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Rank Of A Group
In the mathematical subject of group theory, the rank of a group ''G'', denoted rank(''G''), can refer to the smallest cardinality of a generating set for ''G'', that is : \operatorname(G)=\min\. If ''G'' is a finitely generated group, then the rank of ''G'' is a nonnegative integer. The notion of rank of a group is a group-theoretic analog of the notion of dimension of a vector space. Indeed, for ''p''-groups, the rank of the group ''P'' is the dimension of the vector space ''P''/Φ(''P''), where Φ(''P'') is the Frattini subgroup. The rank of a group is also often defined in such a way as to ensure subgroups have rank less than or equal to the whole group, which is automatically the case for dimensions of vector spaces, but not for groups such as affine groups. To distinguish these different definitions, one sometimes calls this rank the subgroup rank. Explicitly, the subgroup rank of a group ''G'' is the maximum of the ranks of its subgroups: : \operatorname(G)=\max_ \min\ ... [...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] |
Classification Of Finite Simple Groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Lyndon
Roger Conant Lyndon (December 18, 1917 – June 8, 1988) was an American mathematician, for many years a professor at the University of Michigan.. He is known for Lyndon words, the Curtis–Hedlund–Lyndon theorem, Craig–Lyndon interpolation and the Lyndon–Hochschild–Serre spectral sequence. Biography Lyndon was born on December 18, 1917, in Calais, Maine, the son of a Unitarian minister. His mother died when he was two years old, after which he and his father moved several times to towns in Massachusetts and New York. He did his undergraduate studies at Harvard University, originally intending to study literature but eventually settling on mathematics, and graduated in 1939. He took a job as a banker, but soon afterwards returned to graduate school at Harvard, earning a master's degree in 1941. After a brief teaching stint at the Georgia Institute of Technology, he returned to Harvard for the third time in 1942 and while there taught navigation as part of the V-12 Navy ... [...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] |
Primitive Element (co-algebra)
In algebra, a primitive element of a co-algebra ''C'' (over an element ''g'') is an element ''x'' that satisfies :\mu(x) = x \otimes g + g \otimes x where \mu is the co-multiplication and ''g'' is an element of ''C'' that maps to the multiplicative identity 1 of the base field under the co-unit (''g'' is called ''group-like''). If ''C'' is a bi-algebra In mathematics, a bialgebra over a Field (mathematics), field ''K'' is a vector space over ''K'' which is both a unital algebra, unital associative algebra and a coalgebra, counital coassociative coalgebra. The algebraic and coalgebraic structures ..., i.e., a co-algebra that is also an algebra (with certain compatibility conditions satisfied), then one usually takes ''g'' to be 1, the multiplicative identity of ''C''. The bi-algebra ''C'' is said to be primitively generated if it is generated by primitive elements (as an algebra). If ''C'' is a bi-algebra, then the set of primitive elements form a Lie algebra with the usual comm ... [...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] |
Free 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] |
Grushko Theorem
In the mathematical subject of group theory, the Grushko theorem or the Grushko–Neumann theorem is a theorem stating that the rank (that is, the smallest cardinality of a generating set) of a free product of two groups is equal to the sum of the ranks of the two free factors. The theorem was first obtained in a 1940 article of Grushko and then, independently, in a 1943 article of Neumann. Statement of the theorem Let ''A'' and ''B'' be finitely generated groups and let ''A''∗''B'' be the free product of ''A'' and ''B''. Then :rank(''A''∗''B'') = rank(''A'') + rank(''B''). It is obvious that rank(''A''∗''B'') ≤ rank(''A'') + rank(''B'') since if X is a finite generating set of ''A'' and ''Y'' is a finite generating set of ''B'' then ''X''∪''Y'' is a generating set for ''A''∗''B'' and that , ''X'' ∪ ''Y'', ≤ , ''X'', + , ''Y'', . The opposite inequality, rank(''A''∗''B'') ≥ rank(''A'') + rank(''B''), requires proof. Grushko, but not Neumann, proved a mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hanna Neumann Conjecture
In the mathematical subject of group theory, the Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957.Hanna Neumann. ''On the intersection of finitely generated free groups. Addendum.'' Publicationes Mathematicae Debrecen, vol. 5 (1957), p. 128 In 2011, a strengthened version of the conjecture (see below) was proved independently by Joel FriedmanJoel Friedman"Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: With an Appendix by Warren Dicks" Mem. Amer. Math. Soc., 233 (2015), no. 1100. and by Igor Mineyev.Igor Minevev"Submultiplicativity and the Hanna Neumann Conjecture." Ann. of Math., 175 (2012), no. 1, 393-414. In 2017, a third proof of the Strengthened Hanna Neumann conjecture, based on homological arguments inspired by pro-p-group considerations, was published by Andrei Jaikin-Zapirain. History The subject ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Publicationes Mathematicae Debrecen
''Publicationes Mathematicae Debrecen'' is a Hungarian mathematical journal, edited and published in Debrecen, at the Mathematical Institute of the University of Debrecen. It was founded by Alfréd Rényi, Tibor Szele Tibor Szele (Debrecen, 21 June 1918 – Szeged, 5 April 1955) Hungarian mathematician, working in combinatorics and abstract algebra. After graduating at the Debrecen University, he became a researcher at the Szeged University in 1946, then ..., and Ottó Varga in 1950. The current editor-in-chief is Lajos Tamássy. External links * The journal'homepageOn-line papers Mathematics journals University of Debrecen {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hanna Neumann
Johanna (Hanna) Neumann (née von Caemmerer; 12 February 1914 – 14 November 1971) was a German-born mathematician who worked on group theory. Biography Neumann was born on 12 February 1914 in Lankwitz, Steglitz-Zehlendorf (today a district of Berlin), Germany. She was the youngest of three children of Hermann and Katharina von Caemmerer. As a result of her father's death in the first days of the First World War, the family income was small, and from the age of thirteen she was coaching school children. After two years at a private school she entered the Auguste-Viktoria-Schule, a girls' grammar school (Realgymnasium), in 1922. She graduated in early 1932 and then entered the University of Berlin. The lecture courses in mathematics that she took in her first year were: Introduction to Higher Mathematics given by Georg Feigl; Analytical Geometry and Projective Geometry both given by Ludwig Bieberbach, Differential and Integral Calculus given by Erhard Schmidt, and the The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heegaard Genus
In the mathematical field of geometric topology, a Heegaard splitting () is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies. Definitions Let ''V'' and ''W'' be handlebodies of genus ''g'', and let ƒ be an orientation reversing homeomorphism from the boundary of ''V'' to the boundary of ''W''. By gluing ''V'' to ''W'' along ƒ we obtain the compact oriented 3-manifold : M = V \cup_f W. Every closed, orientable three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to Moise. This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory. The decomposition of ''M'' into two handlebodies is called a Heegaard splitting, and their common boundary ''H'' is called the Heegaard surf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
