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Zassenhaus Lemma
In mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Zassenhaus, is a technical result on the lattice of subgroups of a group (mathematics), group or the lattice of submodules of a module (mathematics), module, or more generally for any modular lattice. :Lemma. Suppose G is a group with subgroups A and C. Suppose B\triangleleft A and D\triangleleft C are normal subgroups. Then there is an group isomorphism, isomorphism of quotient groups: ::\frac \cong \frac. This can be generalized to the case of a group with operators (G, \Omega) with stable subgroups A and C, the above statement being the case of \Omega=G group action, acting on itself by conjugation (group theory), conjugation. Zassenhaus mathematical proof, proved this lemma specifically to give the most direct proof of the Schreier refinement theorem. The 'butterfly' becomes apparent when trying to draw the Hasse diagram of the various groups involved. Zassenhaus' lemma for groups can be derived from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Conjugation (group Theory)
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under b = gag^ for all elements g in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set). Functions that are constant for members of the same conjugacy class are called class functions. Definition Let G be a group. Two elements a, b \in G are conjugate if there exists an element g \in G such that gag^ = b, in which case b is called of a and a is called a conjugate of b. In the case of the general linear group \operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Chelsea Publishing
The Chelsea Publishing Company was a publisher of mathematical books, based in New York City New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on one of the world's largest natural harbors. The city comprises five boroughs, each coextensive w ..., founded in 1944 by Aaron Galuten while he was still a graduate student at Columbia. Its initial focus was to republish important European works that were unavailable in the United States because of wartime restrictions, such as Hausdorff's Mengenlehre, or because the works were out of print. This soon expanded to include translations of such works into English, as well as original works by American authors. As of 1985, the company's catalog included more than 200 titles. After Galuten's death in 1994, the company was acquired in 1997 by the AMS, which continues to publish a portion of the company's original catalog under the AMS Chelse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Abhandlungen Aus Dem Mathematischen Seminar Der Universität Hamburg
(English: ''Reports from the Mathematical Seminar of the University of Hamburg'') is a peer-reviewed mathematics journal published by Springer Science+Business Media. It publishes articles on pure mathematics and is scientifically coordinated by the ''Mathematisches Seminar'', an informal cooperation of mathematicians at the Universität Hamburg; its managing editors are professors and Tobias Dyckerhoff. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. History The ''Abhandlungen'' were set up as a new journal by Wilhelm Blaschke in 1922 at the newly created Department of Mathematics (called ''Mathematisches Seminar'') at the newly founded Hamburgische Universität. Blaschke invited Hermann Weyl and David Hilbert to the ''Mathematisches Seminar'' (in 1920 and 1921, respectively) to deliver a talk series on their views concerning the Foundations of Mathematics. These talks formed part of the early history of the Grundlagenkrise der Mathematik, and Hilbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Barbara L
Barbara L (1947–1977) was an American Quarter Horse that raced during the early 1950s and often defeated some of the best racehorses of the time. She earned $32,836 () on the race track in 81 starts and 21 wins, including six wins in stakes races. She set two track records during her racing career. After retiring from racing in 1955, she went on to become a broodmare and had 14 foals, including 11 who earned their Race Register of Merit with the American Quarter Horse Association (AQHA). Her offspring earned more than $200,000 in race money. She died in 1977 and was inducted into the AQHA's American Quarter Horse Hall of Fame in 2007. Early life Barbara L was foaled in 1947, a bay daughter of a Thoroughbred stallion named Patriotic and a Quarter Horse broodmare named Big Bess. She was registered with the AQHA as number 146,954. Her sire, or father, was a grandson of Man o' War, while her dam, or mother, descended from the Quarter Horse Peter McCue. Barbara ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lattice Of Subgroups
In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial ordering being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection. Example The dihedral group Dih4 has ten subgroups, counting itself and the trivial subgroup. Five of the eight group elements generate subgroups of order two, and the other two non- identity elements both generate the same cyclic subgroup of order four. In addition, there are two subgroups of the form Z2 × Z2, generated by pairs of elements. The lattice formed by these ten subgroups is shown in the illustration. This example also shows that the lattice of all subgroups of a group is not a modular lattice in general. Indeed, this particular lattice contains the forbidden "pentagon" N5 as a sublattice. Properties For any ''A'', ''B'', and ''C'' subgroups of a group with '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Goursat's Lemma
Goursat's lemma, named after the French mathematician Édouard Goursat, is an algebraic theorem about subgroups of the direct product of two groups. It can be stated more generally in a Goursat variety (and consequently it also holds in any Maltsev variety), from which one recovers a more general version of Zassenhaus' butterfly lemma. In this form, Goursat's lemma also implies the snake lemma . Groups Goursat's lemma for groups can be stated as follows. : Let G, G' be groups, and let H be a subgroup of G\times G' such that the two projections p_1: H \to G and p_2: H \to G' are surjective (i.e., H is a subdirect product of G and G'). Let N be the kernel of p_2 and N' the kernel of p_1. One can identify N as a normal subgroup of G, and N' as a normal subgroup of G'. Then the image of H in G/N \times G'/N' is the graph of an isomorphism G/N \cong G'/N'. One then obtains a bijection between: :# Subgroups of G\times G' which project onto both factors, :# Triples (N, N', f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hasse Diagram
In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set (S,\le) one represents each element of S as a vertex in the plane and draws a line segment or curve that goes ''upward'' from one vertex x to another vertex y whenever y covers x (that is, whenever x\ne y, x\le y and there is no z distinct from x and y with x\le z\le y). These curves may cross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order. Hasse diagrams are named after Helmut Hasse (1898–1979); according to Garrett Birkhoff, they are so called because of the effective use Hasse made of them. However, Hasse was not the first to use these diagrams. One example that predates Hasse can be found in an 1895 work by Henri Gustave Vogt. Although Hasse diagrams were orig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Schreier Refinement Theorem
In mathematics, the Schreier refinement theorem of group theory states that any two subnormal series of subgroups of a given group have equivalent refinements, where two series are equivalent if there is a bijection between their factor groups that sends each factor group to an isomorphic one. The theorem is named after the Austrian mathematician Otto Schreier who proved it in 1928. It provides an elegant proof of the Jordan–Hölder theorem. It is often proved using the Zassenhaus lemma. gives a short proof by intersecting the terms in one subnormal series with those in the other series. Example Consider \mathbb_2 \times S_3, where S_3 is the symmetric group of degree 3. The alternating group In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ... A_3 is a normal subgroup of S_3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
