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c_oeavd3igdcpn | In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring (R, +, ⋅) is the ring (R, +, ∗) whose multiplication ∗ is defined by a ∗ b = b ⋅ a for ... | Opposite algebra |
c_zmk7r9rpakml | In mathematics, specifically additive number theory, Romanov's theorem is a mathematical theorem proved by Nikolai Pavlovich Romanov. It states that given a fixed base b, the set of numbers that are the sum of a prime and a positive integer power of b has a positive lower asymptotic density. | Romanov's theorem |
c_rrdr8wjvi27h | In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual f... | Donaldson-Thomas theory |
c_jxbm8cqvz0a7 | Donaldson–Thomas invariants have close connections to Gromov–Witten invariants of algebraic three-folds and the theory of stable pairs due to Rahul Pandharipande and Thomas. Donaldson–Thomas theory is physically motivated by certain BPS states that occur in string and gauge theorypg 5. This is due to the fact the invar... | Donaldson-Thomas theory |
c_2sartt5xlbnf | In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map f: X → Y {\displaystyle f:X\rightarrow Y} of varieties is a kind of 'large' subvariety of X {\displaystyle X} which is 'crushed' by f {\displaystyle f} , in a certain definite sense. More strictly, f has an associated exceptional ... | Exceptional curve |
c_3invovem5nwh | A codimension-1 subvariety Z ⊂ X {\displaystyle Z\subset X} is said to be exceptional if f ( Z ) {\displaystyle f(Z)} has codimension at least 2 as a subvariety of Y {\displaystyle Y} . One may then define the exceptional divisor of f {\displaystyle f} to be ∑ i Z i ∈ D i v ( X ) , {\displaystyle \sum _{i}Z_{i}\in Div(... | Exceptional curve |
c_1v4t8e5553jz | In mathematics, specifically algebraic geometry, the valuative criteria are a collection of results that make it possible to decide whether a morphism of algebraic varieties, or more generally schemes, is universally closed, separated, or proper. | Valuative criterion |
c_vul4wnehne3q | In mathematics, specifically algebraic topology, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. Let G be a group and n a positive integer. A connected topological space X is called an Eilenberg–MacLane space of type K ( G , n ) {\displaystyle K(G,n)} , if it has n-th homotopy... | Eilenberg–MacLane spectra |
c_pjabmncjlu3z | Thus, one may consider K ( G , n ) {\displaystyle K(G,n)} as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a K ( G , n ) {\displaystyle K(G,n)} " or as "a model of K ( G , n ) {\displaystyle K(G,n)} ". Moreover, it is common to assume that this space is a CW-c... | Eilenberg–MacLane spectra |
c_tifnfd7u2yr6 | The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s. As such, an Eilenberg–MacLane space is a special kind of topological space that in homotopy theory can be regarded as a building block for CW-complexes via fibrations in a Postnikov system. These spaces are im... | Eilenberg–MacLane spectra |
c_ppmhzuud0la0 | In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space X is semi-locally simply connected if there is a lower bound on the sizes of the “holes” in X. This condition is n... | Semilocally simply connected |
c_mpgp5m6x6mo5 | In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is also present in other... | Cohomology ring |
c_lfnjgyzqmd1r | {\displaystyle H^{k}(X;R)\times H^{\ell }(X;R)\to H^{k+\ell }(X;R).} The cup product gives a multiplication on the direct sum of the cohomology groups H ∙ ( X ; R ) = ⨁ k ∈ N H k ( X ; R ) . | Cohomology ring |
c_9g7qi4oo8it2 | {\displaystyle H^{\bullet }(X;R)=\bigoplus _{k\in \mathbb {N} }H^{k}(X;R).} This multiplication turns H•(X;R) into a ring. In fact, it is naturally an N-graded ring with the nonnegative integer k serving as the degree. | Cohomology ring |
c_r7hfqqopui5b | The cup product respects this grading. The cohomology ring is graded-commutative in the sense that the cup product commutes up to a sign determined by the grading. Specifically, for pure elements of degree k and ℓ; we have ( α k ⌣ β ℓ ) = ( − 1 ) k ℓ ( β ℓ ⌣ α k ) . | Cohomology ring |
c_z7qigaapymn5 | {\displaystyle (\alpha ^{k}\smile \beta ^{\ell })=(-1)^{k\ell }(\beta ^{\ell }\smile \alpha ^{k}).} A numerical invariant derived from the cohomology ring is the cup-length, which means the maximum number of graded elements of degree ≥ 1 that when multiplied give a non-zero result. For example a complex projective spac... | Cohomology ring |
c_o71kdnwwmr9w | In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function f {\displaystyle f} between topological spaces X {\displaystyle X} and Y {\displaystyle Y} is the quotient M f = ( ( × X ) ⨿ Y ) / ∼ {\displaystyle M_{f}=((\times X)\amalg Y)\,/\,\sim } where the ⨿ {\displaystyle \amalg } de... | Mapping cylinder |
c_ydic3bligie7 | It is common to write M f {\displaystyle Mf} for M f {\displaystyle M_{f}} , and to use the notation ⊔ f {\displaystyle \sqcup _{f}} or ∪ f {\displaystyle \cup _{f}} for the mapping cylinder construction. That is, one writes M f = ( × X ) ∪ f Y {\displaystyle Mf=(\times X)\cup _{f}Y} with the subscripted cup symbol de... | Mapping cylinder |
c_pab1uo8o6niz | In mathematics, specifically algebraic topology, there is a resolution analogous to free resolutions of spectra yielding a tool for constructing the Adams spectral sequence. Essentially, the idea is to take a connective spectrum of finite type X {\displaystyle X} and iteratively resolve with other spectra that are in t... | Adams resolution |
c_b0659onnrafj | In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech. | Cocycle condition |
c_mw0kgm70hanu | In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum. | First Feigenbaum constant |
c_az79hzh47ufa | In mathematics, specifically category theory, a family of generators (or family of separators) of a category C {\displaystyle {\mathcal {C}}} is a collection G ⊆ O b ( C ) {\displaystyle {\mathcal {G}}\subseteq Ob({\mathcal {C}})} of objects in C {\displaystyle {\mathcal {C}}} , such that for any two distinct morphisms... | Generator (category theory) |
c_gy3q532oabha | In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between ... | Covariant functor |
c_vds4z5i34o15 | Thus, functors are important in all areas within mathematics to which category theory is applied. The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used functor in a linguistic context; see function word. | Covariant functor |
c_asqw9t685l5p | In mathematics, specifically category theory, a posetal category, or thin category, is a category whose homsets each contain at most one morphism. As such, a posetal category amounts to a preordered class (or a preordered set, if its objects form a set). As suggested by the name, the further requirement that the catego... | Posetal category |
c_fjuyllzqjc0q | When the commutative diagrams of a category are interpreted as a typed equational theory whose objects are the types, a codiscrete posetal category corresponds to an inconsistent theory understood as one satisfying the axiom x = y at all types. Viewing a 2-category as an enriched category whose hom-objects are categori... | Posetal category |
c_mycnv7n4rap5 | For example, under this assumption, a poset may be defined as a small posetal category, a distributive lattice as a small posetal distributive category, a Heyting algebra as a small posetal finitely cocomplete cartesian closed category, and a Boolean algebra as a small posetal finitely cocomplete *-autonomous category.... | Posetal category |
c_oqwkxnt1iwky | In mathematics, specifically category theory, a quasitopos is a generalization of a topos. A topos has a subobject classifier classifying all subobjects, but in a quasitopos, only strong subobjects are classified. Quasitoposes are also required to be finitely cocomplete and locally cartesian closed. A solid quasitopos ... | Quasitopos |
c_7giubjkmbptz | In mathematics, specifically category theory, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and arr... | Subcategory |
c_9xn7o9xeh6j8 | In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right... | Left adjoint |
c_3wri7g0uyvb3 | Naturality here means that there are natural isomorphisms between the pair of functors C ( F − , X ): D → S e t {\displaystyle {\mathcal {C}}(F-,X):{\mathcal {D}}\to \mathrm {Set} } and D ( − , G X ): D → S e t {\displaystyle {\mathcal {D}}(-,GX):{\mathcal {D}}\to \mathrm {Set} } for a fixed X {\displaystyle X} in C {\... | Left adjoint |
c_ypt50skki7id | We write F ⊣ G {\displaystyle F\dashv G} . An adjunction between categories C {\displaystyle {\mathcal {C}}} and D {\displaystyle {\mathcal {D}}} is somewhat akin to a "weak form" of an equivalence between C {\displaystyle {\mathcal {C}}} and D {\displaystyle {\mathcal {D}}} , and indeed every equivalence is an adjunct... | Left adjoint |
c_20wckn9rxo2g | In mathematics, specifically category theory, an essential monomorphism is a monomorphism f in a category C such that for a morphism g in C, the morphism g ∘ f {\displaystyle g\circ f} is a monomorphism only when g is a monomorphism. Essential monomorphisms in a category of modules are those whose image is an essential... | Essential monomorphism |
c_p6aieswr7mie | In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track of data surrounding a fixed object X {\displaystyle X} in some category C... | Overcategory |
c_ikzpe6eli8q0 | In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. WZ pairs are named after Herbert S. Wilf and Doron Zeilberger, and are instrumental in the evaluation of many sums involving binomial coefficients, factori... | Hypergeometric function identities |
c_fqcs87rwwxcs | In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian. | Hilbert's basis theorem |
c_1oeswj667plo | In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form x n / n ! {\displaystyle x^{n}/n!} meaningful even when it is not possible to actually divide by n ! {\displaystyle n!} . | Divided power structure |
c_xw7r91eihxc4 | In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n > 0. For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime number. The notion of primary ideals is... | Primary ideal |
c_5x7int4enb9k | Consequently, an irreducible ideal of a Noetherian ring is primary. Various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity. | Primary ideal |
c_m3grf7gqlqm4 | In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. A simple case arises in taking the square root of a positive real number. For example, 4 has two square roots: 2 and −2; of these the positi... | Principal value |
c_d9xchelezip2 | In mathematics, specifically computability and set theory, an ordinal α {\displaystyle \alpha } is said to be computable or recursive if there is a computable well-ordering of a computable subset of the natural numbers having the order type α {\displaystyle \alpha } . It is easy to check that ω {\displaystyle \omega } ... | Computable ordinal |
c_2yhrkdl3pgku | The Church–Kleene ordinal is a limit ordinal. An ordinal is computable if and only if it is smaller than ω 1 C K {\displaystyle \omega _{1}^{CK}} . | Computable ordinal |
c_k2xigb6jnpg8 | Since there are only countably many computable relations, there are also only countably many computable ordinals. Thus, ω 1 C K {\displaystyle \omega _{1}^{CK}} is countable. The computable ordinals are exactly the ordinals that have an ordinal notation in Kleene's O {\displaystyle {\mathcal {O}}} . | Computable ordinal |
c_2edhb2ymm54d | In mathematics, specifically convex geometry, the normal fan of a convex polytope P is a polyhedral fan that is dual to P. Normal fans have applications to polyhedral combinatorics, linear programming, tropical geometry and other areas of mathematics. | Normal fan |
c_g911agkkofgz | In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnorpg 14 was trying to understand the structure of ( n − 1 ) {\displaystyle (n-1)} -connected manifolds of dimension 2 n {\displaystyle 2n} (since n {\displaystyle n} -connected 2 n {\displaystyle 2n} -manifolds are homeomor... | Milnor's sphere |
c_1kz5x3ywdo9k | In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the invers... | Inverse function theorem |
c_gtiyyodspnco | In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann c... | Riemann curvature |
c_r7zoydmcmpcn | In mathematics, specifically enumerative geometry, the virtual fundamental class E ∙ vir {\displaystyle _{E^{\bullet }}^{\text{vir}}} of a space X {\displaystyle X} is a replacement of the classical fundamental class ∈ A ∗ ( X ) {\displaystyle \in A^{*}(X)} in its chow ring which has better behavior with respect to t... | Virtual fundamental class |
c_ti3dnf84c8pr | One such example is in the moduli space M ¯ 1 , n ( P 2 , 1 ) {\displaystyle {\overline {\mathcal {M}}}_{1,n}(\mathbb {P} ^{2},1)} for H {\displaystyle H} the class of a line in P 2 {\displaystyle \mathbb {P} ^{2}} . The non-compact "smooth" component is empty, but the boundary contains maps of curves f: C → P 2 {\dis... | Virtual fundamental class |
c_px4bqvbz1sfg | In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in (Mercer 1909), is one of the most notable results of the work of James Mercer (1883–1932). It is... | Mercer's theorem |
c_isfth46clo3o | In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true. Every Hilbert space has this property. | Approximation property |
c_iim4tqf8hejb | There are, however, Banach spaces which do not; Per Enflo published the first counterexample in a 1973 article. However, much work in this area was done by Grothendieck (1955). Later many other counterexamples were found. The space of bounded operators on ℓ 2 {\displaystyle \ell ^{2}} does not have the approximation pr... | Approximation property |
c_svregqnt7q6f | In mathematics, specifically functional analysis, a pth Schatten-class operator is a bounded linear operator on a Hilbert space with finite pth Schatten norm. The space of pth Schatten-class operators is a Banach space with respect to the Schatten norm. Via polar decomposition, one can prove that the space of pth Schat... | Schatten class operator |
c_yeysvxjtde86 | {\displaystyle \|ST\|_{S_{1}}\leq \|S\|_{S_{p}}\|T\|_{S_{q}}\ {\mbox{if}}\ S\in S_{p},\ T\in S_{q}{\mbox{ and }}1/p+1/q=1.} If we denote by S ∞ {\displaystyle S_{\infty }} the Banach space of compact operators on H with respect to the operator norm, the above Hölder-type inequality even holds for p ∈ {\displaystyle p\... | Schatten class operator |
c_plpslzr27s7j | From this it follows that ϕ: S p → S q ′ {\displaystyle \phi :S_{p}\rightarrow S_{q}'} , T ↦ t r ( T ⋅ ) {\displaystyle T\mapsto \mathrm {tr} (T\cdot )} is a well-defined contraction. (Here the prime denotes (topological) dual.) Observe that the 2nd Schatten class is in fact the Hilbert space of Hilbert–Schmidt operato... | Schatten class operator |
c_rb51f21eonx5 | In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent if it converges but different orderings do not all converge to that same value. Unconditional convergence is equivalent ... | Unconditional convergence |
c_5fz7ztespmyt | In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear alg... | Trace class operator |
c_v4z715le1mni | In quantum mechanics, mixed states are described by density matrices, which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nu... | Trace class operator |
c_6lskgjv3p3gj | In mathematics, specifically functional analysis, the Schatten norm (or Schatten–von-Neumann norm) arises as a generalization of p-integrability similar to the trace class norm and the Hilbert–Schmidt norm. | Schatten norm |
c_nlo4s40j61hc | In mathematics, specifically functional analysis, the barrier cone is a cone associated to any non-empty subset of a Banach space. It is closely related to the notions of support functions and polar sets. | Barrier cone |
c_2hi6jxuaa4a9 | In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory. The formal state... | Von Neumann bicommutant theorem |
c_9gjyye0rqsan | Then the closures of M in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant M′′ of M.This algebra is called the von Neumann algebra generated by M. There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras ... | Von Neumann bicommutant theorem |
c_eh3rlnhjxznn | One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak, ultrastrong, and *-ultrastrong... | Von Neumann bicommutant theorem |
c_g4vtjnrla8gc | In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1... | Compact space |
c_bzm7xtqx6oq4 | However, the extended real number line would be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. | Compact space |
c_ufx6uxe2ioiy | One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and... | Compact space |
c_kxle96n666d2 | Since neither 0 nor 1 are members of the open unit interval (0, 1), those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded. For example, c... | Compact space |
c_5epk92ds4hp3 | Compactness was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to spaces of functions. The Arzelà–Ascoli theorem and the Peano existence theorem exemplify applications of this notion of compactness to classical analysis. Following its initi... | Compact space |
c_p7oob79hku30 | In general topological spaces, however, these notions of compactness are not necessarily equivalent. The most useful notion — and the standard definition of the unqualified term compactness — is phrased in terms of the existence of finite families of open sets that "cover" the space in the sense that each point of the ... | Compact space |
c_4n1fs4qmothr | In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space. | Geometric group action |
c_t5s92kmnpjtl | In mathematics, specifically geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of equivalence (namely, homotopy equivalence) should... | Borel conjecture |
c_a3jbvdg6lxte | In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain. | Classification of manifolds |
c_pr474gv9yo1e | In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive integer with xp = e, where e is the identity ... | Cauchy's theorem (group theory) |
c_v2dt1otpwqti | In mathematics, specifically group theory, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They were introduced by the group theorist Philip Hall (1928). | Sylow system |
c_ss7kifu3y4wr | In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order p n {\displaystyle p^{n}} , for a fixed prime number p {\displaystyle p} and varying integer exponents n ≥ 0 {\displaysty... | Descendant tree (group theory) |
c_mjaov109pw4y | It turned out that descendant trees of a particular kind, the so-called pruned coclass trees whose infinitely many vertices share a common coclass r {\displaystyle r} , reveal a repeating finite pattern. These two crucial properties of finiteness and periodicity admit a characterization of all members of the tree by fi... | Descendant tree (group theory) |
c_ikbkyjfjtr0i | Consequently, descendant trees play a fundamental role in the classification of finite p-groups. By means of kernels and targets of Artin transfer homomorphisms, descendant trees can be endowed with additional structure. An important question is how the descendant tree T ( R ) {\displaystyle {\mathcal {T}}(R)} can actu... | Descendant tree (group theory) |
c_3rrthzz7nvit | In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, its central series is of finite length or its lower central series terminates with {1}. Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motiva... | Nilpotent Lie group |
c_67xrlw6jt5nr | It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups. Analogous terms ar... | Nilpotent Lie group |
c_s0gwpe300qxm | In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint, equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements (cardinality) as does H. Furthermore, H itself is b... | Right coset |
c_tjkl6hxhixt9 | The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by . Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group G, the numbe... | Right coset |
c_redee0bvr68g | In mathematics, specifically group theory, a subgroup series of a group G {\displaystyle G} is a chain of subgroups: 1 = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G {\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G} where 1 {\displaystyle 1} is the trivial subgroup. Subgroup series can simplify the study of a group to the study of ... | Noetherian group |
c_yibo1jd982a4 | In mathematics, specifically group theory, an abnormal subgroup is a subgroup H of a group G such that for all x in G, x lies in the subgroup generated by H and H x, where H x denotes the conjugate subgroup xHx−1. Here are some facts relating abnormality to other subgroup properties: Every abnormal subgroup is a self-n... | Abnormal subgroup |
c_knj45xova321 | In mathematics, specifically group theory, finite groups of prime power order p n {\displaystyle p^{n}} , for a fixed prime number p {\displaystyle p} and varying integer exponents n ≥ 0 {\displaystyle n\geq 0} , are briefly called finite p-groups. The p-group generation algorithm by M. F. Newman and E. A. O'Brien is a... | P-group generation algorithm |
c_ylijcph0fotj | In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orde... | P-primary group |
c_7my6wvsu972u | In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by Hall (1940) to help classify and understand p-groups, although it is applicable to all groups. Isoclinism also has consequences for the Schur multiplier and the associat... | Isoclinism of groups |
c_bj0nvtwxbep3 | 6.7). The word "isoclinism" comes from the Greek ισοκλινης meaning equal slope. Some textbooks discussing isoclinism include Berkovich (2008, §29) and Blackburn, Neumann & Venkataraman (2007, §21.2) and Suzuki (1986, pp. 92–95). | Isoclinism of groups |
c_u6cus6q8hqpx | In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. 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 homo... | Free product |
c_ssif5zcqey8h | Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial group. Therefore, the free product is not the coproduct in the category of abelian groups. The free product is important in algebraic topology because of van Kampen's theorem, which states that the fundamental gro... | Free product |
c_6n9lws5m1n0w | In particular, the fundamental group of the wedge sum of two spaces (i.e. the space obtained by joining two spaces together at a single point) is, under certain conditions given in the Seifert van-Kampen theorem, the free product of the fundamental groups of the spaces. Free products are also important in Bass–Serre th... | Free product |
c_2zuuuk7g4p8q | In mathematics, specifically group theory, the identity component of a group G refers to several closely related notions of the largest connected subgroup of G containing the identity element. In point set topology, the identity component of a topological group G is the connected component G0 of G that contains the ide... | Component group |
c_r3uaajuplegx | In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted | G: H | {\displaystyle |G:H|} or {\displaystyle } or ( G: H ) {\displaystyle (G:H)} . Because G is the disjoint union of... | Finite index |
c_bgdvmflh3uhm | More generally, | Z: n Z | = n {\displaystyle |\mathbb {Z} :n\mathbb {Z} |=n} for any positive integer n. When G is finite, the formula may be written as | G: H | = | G | / | H | {\displaystyle |G:H|=|G|/|H|} , and it implies Lagrange's theorem that | H | {\displaystyle |H|} divides | G | {\displaystyle |G|} . When G i... | Finite index |
c_e6tfpli4qtwo | In mathematics, specifically homotopical algebra, an H-object is a categorical generalization of an H-space, which can be defined in any category C {\displaystyle {\mathcal {C}}} with a product × {\displaystyle \times } and an initial object ∗ {\displaystyle *} . These are useful constructions because they help export ... | H-object |
c_2g8j64ft1u0s | In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements, a concept which is the same in UFD... | Prime element |
c_cbw5v4h0kqnt | In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order. While finitely generated abelian groups are completel... | Torsion-free abelian group |
c_3i7mpg4fwzxg | In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. | Power associative |
c_qg1eo7iu0vtx | In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete homogeneous symmetric polynomials. | Complete homogeneous symmetric polynomial |
c_if2sn8dw1ej4 | In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hy... | Weak Lefschetz theorem |
c_2s5ovsf3348m | In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this reason, formal schemes frequently appear in topics ... | Formal algebraic geometry |
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