In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. In other words, the order of elements in a product doesn't matter. Such groups are generally easier to understand. Abelian groups are named after Niels Henrik Abel. Groups that are not commutative are called non-abelian (rather than non-commutative). Abelian group - NotationIncluding:

Abelian group - Notation Abelian group - Examples Abelian group - Multiplication table Abelian group - Properties Abelian group - Finite abelian groups Abelian group - List of small abelian groups Abelian group - Relation to other mathematical topics Abelian group - A note on the typography

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Integer - Algebraic properties Integer - Order-theoretic properties Integer - Integers in computing Integer - Quotations

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In mathematics, a characteristic subgroup of a group G is a subgroup H that is invariant under each automorphism of G. That is, if φ : G → G is a group automorphism (a bijective homomorphism from the group G to itself), then for every x in H we have φ(x) ∈ H: It follows that In symbols, one denotes the fact that H is ...

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Commutative operation - Mathematical meaning. In mathematics, especially abstract algebra, a binary operation on a set S is commutative if for all x and y in S. Otherwise, the operation is noncommutative. Additionally, if for a particular pair of elements x and y, then x and y are said to commute. Every element commutes with itself and, in a group, every elemen ...

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Commutative operation - Mathematical meaning Commutative operation - Neurophysiological meaning

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In category theory, a zero morphism is a special kind of "trivial" morphism. Suppose C is a category, and for any two objects X and Y in C we are given a morphism 0XY : X → Y with the following property: for any two morphism f : R → S and g : U → V we obtain a commutative diagram: Then the morphisms 0XY are c ...

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Zero morphism - Examples

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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. In other words, it is a unital semigroup. Monoid - Definition. A monoid is a magma (M,*), i.e. a set M with binary operation * : M × M → M, obeying the following axioms: Associativity: for all a, b, c in M, (a*b)*c = a*(b*c) Identity ...

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Monoid - Definition Monoid - Examples Monoid - Properties Monoid - Monoid homomorphisms Monoid - Relation to category theory

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In graph theory, certain vector spaces over the two-element field Z2 are associated with an undirected graph; this allows one to use the tools of linear algebra to study graphs. Let G be a finite simple undirected graph with edge set E. The power set of E becomes a Z2-vector space if we take the symmetric difference as addition. Every element of this vector space can be thought of as a linear combination of edges with coefficient from Z2. In yet another interpr ...

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Cycle space - The binary cycle space Cycle space - The integral cycle space Cycle space - The cycle space over a field or commutative ring

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In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories. Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories. Functor - Definition. Let C and D be ca ...

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Functor - Definition
Functor - Covariance and contravariance
Functor - Examples Functor - Properties Functor - Relation to other categorical concepts

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In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. Constructible polygon - Conditions for constructibility. Some regular polygons are easy to construct with compass and straightedge; others are not. This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not ...

Including:

Constructible polygon - Conditions for constructibility Constructible polygon - General theory Constructible polygon - Detailed results in terms of Fermat primes Constructible polygon - Compass-and-straightedge constructions Constructible polygon - Other constructions

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In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. Adjoint functors are studied in a branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians. Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some ...

Including:

Adjoint functors - Motivation
Adjoint functors - Ubiquity of adjoint functors Adjoint functors - Deep problems formulated with adjoint functors Adjoint functors - Adjoint functors as solving optimization problems Adjoint functors - The case of partial orders
Adjoint functors - Formal definitions Adjoint functors - Examples Adjoint functors - Properties
Adjoint functors - Uniqueness of adjoints Adjoint functors - Relation to universal constructions Adjoint functors - Characterization via unit and co-unit Adjoint functors - Adjoints preserve certain limits Adjoint functors - Additivity Adjoint functors - Composition Adjoint functors - Adjoint pairs extend equivalences Adjoint functors - General existence theorem

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In mathematics, an additive group may be a group written using the symbol + for its binary operation (always an abelian group) the underlying group under addition of a field, ring, vector space or other structure having addition as one of its operations a group scheme representing the underlying-additive-group functor. Other related archivesabelian group, addition, binary operation, field, functor, group, group scheme, mathematics, ring, vec

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In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. It is in different, definite senses dual both to singular homology, and to Alexander-Spanier cohomology. De Rham cohomology - Defini ...

Including:

De Rham cohomology - Definition De Rham cohomology - de Rham cohomology computed De Rham cohomology - de Rham's theorem De Rham cohomology - Sheaf-theoretic de Rham isomorphism
De Rham cohomology - Proof
De Rham cohomology - Related ideas
De Rham cohomology - Harmonic forms De Rham cohomology - Hodge decomposition

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In mathematics, the term abelian is used in many different definitions: Abelian - In group theory. An abelian group is a group in which the binary operation is commutative. The category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms. A metabelian group is a group where the commutator subgroup is contained in the center. Any group is "made abelian" by its abelianisation. Abelian - In Galois theoryIncluding:

Abelian - In group theory Abelian - In Galois theory Abelian - In real analysis Abelian - In functional analysis Abelian - In topology and number theory Abelian - In category theory

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Group mathematics - An abelian group: the integers under addition. A group that we are introduced to in elementary school is the integers under addition. For this example, let Z be the set of integers, {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}, and let the symbol "+" indicate the operation of addition. Then (Z,+) is a group (written additively). Proof: If a and b are integers then a + b is an integer. (Closure; + really is a binary operation)See also:

Group mathematics, Group mathematics - History, Group mathematics - Basic definitions, Group mathematics - Notation for groups, Group mathematics - Some elementary examples and nonexamples, Group mathematics - An abelian group: the integers under addition, Group mathematics - Not a group: the integers under multiplication, Group mathematics - An abelian group: the nonzero rational numbers under multiplication, Group mathematics - A finite nonabelian group: permutations of a set, Group mathematics - Further examples, Group mathematics - Simple theorems, Group mathematics - Constructing new groups from given ones

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The collection of all G-modules is a category (the morphisms are group homomorphisms f with the property f(gx) = g(f(x)) for all g in G and x in M). This category of G-modules is an abelian category with enough injectives (since it is isomorphic to the category of all modules over the group ring ZG). Sending each module M to the group of invariants MG yields a functor from this category to the category A ...

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Group cohomology, Group cohomology - Motivation, Group cohomology - Formal constructions, Group cohomology - Properties, Group cohomology - History and relation to other fields

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To construct the Grothendieck group of a commutative monoid M, one forms the Cartesian product M×M. The two coordinates are meant to represent a positive part and a negative part: (m, n) is meant to correspond to m − n. Addition is defined coordinate-wise: (m1, m2) + (n1, n2) = (m1 + n1, m ...

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Grothendieck group, Grothendieck group - Explicit construction, Grothendieck group - Generalization, Grothendieck group - Splitting principle, Grothendieck group - Example

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A field is a commutative ring (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse. Spelled out, this means that the following hold: Closure of F under + and *  For all a, b belonging to F, both a + b and a * b belong to F (or more formally, + and * are binary operations on F). Both + and * are associative  For all a, b, c ...

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Field mathematics, Field mathematics - Introduction, Field mathematics - Definition, Field mathematics - Examples of fields, Field mathematics - Some first theorems

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Any product of (even infinitely many) injective modules is injective. Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules or infinite direct sums of injective modules need not be injective. In Baer's original paper, he proved a useful result, usually known as Baer's Criterion, for checking whether a module is injective: a left R-module Q is injective if and only if any homomorphism g : I → Q defined on a left ideal I of See also:

Injective module, Injective module - Definition, Injective module - Examples, Injective module - Facts, Injective module - Generalization

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If S is finite, then a group G = <S> is called finitely generated. The structure of finitely generated abelian groups in particular is easily described. Many theorems that are true for finitely generated groups fail for groups in general. Every finite group is finitely generated since <G> = G. The integers under addition are an example of an infinite group which is finitely generated by both <1> and <−1>, but the group of rationals under addition cannot ...

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Generating set of a group, Generating set of a group - Finitely generated group, Generating set of a group - Free group, Generating set of a group - Frattini subgroup, Generating set of a group - Examples

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In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. Much of the theory of modules consists of extending as many as possible of the desirable properties of vector spaces to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicate ...

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Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

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