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c_ucbdkdan2qdy | Together with an important generalization, Schunck classes, and an important dualization, Fischer classes, formations formed the major research themes of the late 20th century in the theory of finite soluble groups. A dual notion to Carter subgroups was introduced by Bernd Fischer in (Fischer 1966). A Fischer subgroup of a group is a nilpotent subgroup containing every other nilpotent subgroup it normalizes. A Fischer subgroup is a maximal nilpotent subgroup, but not every maximal nilpotent subgroup is a Fischer subgroup: again the nonabelian group of order six provides an example as every non-identity proper subgroup is a maximal nilpotent subgroup, but only the subgroup of order three is a Fischer subgroup (Wehrfritz 1999, p. 98). | Carter subgroup |
c_6yqboe9mt3qe | In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups. | Reduced abelian group |
c_3mjplhgdmy4p | In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and abnormal subgroups such as Sylow subgroups, (Doerk & Hawkes 1992, I.§6). A subgroup is pronormal if each of its conjugates is conjugate to it already in the subgroup generated by it and its conjugate. | Pronormal subgroup |
c_yppagxkl68tv | That is, H is pronormal in G if for every g in G, there is some k in the subgroup generated by H and Hg such that Hk = Hg. (Here Hg denotes the conjugate subgroup gHg-1.) Here are some relations with other subgroup properties: Every normal subgroup is pronormal. | Pronormal subgroup |
c_wzpygoeujlad | Every Sylow subgroup is pronormal. Every pronormal subnormal subgroup is normal. Every abnormal subgroup is pronormal. Every pronormal subgroup is weakly pronormal, that is, it has the Frattini property. Every pronormal subgroup is paranormal, and hence polynormal. | Pronormal subgroup |
c_y9kg6v8pq7ml | In mathematics, especially in the field of group theory, the central product is one way of producing a group from two smaller groups. The central product is similar to the direct product, but in the central product two isomorphic central subgroups of the smaller groups are merged into a single central subgroup of the product. Central products are an important construction and can be used for instance to classify extraspecial groups. | Central product |
c_1cdhfkmzlivb | In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those modules which leave short exact sequences exact after tensoring, a pure submodule defines a short exact sequence (known as a pure exact sequence) that remains exact after tensoring with any module. Similarly a flat module is a direct limit of projective modules, and a pure exact sequence is a direct limit of split exact sequences. | Pure subring |
c_bm9wzuq7kk3q | In mathematics, especially in the field of representation theory, Schur functors (named after Issai Schur) are certain functors from the category of modules over a fixed commutative ring to itself. They generalize the constructions of exterior powers and symmetric powers of a vector space. Schur functors are indexed by Young diagrams in such a way that the horizontal diagram with n cells corresponds to the nth symmetric power functor, and the vertical diagram with n cells corresponds to the nth exterior power functor. If a vector space V is a representation of a group G, then S λ V {\displaystyle \mathbb {S} ^{\lambda }V} also has a natural action of G for any Schur functor S λ ( − ) {\displaystyle \mathbb {S} ^{\lambda }(-)} . | Schur functor |
c_131bn8jtdune | In mathematics, especially in the field of ring theory, a (right) free ideal ring, or fir, is a ring in which all right ideals are free modules with unique rank. A ring such that all right ideals with at most n generators are free and have unique rank is called an n-fir. A semifir is a ring in which all finitely generated right ideals are free modules of unique rank. (Thus, a ring is semifir if it is n-fir for all n ≥ 0.) The semifir property is left-right symmetric, but the fir property is not. | Free ideal ring |
c_3zt6iv50y5jq | In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways. A (meet-)irreducible ring is a ring in which the intersection of two non-zero ideals is always non-zero. A directly irreducible ring is a ring which cannot be written as the direct sum of two non-zero rings. A subdirectly irreducible ring is a ring with a unique, non-zero minimum two-sided ideal. | Irreducible ring |
c_22zjxxf9un9f | A ring with an irreducible spectrum is a ring whose spectrum is irreducible as a topological space. "Meet-irreducible" rings are referred to as "irreducible rings" in commutative algebra. | Irreducible ring |
c_qq8knyjdnjj4 | This article adopts the term "meet-irreducible" in order to distinguish between the several types being discussed. Meet-irreducible rings play an important part in commutative algebra, and directly irreducible and subdirectly irreducible rings play a role in the general theory of structure for rings. Subdirectly irreducible algebras have also found use in number theory. This article follows the convention that rings have multiplicative identity, but are not necessarily commutative. | Irreducible ring |
c_a8iqdaknhzcg | In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup N and the quotient group G/N to the cohomology of the total group G. The spectral sequence is named after Roger Lyndon, Gerhard Hochschild, and Jean-Pierre Serre. | Lyndon–Hochschild–Serre spectral sequence |
c_t5jcsh2lgqa2 | In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras), it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal. This article uses the language of group theory; analogous terms are used for Lie algebras. A general group possesses a lower central series and upper central series (also called the descending central series and ascending central series, respectively), but these are central series in the strict sense (terminating in the trivial subgroup) if and only if the group is nilpotent. A related but distinct construction is the derived series, which terminates in the trivial subgroup whenever the group is solvable. | Upper central series |
c_j8sp3kk82oqm | In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group G that is constant on the conjugacy classes of G. In other words, it is invariant under the conjugation map on G. Such functions play a basic role in representation theory. | Class function |
c_nh4yi1ia0lr2 | In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Georg Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory (Nakayama 1939), (Nakayama 1941). | Frobenius ring |
c_4ba3ufpdtzbk | Jean Dieudonné used this to characterize Frobenius algebras (Dieudonné 1958). Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory. | Frobenius ring |
c_oh0d7iksbukf | In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced by McNulty and Shallon, and has seen many uses in the field of universal algebra since then. | Graph algebra |
c_pvspxzhgk9u8 | In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group PGL(V) = GL(V)/Z(V)where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of the general linear group. The projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly: PSL(V) = SL(V)/SZ(V)where SL(V) is the special linear group over V and SZ(V) is the subgroup of scalar transformations with unit determinant. | Projective linear group |
c_fwecaibj7gm2 | Here SZ is the center of SL, and is naturally identified with the group of nth roots of unity in F (where n is the dimension of V and F is the base field). PGL and PSL are some of the fundamental groups of study, part of the so-called classical groups, and an element of PGL is called projective linear transformation, projective transformation or homography. | Projective linear group |
c_ue0hantigspp | If V is the n-dimensional vector space over a field F, namely V = Fn, the alternate notations PGL(n, F) and PSL(n, F) are also used. Note that PGL(n, F) and PSL(n, F) are isomorphic if and only if every element of F has an nth root in F. As an example, note that PGL(2, C) = PSL(2, C), but that PGL(2, R) > PSL(2, R); this corresponds to the real projective line being orientable, and the projective special linear group only being the orientation-preserving transformations. PGL and PSL can also be defined over a ring, with an important example being the modular group, PSL(2, Z). | Projective linear group |
c_ob6s3iwlafzj | In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point. It roughly states that the existence of a local diffeomorphism near a fixed point implies the existence of a local stable center manifold containing that fixed point. This manifold has dimension equal to the number of eigenvalues of the Jacobian matrix of the fixed point that are less than 1. | Stable manifold theorem |
c_abqoi9rngqfr | In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system. A system that has reached its limiting set is said to be at equilibrium. | Limit set |
c_sb2d86gcot90 | In mathematics, especially in the study of infinite groups, the Hirsch–Plotkin radical is a subgroup describing the normal locally nilpotent subgroups of the group. It was named by Gruenberg (1961) after Kurt Hirsch and Boris I. Plotkin, who proved that the join of normal locally nilpotent subgroups is locally nilpotent; this fact is the key ingredient in its construction.The Hirsch–Plotkin radical is defined as the subgroup generated by the union of the normal locally nilpotent subgroups (that is, those normal subgroups such that every finitely generated subgroup is nilpotent). The Hirsch–Plotkin radical is itself a locally nilpotent normal subgroup, so is the unique largest such. | Hirsch–Plotkin radical |
c_hzka9xg85123 | In a finite group, the Hirsch–Plotkin radical coincides with the Fitting subgroup but for infinite groups the two subgroups can differ. The subgroup generated by the union of infinitely many normal nilpotent subgroups need not itself be nilpotent, so the Fitting subgroup must be modified in this case. == References == | Hirsch–Plotkin radical |
c_2uiebxtud1g3 | In mathematics, especially in topology, a Kuranishi structure is a smooth analogue of scheme structure. If a topological space is endowed with a Kuranishi structure, then locally it can be identified with the zero set of a smooth map ( f 1 , … , f k ): R n + k → R k {\displaystyle (f_{1},\ldots ,f_{k})\colon \mathbb {R} ^{n+k}\to \mathbb {R} ^{k}} , or the quotient of such a zero set by a finite group. Kuranishi structures were introduced by Japanese mathematicians Kenji Fukaya and Kaoru Ono in the study of Gromov–Witten invariants and Floer homology in symplectic geometry, and were named after Masatake Kuranishi. | Kuranishi theory |
c_1nd9zi2anw8b | In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat). A basic example is a subset of a smooth manifold that admits a Whitney stratification. But there is also an abstract stratified space such as a Thom–Mather stratified space. On a stratified space, a constructible sheaf can be defined as a sheaf that is locally constant on each stratum. Among the several ideals, Grothendieck's Esquisse d’un programme considers (or proposes) a stratified space with what he calls the tame topology. | Mather stratified space |
c_7vhz9avpjrxd | In mathematics, especially in topology, a topological group G {\displaystyle G} is said to have no small subgroup if there exists a neighborhood U {\displaystyle U} of the identity that contains no nontrivial subgroup of G . {\displaystyle G.} An abbreviation '"NSS"' is sometimes used. | No small subgroup |
c_tydivqb83p6z | A basic example of a topological group with no small subgroup is the general linear group over the complex numbers. A locally compact, separable metric, locally connected group with no small subgroup is a Lie group. (cf. Hilbert's fifth problem.) | No small subgroup |
c_59zoyvttwe6d | In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere. | Equidimensionality |
c_sag3gxlejnte | In mathematics, especially linear algebra, a matrix is called Metzler, quasipositive (or quasi-positive) or essentially nonnegative if all of its elements are non-negative except for those on the main diagonal, which are unconstrained. That is, a Metzler matrix is any matrix A which satisfies A = ( a i j ) ; a i j ≥ 0 , i ≠ j . {\displaystyle A=(a_{ij});\quad a_{ij}\geq 0,\quad i\neq j.} Metzler matrices are also sometimes referred to as Z ( − ) {\displaystyle Z^{(-)}} -matrices, as a Z-matrix is equivalent to a negated quasipositive matrix. | Quasipositive matrix |
c_3zcuv49lbq93 | In mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices). The name M-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive. | M-matrix |
c_us0qo70umli1 | In mathematics, especially linear algebra, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of permutation matrices, where the 1 elements reside on the antidiagonal and all other elements are zero. In other words, they are 'row-reversed' or 'column-reversed' versions of the identity matrix. J 2 = ( 0 1 1 0 ) ; J 3 = ( 0 0 1 0 1 0 1 0 0 ) ; J n = ( 0 0 ⋯ 0 0 1 0 0 ⋯ 0 1 0 0 0 ⋯ 1 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 1 ⋯ 0 0 0 1 0 ⋯ 0 0 0 ) . {\displaystyle J_{2}={\begin{pmatrix}0&1\\1&0\end{pmatrix}};\quad J_{3}={\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}};\quad J_{n}={\begin{pmatrix}0&0&\cdots &0&0&1\\0&0&\cdots &0&1&0\\0&0&\cdots &1&0&0\\\vdots &\vdots &&\vdots &\vdots &\vdots \\0&1&\cdots &0&0&0\\1&0&\cdots &0&0&0\end{pmatrix}}.} | Exchange matrix |
c_vho77or5pe2m | In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line R ∪ { ± ∞ } , {\displaystyle \mathbb {R} \cup \{\pm \infty \},} which consists of the real numbers R {\displaystyle \mathbb {R} } and ± ∞ . {\displaystyle \pm \infty .} A set function generally aims to measure subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning. | Set function |
c_veumalktit6w | In mathematics, especially operator theory, a convexoid operator is a bounded linear operator T on a complex Hilbert space H such that the closure of the numerical range coincides with the convex hull of its spectrum. An example of such an operator is a normal operator (or some of its generalization). A closely related operator is a spectraloid operator: an operator whose spectral radius coincides with its numerical radius. In fact, an operator T is convexoid if and only if T − λ {\displaystyle T-\lambda } is spectraloid for every complex number λ {\displaystyle \lambda } . | Convexoid operator |
c_4yei0nb4zecf | In mathematics, especially operator theory, a hyponormal operator is a generalization of a normal operator. In general, a bounded linear operator T on a complex Hilbert space H is said to be p-hyponormal ( 0 < p ≤ 1 {\displaystyle 0 | Hyponormal operator |
c_md8sfpm9543d | In mathematics, especially operator theory, a paranormal operator is a generalization of a normal operator. More precisely, a bounded linear operator T on a complex Hilbert space H is said to be paranormal if: ‖ T 2 x ‖ ≥ ‖ T x ‖ 2 {\displaystyle \|T^{2}x\|\geq \|Tx\|^{2}} for every unit vector x in H. The class of paranormal operators was introduced by V. Istratescu in 1960s, though the term "paranormal" is probably due to Furuta.Every hyponormal operator (in particular, a subnormal operator, a quasinormal operator and a normal operator) is paranormal. If T is a paranormal, then Tn is paranormal. | Paranormal operator |
c_rydi5dp7dvvf | On the other hand, Halmos gave an example of a hyponormal operator T such that T2 isn't hyponormal. Consequently, not every paranormal operator is hyponormal.A compact paranormal operator is normal. == References == | Paranormal operator |
c_02yqinect3cb | In mathematics, especially operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators. Some examples of subnormal operators are isometries and Toeplitz operators with analytic symbols. | Subnormal operator |
c_hxrfbeuzqh9u | In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, transitive and antisymmetric. A partially ordered set (poset for short) is a set on which a partial order is defined. | Poset category |
c_6ngpukf04h6e | In mathematics, especially order theory, a prefix ordered set generalizes the intuitive concept of a tree by introducing the possibility of continuous progress and continuous branching. Natural prefix orders often occur when considering dynamical systems as a set of functions from time (a totally-ordered set) to some phase space. In this case, the elements of the set are usually referred to as executions of the system. The name prefix order stems from the prefix order on words, which is a special kind of substring relation and, because of its discrete character, a tree. | Prefix order |
c_5spwkt027x0v | In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders.There are several common ways of formalizing weak orderings, that are different from each other but cryptomorphic (interconvertable with no loss of information): they may be axiomatized as strict weak orderings (strictly partially ordered sets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at least one of the two possible relations exists between every pair of elements), or as ordered partitions (partitions of the elements into disjoint subsets, together with a total order on the subsets). In many cases another representation called a preferential arrangement based on a utility function is also possible. Weak orderings are counted by the ordered Bell numbers. They are used in computer science as part of partition refinement algorithms, and in the C++ Standard Library. | Weak order |
c_wku5ev5a6c0c | In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering relation is commonly used to graphically express the partial order by means of the Hasse diagram. | Covering relation |
c_vbxpnnr3xnu6 | In mathematics, especially order theory, the interval order for a collection of intervals on the real line is the partial order corresponding to their left-to-right precedence relation—one interval, I1, being considered less than another, I2, if I1 is completely to the left of I2. More formally, a countable poset P = ( X , ≤ ) {\displaystyle P=(X,\leq )} is an interval order if and only if there exists a bijection from X {\displaystyle X} to a set of real intervals, so x i ↦ ( ℓ i , r i ) {\displaystyle x_{i}\mapsto (\ell _{i},r_{i})} , such that for any x i , x j ∈ X {\displaystyle x_{i},x_{j}\in X} we have x i < x j {\displaystyle x_{i} b {\displaystyle a>b} and c > d {\displaystyle c>d} one must have a > d {\displaystyle a>d} or c > b {\displaystyle c>b} . The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form ( ℓ i , ℓ i + 1 ) {\displaystyle (\ell _{i},\ell _{i}+1)} , is precisely the semiorders. The complement of the comparability graph of an interval order ( X {\displaystyle X} , ≤) is the interval graph ( X , ∩ ) {\displaystyle (X,\cap )} . Interval orders should not be confused with the interval-containment orders, which are the inclusion orders on intervals on the real line (equivalently, the orders of dimension ≤ 2). | Interval dimension |
c_frus3f15je7c | In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem. In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space R n {\displaystyle R^{n}} , n ≥ 2 {\displaystyle n\geq 2} is the probability that a Brownian motion started inside a domain hits that subset of the boundary. More generally, harmonic measure of an Itō diffusion X describes the distribution of X as it hits the boundary of D. In the complex plane, harmonic measure can be used to estimate the modulus of an analytic function inside a domain D given bounds on the modulus on the boundary of the domain; a special case of this principle is Hadamard's three-circle theorem. | Harmonic measure |
c_16mwfc5h9u22 | On simply connected planar domains, there is a close connection between harmonic measure and the theory of conformal maps. The term harmonic measure was introduced by Rolf Nevanlinna in 1928 for planar domains, although Nevanlinna notes the idea appeared implicitly in earlier work by Johansson, F. Riesz, M. Riesz, Carleman, Ostrowski and Julia (original order cited). The connection between harmonic measure and Brownian motion was first identified by Kakutani ten years later in 1944. | Harmonic measure |
c_i7cwootj4ina | In mathematics, especially real analysis, a flat function is a smooth function f: R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } all of whose derivatives vanish at a given point x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} } . The flat functions are, in some sense, the antitheses of the analytic functions. An analytic function f: R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is given by a convergent power series close to some point x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} }: f ( x ) ∼ lim n → ∞ ∑ k = 0 n f ( k ) ( x 0 ) k ! ( x − x 0 ) k . | Flat function |
c_b3wrjmcujijr | {\displaystyle f(x)\sim \lim _{n\to \infty }\sum _{k=0}^{n}{\frac {f^{(k)}(x_{0})}{k!}}(x-x_{0})^{k}.} In the case of a flat function, all derivatives vanish at x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} } , i.e. f ( k ) ( x 0 ) = 0 {\displaystyle f^{(k)}(x_{0})=0} for all k ∈ N {\displaystyle k\in \mathbb {N} } . This means that a meaningful Taylor series expansion in a neighbourhood of x 0 {\displaystyle x_{0}} is impossible. | Flat function |
c_j1jf07wvnmj1 | In the language of Taylor's theorem, the non-constant part of the function always lies in the remainder R n ( x ) {\displaystyle R_{n}(x)} for all n ∈ N {\displaystyle n\in \mathbb {N} } . The function need not be flat at just one point. Trivially, constant functions on R {\displaystyle \mathbb {R} } are flat everywhere. But there are also other, less trivial, examples. | Flat function |
c_gbtiigg7qbph | In mathematics, especially representation theory and combinatorics, a Frobenius characteristic map is an isometric isomorphism between the ring of characters of symmetric groups and the ring of symmetric functions. It builds a bridge between representation theory of the symmetric groups and algebraic combinatorics. This map makes it possible to study representation problems with help of symmetric functions and vice versa. This map is named after German mathematician Ferdinand Georg Frobenius. | Frobenius characteristic map |
c_rdsmisfoa46e | In mathematics, especially ring theory, a regular ideal can refer to multiple concepts. In operator theory, a right ideal i {\displaystyle {\mathfrak {i}}} in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that e x − x ∈ i {\displaystyle ex-x\in {\mathfrak {i}}} for every x ∈ A {\displaystyle x\in A} .In commutative algebra a regular ideal refers to an ideal containing a non-zero divisor. This article will use "regular element ideal" to help distinguish this type of ideal. A two-sided ideal i {\displaystyle {\mathfrak {i}}} of a ring R can also be called a (von Neumann) regular ideal if for each element x of i {\displaystyle {\mathfrak {i}}} there exists a y in i {\displaystyle {\mathfrak {i}}} such that xyx=x.Finally, regular ideal has been used to refer to an ideal J of a ring R such that the quotient ring R/J is von Neumann regular ring. This article will use "quotient von Neumann regular" to refer to this type of regular ideal. Since the adjective regular has been overloaded, this article adopts the alternative adjectives modular, regular element, von Neumann regular, and quotient von Neumann regular to distinguish between concepts. | Regular ideal |
c_i384hjrnl5v3 | In mathematics, especially ring theory, the class of Frobenius rings and their generalizations are the extension of work done on Frobenius algebras. Perhaps the most important generalization is that of quasi-Frobenius rings (QF rings), which are in turn generalized by right pseudo-Frobenius rings (PF rings) and right finitely pseudo-Frobenius rings (FPF rings). Other diverse generalizations of quasi-Frobenius rings include QF-1, QF-2 and QF-3 rings. These types of rings can be viewed as descendants of algebras examined by Georg Frobenius. A partial list of pioneers in quasi-Frobenius rings includes R. Brauer, K. Morita, T. Nakayama, C. J. Nesbitt, and R. M. Thrall. | Quasi-Frobenius ring |
c_01x2qlr7jcby | In mathematics, especially several complex variables, an analytic polyhedron is a subset of the complex space Cn of the form P = { z ∈ D: | f j ( z ) | < 1 , 1 ≤ j ≤ N } {\displaystyle P=\{z\in D:|f_{j}(z)|<1,\;\;1\leq j\leq N\}} where D is a bounded connected open subset of Cn, f j {\displaystyle f_{j}} are holomorphic on D and P is assumed to be relatively compact in D. If f j {\displaystyle f_{j}} above are polynomials, then the set is called a polynomial polyhedron. Every analytic polyhedron is a domain of holomorphy and it is thus pseudo-convex. The boundary of an analytic polyhedron is contained in the union of the set of hypersurfaces σ j = { z ∈ D: | f j ( z ) | = 1 } , 1 ≤ j ≤ N . {\displaystyle \sigma _{j}=\{z\in D:|f_{j}(z)|=1\},\;1\leq j\leq N.} An analytic polyhedron is a Weil polyhedron, or Weil domain if the intersection of any k of the above hypersurfaces has dimension no greater than 2n-k. | Analytic polyhedron |
c_icc6ybou2abm | In mathematics, especially several complex variables, the Behnke–Stein theorem states that a connected, non-compact (open) Riemann surface is a Stein manifold. In other words, it states that there is a nonconstant single-valued holomorphic function (univalent function) on such a Riemann surface. It is a generalization of the Runge approximation theorem and was proved by Heinrich Behnke and Karl Stein in 1948. | Behnke–Stein theorem on Stein manifolds |
c_zuwrfvwg75nm | In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence G k ⊂ C n {\displaystyle G_{k}\subset \mathbb {C} ^{n}} (i.e., G k ⊂ G k + 1 {\displaystyle G_{k}\subset G_{k+1}} ) of domains of holomorphy is again a domain of holomorphy. It was proved by Heinrich Behnke and Karl Stein in 1938.This is related to the fact that an increasing union of pseudoconvex domains is pseudoconvex and so it can be proven using that fact and the solution of the Levi problem. Though historically this theorem was in fact used to solve the Levi problem, and the theorem itself was proved using the Oka–Weil theorem. This theorem again holds for Stein manifolds, but it is not known if it holds for Stein space. | Behnke–Stein theorem |
c_d2zlc7mgpr4j | In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the d = 2 , 3 {\displaystyle d=2,3} case) by Hermann Weyl for eigenvalues for the Laplace–Beltrami operator acting on functions that vanish at the boundary of a bounded domain Ω ⊂ R d {\displaystyle \Omega \subset \mathbb {R} ^{d}} . In particular, he proved that the number, N ( λ ) {\displaystyle N(\lambda )} , of Dirichlet eigenvalues (counting their multiplicities) less than or equal to λ {\displaystyle \lambda } satisfies lim λ → ∞ N ( λ ) λ d / 2 = ( 2 π ) − d ω d v o l ( Ω ) {\displaystyle \lim _{\lambda \rightarrow \infty }{\frac {N(\lambda )}{\lambda ^{d/2}}}=(2\pi )^{-d}\omega _{d}\mathrm {vol} (\Omega )} where ω d {\displaystyle \omega _{d}} is a volume of the unit ball in R d {\displaystyle \mathbb {R} ^{d}} . In 1912 he provided a new proof based on variational methods. | Weyl's law |
c_2gq1n9o5vn7i | In mathematics, especially the field of group theory, the Parker vector is an integer vector that describes a permutation group in terms of the cycle structure of its elements. | Parker vector |
c_292n0ub824gq | In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil. | Oka–Weil theorem |
c_hhjbhmler4ro | In mathematics, especially the usage of linear algebra in mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916. | Einstein summation notation |
c_1zqyjpurk21h | In mathematics, especially topology, a perfect map is a particular kind of continuous function between topological spaces. Perfect maps are weaker than homeomorphisms, but strong enough to preserve some topological properties such as local compactness that are not always preserved by continuous maps. | Perfect map |
c_nhbszgw8f7xs | In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d. For an exact form α, α = dβ for some differential form β of degree one less than that of α. The form β is called a "potential form" or "primitive" for α. Since the exterior derivative of a closed form is zero, β is not unique, but can be modified by the addition of any closed form of degree one less than that of α. Because d2 = 0, every exact form is necessarily closed. The question of whether every closed form is exact depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. More general questions of this kind on an arbitrary differentiable manifold are the subject of de Rham cohomology, which allows one to obtain purely topological information using differential methods. | Closed differential form |
c_ax6jc68micif | In mathematics, essential dimension is an invariant defined for certain algebraic structures such as algebraic groups and quadratic forms. It was introduced by J. Buhler and Z. Reichstein and in its most generality defined by A. Merkurjev.Basically, essential dimension measures the complexity of algebraic structures via their fields of definition. For example, a quadratic form q: V → K over a field K, where V is a K-vector space, is said to be defined over a subfield L of K if there exists a K-basis e1,...,en of V such that q can be expressed in the form q ( ∑ x i e i ) = ∑ a i j x i x j {\displaystyle q\left(\sum x_{i}e_{i}\right)=\sum a_{ij}x_{i}x_{j}} with all coefficients aij belonging to L. If K has characteristic different from 2, every quadratic form is diagonalizable. Therefore, q has a field of definition generated by n elements. Technically, one always works over a (fixed) base field k and the fields K and L in consideration are supposed to contain k. The essential dimension of q is then defined as the least transcendence degree over k of a subfield L of K over which q is defined. | Essential dimension |
c_8eugrrdnac9v | In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers. They are useful in some transfinite induction proofs. The literature contains a few equivalent definitions of the parity of an ordinal α: Every limit ordinal (including 0) is even. The successor of an even ordinal is odd, and vice versa. | Even ordinal |
c_bf26c6ed35mb | Let α = λ + n, where λ is a limit ordinal and n is a natural number. The parity of α is the parity of n. Let n be the finite term of the Cantor normal form of α. The parity of α is the parity of n. Let α = ωβ + n, where n is a natural number. The parity of α is the parity of n. If α = 2β, then α is even. | Even ordinal |
c_cbwnv0s7qekk | Otherwise α = 2β + 1 and α is odd.Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form β2 = β + β. Ordinal multiplication is not commutative, so in general 2β ≠ β2. In fact, the even ordinal ω + 4 cannot be expressed as β + β, and the ordinal number (ω + 3)2 = (ω + 3) + (ω + 3) = ω + (3 + ω) + 3 = ω + ω + 3 = ω2 + 3is not even. A simple application of ordinal parity is the idempotence law for cardinal addition (given the well-ordering theorem). | Even ordinal |
c_zedxdjw7waj9 | Given an infinite cardinal κ, or generally any limit ordinal κ, κ is order-isomorphic to both its subset of even ordinals and its subset of odd ordinals. Hence one has the cardinal sum κ + κ = κ. == References == | Even ordinal |
c_mjmn0fp47jqr | In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function f ( x ) = x n {\displaystyle f(x)=x^{n}} is an even function if n is an even integer, and it is an odd function if n is an odd integer. | Even function |
c_awmqs6lwgoeu | In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations. | Analytic function of a matrix |
c_du6t6csmyuph | In mathematics, exponential equivalence of measures is how two sequences or families of probability measures are "the same" from the point of view of large deviations theory. | Exponentially equivalent measures |
c_9wlmnhvibo4w | In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function. | Exponential polynomial |
c_atie1vcm9fqu | In mathematics, exponentiation is an operation involving two numbers, the base and the exponent or power. Exponentiation is written as bn, where b is the base and n is the power; this is pronounced as "b (raised) to the (power of) n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases: The exponent is usually shown as a superscript to the right of the base. In that case, bn is called "b raised to the nth power", "b (raised) to the power of n", "the nth power of b", "b to the nth power", or most briefly as "b to the nth". | Laws of exponents |
c_es6xk92577zr | Starting from the basic fact stated above that, for any positive integer n {\displaystyle n} , b n {\displaystyle b^{n}} is n {\displaystyle n} occurrences of b {\displaystyle b} all multiplied by each other, several other properties of exponentiation directly follow. In particular: In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that b 0 {\displaystyle b^{0}} must be equal to 1 for any b ≠ 0 {\displaystyle b\neq 0} , as follows. | Laws of exponents |
c_sx1ucqkexq0g | For any n {\displaystyle n} , b 0 × b n = b 0 + n = b n {\displaystyle b^{0}\times b^{n}=b^{0+n}=b^{n}} . Dividing both sides by b n {\displaystyle b^{n}} gives b 0 = b n / b n = 1 {\displaystyle b^{0}=b^{n}/b^{n}=1} . The fact that b 1 = b {\displaystyle b^{1}=b} can similarly be derived from the same rule. | Laws of exponents |
c_yxwfvcibun7b | For example, ( b 1 ) 3 = b 1 × b 1 × b 1 = b 1 + 1 + 1 = b 3 {\displaystyle (b^{1})^{3}=b^{1}\times b^{1}\times b^{1}=b^{1+1+1}=b^{3}} . Taking the cube root of both sides gives b 1 = b {\displaystyle b^{1}=b} . The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. | Laws of exponents |
c_x7zr70wjagmj | Consider the question of what b − 1 {\displaystyle b^{-1}} should mean. In order to respect the "exponents add" rule, it must be the case that b − 1 × b 1 = b − 1 + 1 = b 0 = 1 {\displaystyle b^{-1}\times b^{1}=b^{-1+1}=b^{0}=1} . Dividing both sides by b 1 {\displaystyle b^{1}} gives b − 1 = 1 / b 1 {\displaystyle b^{-1}=1/b^{1}} , which can be more simply written as b − 1 = 1 / b {\displaystyle b^{-1}=1/b} , using the result from above that b 1 = b {\displaystyle b^{1}=b} . | Laws of exponents |
c_e82wzjx1qy3d | By a similar argument, b − n = 1 / b n {\displaystyle b^{-n}=1/b^{n}} . The properties of fractional exponents also follow from the same rule. For example, suppose we consider b {\displaystyle {\sqrt {b}}} and ask if there is some suitable exponent, which we may call r {\displaystyle r} , such that b r = b {\displaystyle b^{r}={\sqrt {b}}} . | Laws of exponents |
c_7gjumyg922vh | From the definition of the square root, we have that b × b = b {\displaystyle {\sqrt {b}}\times {\sqrt {b}}=b} . Therefore, the exponent r {\displaystyle r} must be such that b r × b r = b {\displaystyle b^{r}\times b^{r}=b} . Using the fact that multiplying makes exponents add gives b r + r = b {\displaystyle b^{r+r}=b} . | Laws of exponents |
c_66a82mxgto1u | The b {\displaystyle b} on the right-hand side can also be written as b 1 {\displaystyle b^{1}} , giving b r + r = b 1 {\displaystyle b^{r+r}=b^{1}} . Equating the exponents on both sides, we have r + r = 1 {\displaystyle r+r=1} . | Laws of exponents |
c_2onofeiot9tf | Therefore, r = 1 2 {\displaystyle r={\frac {1}{2}}} , so b = b 1 / 2 {\displaystyle {\sqrt {b}}=b^{1/2}} . The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices. Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography. | Laws of exponents |
c_7yon0dz7hzge | In mathematics, extendible cardinals are large cardinals introduced by Reinhardt (1974), who was partly motivated by reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces of the universe of sets start to look similar, in the sense that each is elementarily embeddable into a later one. | Extendible cardinal |
c_eax9sv68g6oj | In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results. Extrapolation may also mean extension of a method, assuming similar methods will be applicable. Extrapolation may also apply to human experience to project, extend, or expand known experience into an area not known or previously experienced so as to arrive at a (usually conjectural) knowledge of the unknown (e.g. a driver extrapolates road conditions beyond his sight while driving). The extrapolation method can be applied in the interior reconstruction problem. | Extrapolation method |
c_pvy8jxca09pu | In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x2 – 4. Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any x {\displaystyle x} can be trivially written as ( x y ) × ( 1 / y ) {\displaystyle (xy)\times (1/y)} whenever y {\displaystyle y} is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. | Factorization |
c_xd52xvw7ob5p | Factorization was first considered by ancient Greek mathematicians in the case of integers. They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1. | Factorization |
c_c9s2c41zaqaj | Moreover, this factorization is unique up to the order of the factors. Although integer factorization is a sort of inverse to multiplication, it is much more difficult algorithmically, a fact which is exploited in the RSA cryptosystem to implement public-key cryptography. Polynomial factorization has also been studied for centuries. | Factorization |
c_f33o1vv7xlt8 | In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials. In particular, a univariate polynomial with complex coefficients admits a unique (up to ordering) factorization into linear polynomials: this is a version of the fundamental theorem of algebra. | Factorization |
c_p3z2sra0o21u | In this case, the factorization can be done with root-finding algorithms. The case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing (complete) factorizations within the ring of polynomials with rational number coefficients (see factorization of polynomials). | Factorization |
c_baeynpca5kbf | A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor uniquely into prime ideals. | Factorization |
c_0dz042o0qg7b | Factorization may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix L with all diagonal entries equal to one, an upper triangular matrix U, and a permutation matrix P; this is a matrix formulation of Gaussian elimination. | Factorization |
c_xl3yugqbatdm | In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a field and its absolute Galois group. It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups. | Field Arithmetic |
c_1oy67nm6ijnl | In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers. There are infinitely many different finite fields. Their number of elements is necessarily of the form pn where p is a prime number and n is a positive integer, and two finite fields of the same size are isomorphic. The prime p is called the characteristic of the field, and the positive integer n is called the dimension of the field over its prime field. Finite fields are used in a variety of applications, including in classical coding theory in linear block codes such as BCH codes and Reed–Solomon error correction, in cryptography algorithms such as the Rijndael (AES) encryption algorithm, in tournament scheduling, and in the design of experiments. | Finite field arithmetic |
c_tum1a0h2qowp | In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional vector space (or finite collection of times). | Finite-dimensional distribution |
c_wmdvpgx6dnum | In mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. It is mostly of interest for the study of infinite groups. Special cases of groups with finiteness properties are finitely generated and finitely presented groups. | Finiteness properties of groups |
c_iba5nxdo3rkj | In mathematics, five-term exact sequence or exact sequence of low-degree terms is a sequence of terms related to the first step of a spectral sequence. More precisely, let E 2 p , q ⇒ H n ( A ) {\displaystyle E_{2}^{p,q}\Rightarrow H^{n}(A)} be a first quadrant spectral sequence, meaning that E 2 p , q {\displaystyle E_{2}^{p,q}} vanishes except when p and q are both non-negative. Then there is an exact sequence 0 → E21,0 → H 1(A) → E20,1 → E22,0 → H 2(A).Here, the map E 2 0 , 1 → E 2 2 , 0 {\displaystyle E_{2}^{0,1}\to E_{2}^{2,0}} is the differential of the E 2 {\displaystyle E_{2}} -term of the spectral sequence. | Five-term exact sequence |
c_1mwme1p3ob7e | In mathematics, flat convergence is a notion for convergence of submanifolds of Euclidean space. It was first introduced by Hassler Whitney in 1957, and then extended to integral currents by Federer and Fleming in 1960. It forms a fundamental part of the field of geometric measure theory. The notion was applied to find solutions to Plateau's problem. In 2001 the notion of an integral current was extended to arbitrary metric spaces by Ambrosio and Kirchheim. | Flat convergence |
c_mbzcm2ibg75y | In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient R ( M , x ) , {\displaystyle R(M,\mathbf {x} ),} is defined as:: p. 234 For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x H {\displaystyle \mathbf {x} ^{\mathsf {H}}} to the usual transpose x T . {\displaystyle \mathbf {x} ^{\mathsf {T}}.} R ( M , c x ) = R ( M , x ) {\displaystyle R(M,c\mathbf {x} )=R(M,\mathbf {x} )} for any non-zero real scalar c . | Hermitian matrix |
c_r0senofqlu58 | {\displaystyle c.} Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues. It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value λ min {\displaystyle \lambda _{\min }} (the smallest eigenvalue of M) when x {\displaystyle \mathbf {x} } is v min {\displaystyle \mathbf {v} _{\min }} (the corresponding eigenvector). | Hermitian matrix |
c_ot2zgdrayxpd | Similarly, R ( M , x ) ≤ λ max {\displaystyle R(M,\mathbf {x} )\leq \lambda _{\max }} and R ( M , v max ) = λ max . {\displaystyle R(M,\mathbf {v} _{\max })=\lambda _{\max }.} The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. | Hermitian matrix |
c_uw0abzy6h86p | It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration. The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). | Hermitian matrix |
c_dgjqfjzsz0y5 | When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, λ max {\displaystyle \lambda _{\max }} is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to M associates the Rayleigh quotient R(M, x) for a fixed x and M varying through the algebra would be referred to as "vector state" of the algebra. | Hermitian matrix |
c_9xiqyu9s2i50 | In mathematics, for a natural number n ≥ 2 {\displaystyle n\geq 2} , the nth Fibonacci group, denoted F ( 2 , n ) {\displaystyle F(2,n)} or sometimes F ( n ) {\displaystyle F(n)} , is defined by n generators a 1 , a 2 , … , a n {\displaystyle a_{1},a_{2},\dots ,a_{n}} and n relations: a 1 a 2 = a 3 , {\displaystyle a_{1}a_{2}=a_{3},} a 2 a 3 = a 4 , {\displaystyle a_{2}a_{3}=a_{4},} … {\displaystyle \dots } a n − 2 a n − 1 = a n , {\displaystyle a_{n-2}a_{n-1}=a_{n},} a n − 1 a n = a 1 , {\displaystyle a_{n-1}a_{n}=a_{1},} a n a 1 = a 2 {\displaystyle a_{n}a_{1}=a_{2}} .These groups were introduced by John Conway in 1965. The group F ( 2 , n ) {\displaystyle F(2,n)} is of finite order for n = 2 , 3 , 4 , 5 , 7 {\displaystyle n=2,3,4,5,7} and infinite order for n = 6 {\displaystyle n=6} and n ≥ 8 {\displaystyle n\geq 8} . The infinitude of F ( 2 , 9 ) {\displaystyle F(2,9)} was proved by computer in 1990. | Fibonacci group |
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