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Group (mathematics)

In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition operation form a group.

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Overview

Definition and illustration

History

Elementary consequences of the group axioms

Basic concepts

Examples and applications

Finite groups

Groups with additional structure

Generalizations

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Map overview Semantic statistics

Group (mathematics)

Nodes310
Edges309
Triples33
Avg. degree1.99
Density0.006452
Components1

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Important terminology Word statistics

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Important terminology

group displaystyle groups element elements theory symmetry example operation mathrm identity cdot symmetries addition axioms inverse two integers multiplication mathbb

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
numbersinstance ofmany mathematical structures0.80text
geometric shapesinstance ofmany mathematical structures0.80text
polynomial rootsinstance ofmany mathematical structures0.80text
number theoryinstance ofAfter contributions from other fields0.80text
geometryinstance ofAfter contributions from other fields0.80text
the group notion was generalizedinstance ofAfter contributions from other fields0.80text
firmly established around 1870instance ofAfter contributions from other fields0.80text
hyperbolicinstance ofAfter novel geometries0.80text
projective geometry had emergedinstance ofAfter novel geometries0.80text
Klein used group theory to organize them in a more coherent wayinstance ofAfter novel geometries0.80text
Daniel Gorensteininstance of61 Group Theory Year brought together group theorists0.80text
John Ginstance of61 Group Theory Year brought together group theorists0.80text

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