeta functions. Life Hasse was born in Kassel, Province of Hesse-Nassau, the son of Judge Paul Reinhard Hasse, also written Haße (12 April 1868 – 1 June 1940, son of Friedrich Ernst Hasse and his wife Anna Von Reinhard) and his wife Margarethe Louise Adolphine Quentin (born 5 July 1872 in Milwaukee, daughter of retail toy merchant Adolph Quentin (b. May 1832, probably Berlin, Kingdom of Prussia) and Margarethe Wehr (b. about 1840, Prussia), then raised in Kassel). After serving in the Imperial German Navy in World War I, he studied at the University of Göttingen, and then at the University of Marburg under Kurt Hensel, writing a dissertation in 1921 containing the Hasse–Min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying Space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free action of ''G''. It has the property that any ''G'' principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle ''EG'' → ''BG''. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete group ''G'', ''BG'' is, roughly speaking, a path-connected topological space ''X'' such that the fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
