In mathematics, a group is an algebraic structure which is made by a set of elements equipped with an operation that combines any two elements to have a third element as a result. This outcome satisfies the four group axioms (closure, associativity, identity and invertibility).
One of the most familiar group is the set of integers which consist of only numbers (e.g. -2, -1, 0, 1, 2) that have different properties which have to be respected. As well as this, another popular group is the symmetry group; here, two figures in a plane are congruent if one can be changed into the other by adopting a combination of rotations, reflections and translations.
Groups can be divided into different classes; the most important ones are premutation groups, transformation groups, matrix groups, abstract groups, and groups with additional structure.
The group theory has three main historical sources: number theory, geometry and the theory of algebraic equations. Nevertheless, in 1880, it becomes a unified theory which, by growing over the years, has permitted the development of abstract algebra in the 20th century.
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