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Virtually Nilpotent
In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group ''G'' is said to be ''virtually P'' if there is a finite index subgroup H \le G such that ''H'' has property P. Common uses for this would be when P is abelian, nilpotent, solvable or free. For example, virtually solvable groups are one of the two alternatives in the Tits alternative, while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups. This terminology is also used when P is just another group. That is, if ''G'' and ''H'' are groups then ''G'' is ''virtually'' ''H'' if ''G'' has a subgroup ''K'' of finite index in ''G'' such that ''K'' is isomorphic to ''H''. In particular, a group is virtually trivial if and only if it is finite. Two groups are v ... [...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] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schreier Index Formula
Schreier is a surname of German origin. Notable people with the surname include: *Christian Schreier (born 1959), German footballer *Dan Moses Schreier, American sound designer and composer *Jake Schreier, American director *Józef Schreier, Polish mathematician *Otto Schreier (1901-1929), Austrian mathematician *Peter Schreier (1935–2019), German tenor and conductor *Richard Schreier, Canadian engineer *Sandy Schreier, American fashion historian and collector *Jason Schreier, American journalist and author See also * Schreyer * Shrayer Shrayer is a surname. Notable people with the surname include: * David Shrayer-Petrov (born 1936), Russian American novelist, poet, memoirist, translator and medical scientist * Maxim D. Shrayer (born 1967), Russian-American author, translator, an ... * Shroyer, Pennsylvania German form * Shrier, Americanized version {{surname German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nielsen–Schreier Theorem
In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free. It is named after Jakob Nielsen and Otto Schreier. Statement of the theorem A free group may be defined from a group presentation consisting of a set of generators with no relations. That is, every element is a product of some sequence of generators and their inverses, but these elements do not obey any equations except those trivially following from = 1. The elements of a free group may be described as all possible reduced words, those strings of generators and their inverses in which no generator is adjacent to its own inverse. Two reduced words may be multiplied by concatenating them and then removing any generator-inverse pairs that result from the concatenation. The Nielsen–Schreier theorem states that if ''H'' is a subgroup of a free group ''G'', then ''H'' is itself isomorphic to a free group. That is, there exists a set ''S'' of elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stallings Theorem About Ends Of Groups
In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group ''G'' has more than one end if and only if the group ''G'' admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. In the modern language of Bass–Serre theory the theorem says that a finitely generated group ''G'' has more than one end if and only if ''G'' admits a nontrivial (that is, without a global fixed point) action on a simplicial tree with finite edge-stabilizers and without edge-inversions. The theorem was proved by John R. Stallings, first in the torsion-free case (1968) and then in the general case (1971). Ends of graphs Let Γ be a connected graph where the degree of every vertex is finite. One can view Γ as a topological space by giving it the natural structure of a one-dimensional cell complex. Then the ends of Γ are the ends of this topological space. A more explicit definition of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. Definition The modular group is the group of linear fractional transformations of the upper half of the complex plane, which have the form :z\mapsto\frac, where , , , are integers, and . The group operation is function composition. This group of transformations is isomorphic to the projective special linear group , which is the quotient of the 2-dimensional special linear group over the integers by its center . In other words, consists of all matrices :\begin a & b \\ c & d \end where , , , are integers, , and pairs of matrices and are considered to be identical. The group operation is the usual mult ... [...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] |
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 Free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Dihedral Group
In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group ''O''(2). Dihedral groups play an important role in group theory, geometry, and chemistry. Definition For any abelian group ''H'', the generalized dihedral group of ''H'', written Dih(''H''), is the semidirect product of ''H'' and Z2, with Z2 acting on ''H'' by inverting elements. I.e., \mathrm(H) = H \rtimes_\phi Z_2 with φ(0) the identity and φ(1) inversion. Thus we get: :(''h''1, 0) * (''h''2, ''t''2) = (''h''1 + ''h''2, ''t''2) :(''h''1, 1) * (''h''2, ''t''2) = (''h''1 − ''h''2, 1 + ''t''2) for all ''h''1, ''h''2 in ''H'' and ''t''2 in Z2. (Writing Z2 multiplicatively, we have (''h''1, ''t''1) * (''h''2, ''t''2) = (''h''1 + ''t''1''h''2, ''t''1''t''2) .) Note that (''h'', 0) * (0,1) = (''h'',1), i.e. first the inversion and then the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semidirect Product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup. * an ''outer'' semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as ''semidirect products''. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension). Inner semidirect product definitions Given a group with identity element , a subgroup , and a normal subgroup , the following statements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commensurability (group Theory)
In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to the normalizer. Commensurability in group theory Two groups ''G''1 and ''G''2 are said to be (abstractly) commensurable if there are subgroups ''H''1 ⊂ ''G''1 and ''H''2 ⊂ ''G''2 of finite index such that ''H''1 is isomorphic to ''H''2. For example: *A group is finite if and only if it is commensurable with the trivial group. *Any two finitely generated free groups on at least 2 generators are commensurable with each other. The group ''SL''(2,Z) is also commensurable with these free groups. *Any two surface groups of genus at least 2 are commensurable with each other. A different but related notion is used for subgroups of a given group. Namely, two subgroups Γ1 and Γ2 of a group ''G'' are said to be commensurable if the intersection Γ1 ∩ Γ2 is of finite index in both ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Growth Rate (group Theory)
In the mathematical subject of geometric group theory, the growth rate of a group with respect to a symmetric generating set describes how fast a group grows. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length ''n''. Definition Suppose ''G'' is a finitely generated group; and ''T'' is a finite ''symmetric'' set of generators (symmetric means that if x \in T then x^ \in T ). Any element x \in G can be expressed as a word in the ''T''-alphabet : x = a_1 \cdot a_2 \cdots a_k \text a_i\in T. Consider the subset of all elements of ''G'' that can be expressed by such a word of length ≤ ''n'' :B_n(G,T) = \. This set is just the closed ball of radius ''n'' in the word metric ''d'' on ''G'' with respect to the generating set ''T'': :B_n(G,T) = \. More geometrically, B_n(G,T) is the set of vertices in the Cayley graph with respect to ''T'' that are within distan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
