group

Also known as group (algebra), group (math), mathematical group, algebraic group

algebraic set with an invertible, associative internal operation admitting a neutral element

AI overview

A group is a mathematical structure where you have a collection of elements and a way to combine any two of them that always produces another element in the collection, with the combination process being reversible and orderly. It matters because groups appear throughout mathematics and science as a fundamental way to describe symmetries, transformations, and other patterns that follow consistent rules.

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Commons category: Group theory

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Article

The manipulations of the Rubik's Cube form the Rubik's Cube group. 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.

The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. 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.