Abelian group

In the mathematical field of abstract algebra, an abelian group is a type of group in which the group operation is commutative – abelian groups are also known as commutative groups. Abelian groups are named for the mathematician Niels Henrik Abel; even so, most modern authors leave the term "abelian" uncapitalized.

Many common number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers are abelian groups with the group operation being addition. In contrast, symmetry groups and permutation groups, which describe the symmetry of a figure and the ways to rearrange the elements in a list respectively, are often non-commutative. Symmetry groups and permutation groups consist of maps; a group consisting of maps is commutative if and only if the equality $\scriptstyle f\circ g=g\circ f$ (it means $f(g(x))=g(f(x))$ for all x) holds for all maps f, g in the group.

Computations with an abelian group operation are usually easier than computations with non-abelian operations because one is allowed to rearrange the group elements in the computation and collect "like terms". The structure of abelian groups is generally easier both to obtain and to describe than that of non-abelian groups. When an abelian group is also finitely generated, the possible group structure is exceptionally simple, being described by the fundamental theorem of finitely generated abelian groups.