Editing Group (mathematics) (section)

{{Short description|Set with associative invertible operation}}
{{about|basic notions of groups in mathematics|a more advanced treatment|Group theory}}
{{Use shortened footnotes|date=September 2024}}
[[Image:Rubik's cube.svg|thumb|right|The manipulations of the [[Rubik's Cube]] form the [[Rubik's Cube group]].|alt=A Rubik's cube with one side rotated]]
In [[mathematics]], a '''group''' is a [[Set (mathematics)|set]] with a [[binary operation]] that satisfies the following constraints: the operation is [[Associative property|associative]], it has an [[identity element]], and every element of the set has an [[inverse element]]. 

Many [[mathematical structure]]s are groups endowed with other properties. For example, the [[integer]]s with the [[addition|addition operation]] form an [[infinite set|infinite]] group that is [[generating set|generated by a single element]] called {{tmath|1= 1 }} (these properties fully characterize the integers).

The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, [[geometric shape]]s and [[polynomial root]]s. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.{{sfn|Herstein|1975|p=26|loc=§2}}{{sfn|Hall|1967|p=1|loc=§1.1: "The idea of a group is one which pervades the whole of mathematics both [[pure mathematics|pure]] and [[applied mathematics|applied]]."|ps=}}

In [[geometry]], groups arise naturally in the study of [[symmetries]] and [[geometric transformation]]s: The symmetries of an object form a group, called the [[symmetry group]] of the object, and the transformations of a given type form a general group.  [[Lie group]]s appear in symmetry groups in geometry, and also in the [[Standard Model]] of [[particle physics]]. The [[Poincaré group]] is a Lie group consisting of the symmetries of [[spacetime]] in [[special relativity]]. [[Point group]]s describe [[Molecular symmetry|symmetry in molecular chemistry]].

The concept of a group arose in the study of [[polynomial equation]]s, starting with [[Évariste Galois]] in the 1830s, who introduced the term ''group'' (French: {{lang|fr|groupe}}) for the symmetry group of the [[zero of a function|roots]] of an equation, now called a [[Galois group]]. After contributions from other fields such as [[number theory]] and geometry, the group notion was generalized and firmly established around 1870. Modern [[group theory]]—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as [[subgroup]]s, [[quotient group]]s and [[simple group]]s. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of [[representation theory]] (that is, through the [[group representation|representations of the group]]) and of [[computational group theory]]. A theory has been developed for [[finite group]]s, which culminated with the [[classification of finite simple groups]], completed in 2004. Since the mid-1980s, [[geometric group theory]], which studies [[finitely generated group]]s as geometric objects, has become an active area in group theory.

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{{Algebraic structures |Group}}

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