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Abelian group

Index Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. [1]

128 relations: Abelian category, Abstract algebra, Academic Press, Addition, Additive identity, Additive inverse, Adjective, Algebraic structure, Algebraically compact module, American Mathematical Monthly, Arithmetic, Automorphism, Basic subgroup, Binary operation, Boolean algebra (structure), Camille Jordan, Cardinal number, Category of abelian groups, Cayley table, Center (group theory), Characteristic subgroup, Class field theory, Cokernel, Commutative property, Commutative ring, Commutator subgroup, Computational group theory, Constructible universe, Continuum hypothesis, Coprime integers, Cotorsion group, Cyclic group, David Arnold (mathematician), Decidability (logic), Dihedral group of order 6, Dimension, Direct product, Direct sum, Direct sum of groups, Divisible group, Divisor, Dover Publications, Element (mathematics), Elementary abelian group, Ferdinand Georg Frobenius, Finite group, Finitely generated abelian group, Free abelian group, Fundamenta Mathematicae, Galois theory, ..., General linear group, Generalization, Group (mathematics), Group homomorphism, Group isomorphism, Height (abelian group), Helmut Ulm, Injective module, Integer, Integer matrix, Irving Kaplansky, John Wiley & Sons, Kernel (linear algebra), László Fuchs, Lecture Notes in Mathematics, Leopold Kronecker, Linear algebra, Linear combination, Linear independence, List of small groups, List of statements independent of ZFC, Ludwig Stickelberger, Mathematician, Matrix (mathematics), Modular arithmetic, Module (mathematics), Multiplication table, Multiplicative group, Natural number, Near-ring, Niels Henrik Abel, Noetherian ring, Non-abelian group, Normal subgroup, Norway, Operation (mathematics), Order (group theory), P-adic number, Partially ordered group, Phillip Griffith, Polynomial, Pontryagin duality, Prüfer group, Prüfer theorems, Prime number, Principal ideal domain, Proper noun, Pure subgroup, Quotient group, Rank of an abelian group, Rational number, Real number, Ring (mathematics), Rotation matrix, Saharon Shelah, Set (mathematics), Set theory, Simple group, Slender group, Smith normal form, Statements true in L, Structure theorem for finitely generated modules over a principal ideal domain, Subgroup, Sylow theorems, Tensor product, Topological group, Topology, Torsion (algebra), Torsion group, Torsion subgroup, Torsion-free abelian group, Torsion-free abelian groups of rank 1, Unimodular matrix, Unit (ring theory), University of Chicago Press, Vector space, Whitehead problem, Zermelo–Fraenkel set theory. Expand index (78 more) »

Abelian category

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.

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Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

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Academic Press

Academic Press is an academic book publisher.

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Addition

Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

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Additive identity

In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

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Additive inverse

In mathematics, the additive inverse of a number is the number that, when added to, yields zero.

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Adjective

In linguistics, an adjective (abbreviated) is a describing word, the main syntactic role of which is to qualify a noun or noun phrase, giving more information about the object signified.

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Algebraic structure

In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.

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Algebraically compact module

In mathematics, especially in the area of abstract algebra known as module theory and in model theory, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means.

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American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

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Arithmetic

Arithmetic (from the Greek ἀριθμός arithmos, "number") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.

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Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

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Basic subgroup

In abstract algebra, a basic subgroup is a subgroup of an abelian group which is a direct sum of cyclic subgroups and satisfies further technical conditions.

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Binary operation

In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set.

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Boolean algebra (structure)

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

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Camille Jordan

Marie Ennemond Camille Jordan (5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse.

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Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.

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Category of abelian groups

In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms.

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Cayley table

A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table.

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Center (group theory)

In abstract algebra, the center of a group,, is the set of elements that commute with every element of.

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Characteristic subgroup

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group.

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Class field theory

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of local fields (one-dimensional local fields) and "global fields" (one-dimensional global fields) such as number fields and function fields of curves over finite fields in terms of abelian topological groups associated to the fields.

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Cokernel

In mathematics, the cokernel of a linear mapping of vector spaces f: X → Y is the quotient space Y/im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f. Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain).

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Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

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Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

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Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

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Computational group theory

In mathematics, computational group theory is the study of groups by means of computers.

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Constructible universe

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted L, is a particular class of sets that can be described entirely in terms of simpler sets.

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Continuum hypothesis

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets.

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Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

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Cotorsion group

In abelian group theory, an abelian group is said to be cotorsion if every extension of it by a torsion-free group splits.

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Cyclic group

In algebra, a cyclic group or monogenous group is a group that is generated by a single element.

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David Arnold (mathematician)

David M. Arnold is an American mathematician, currently the Ralph and Jean Storm Professor of Mathematics at Baylor University, and also a published author of 10 books, currently held in 1,886 libraries.

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Decidability (logic)

In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas, or, more precisely, an algorithm that can and will return a boolean true or false value that is correct (instead of looping indefinitely, crashing, returning "don't know" or returning a wrong answer).

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Dihedral group of order 6

In mathematics, the smallest non-abelian group has 6 elements.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one.

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Direct sum

The direct sum is an operation from abstract algebra, a branch of mathematics.

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Direct sum of groups

In mathematics, a group G is called the direct sum of two subgroups H1 and H2 if.

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Divisible group

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.

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Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.

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Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.

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Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.

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Elementary abelian group

In group theory, an elementary abelian group (or elementary abelian p-group) is an abelian group in which every nontrivial element has order p. The number p must be prime, and the elementary abelian groups are a particular kind of ''p''-group.

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Ferdinand Georg Frobenius

Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.

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Finite group

In abstract algebra, a finite group is a mathematical group with a finite number of elements.

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Finitely generated abelian group

In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,..., xs in G such that every x in G can be written in the form with integers n1,..., ns.

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Free abelian group

In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis.

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Fundamenta Mathematicae

Fundamenta Mathematicae is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems.

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Galois theory

In the field of algebra within mathematics, Galois theory, provides a connection between field theory and group theory.

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General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

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Generalization

A generalization (or generalisation) is the formulation of general concepts from specific instances by abstracting common properties.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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Group homomorphism

In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".

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Group isomorphism

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.

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Height (abelian group)

In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such that the equation Nx.

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Helmut Ulm

Helmut Ulm (born June 21, 1908 in Gelsenkirchen; died June 13, 1975) was a German mathematician who established the classification of countable periodic abelian groups by means of their Ulm invariants.

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Injective module

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers.

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Integer

An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

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Integer matrix

In mathematics, an integer matrix is a matrix whose entries are all integers.

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Irving Kaplansky

Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and musician.

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John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

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Kernel (linear algebra)

In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map between two vector spaces V and W, is the set of all elements v of V for which, where 0 denotes the zero vector in W. That is, in set-builder notation,.

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László Fuchs

László Fuchs (born June 24, 1924 in Budapest) is a Hungarian-American mathematician, the Evelyn and John G. Phillips Distinguished Professor Emeritus in Mathematics at Tulane University.

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Lecture Notes in Mathematics

Lecture Notes in Mathematics (LNM) is a book series in the field of mathematics, including articles related to both research and teaching.

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Leopold Kronecker

Leopold Kronecker (7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic.

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Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

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Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

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Linear independence

In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.

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List of small groups

The following list in mathematics contains the finite groups of small order up to group isomorphism.

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List of statements independent of ZFC

The mathematical statements discussed below are independent of ZFC (the Zermelo–Fraenkel axioms plus the axiom of choice, the canonical axiomatic set theory of contemporary mathematics), assuming that ZFC is consistent.

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Ludwig Stickelberger

Ludwig Stickelberger (May 18, 1850 – April 11, 1936) was a Swiss mathematician who made important contributions to linear algebra (theory of elementary divisors) and algebraic number theory (Stickelberger relation in the theory of cyclotomic fields).

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Mathematician

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

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Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

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Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

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Module (mathematics)

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.

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Multiplication table

In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.

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Multiplicative group

In mathematics and group theory, the term multiplicative group refers to one of the following concepts.

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Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

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Near-ring

In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms.

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Niels Henrik Abel

Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields.

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Noetherian ring

In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any chain of left (or right) ideals: there exists an n such that: Noetherian rings are named after Emmy Noether.

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Non-abelian group

In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a.

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Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.

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Norway

Norway (Norwegian: (Bokmål) or (Nynorsk); Norga), officially the Kingdom of Norway, is a unitary sovereign state whose territory comprises the western portion of the Scandinavian Peninsula plus the remote island of Jan Mayen and the archipelago of Svalbard.

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Operation (mathematics)

In mathematics, an operation is a calculation from zero or more input values (called operands) to an output value.

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Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

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P-adic number

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

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Partially ordered group

In abstract algebra, a partially ordered group is a group (G,+) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b.

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Phillip Griffith

Phillip Alan Griffith (born December 29, 1940) is a mathematician and professor emeritus at University of Illinois at Urbana-Champaign who works on commutative algebra and ring theory.

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Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

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Pontryagin duality

In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as \R, the circle, or finite cyclic groups.

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Prüfer group

In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique ''p''-group in which every element has p different p-th roots.

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Prüfer theorems

In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups.

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Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

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Principal ideal domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element.

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Proper noun

A proper noun is a noun that in its primary application refers to a unique entity, such as London, Jupiter, Sarah, or Microsoft, as distinguished from a common noun, which usually refers to a class of entities (city, planet, person, corporation), or non-unique instances of a specific class (a city, another planet, these persons, our corporation).

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Pure subgroup

In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand.

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Quotient group

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

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Rank of an abelian group

In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset.

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Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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Rotation matrix

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space.

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Saharon Shelah

Saharon Shelah (שהרן שלח) is an Israeli mathematician.

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Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

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Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

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Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

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Slender group

In mathematics, a slender group is a torsion-free abelian group that is "small" in a sense that is made precise in the definition below.

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Smith normal form

In mathematics, the Smith normal form is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID).

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Statements true in L

Here is a list of propositions that hold in the constructible universe (denoted L).

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Structure theorem for finitely generated modules over a principal ideal domain

In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain can be uniquely decomposed in much the same way that integers have a prime factorization.

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Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

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Sylow theorems

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains.

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Tensor product

In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.

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Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.

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Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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Torsion (algebra)

In abstract algebra, the term torsion refers to elements of finite order in groups and to elements of modules annihilated by regular elements of a ring.

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Torsion group

In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which each element has finite order.

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Torsion subgroup

In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order (the torsion elements of A).

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Torsion-free abelian group

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.

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Torsion-free abelian groups of rank 1

Infinitely generated abelian groups have very complex structure and are far less well understood than finitely generated abelian groups.

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Unimodular matrix

In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1.

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Unit (ring theory)

In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.

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University of Chicago Press

The University of Chicago Press is the largest and one of the oldest university presses in the United States.

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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Whitehead problem

In group theory, a branch of abstract algebra, the Whitehead problem is the following question: Shelah (1974) proved that Whitehead's problem is undecidable within standard ZFC set theory.

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Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.

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Abelian Group, Abelian groups, Abelian subgroup, Additive notation, Classification of finite Abelian groups, Commutative Group, Commutative group, Finite abelian group, Fundamental theorem of finite abelian groups.

References

[1] https://en.wikipedia.org/wiki/Abelian_group

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