# Abelian group

group whose group operation is commutative

In group theory, an abelian group is a group with operation that is commutative. Because of that, an abelian group is sometimes called a ‘commutative group’.

A group in which the group operation is not commutative is called a ‘non-abelian group’ or ‘non-commutative group’.

## Definition

An abelian group is a set, A, together with an operation "•". It combines any two elements a and b to form another element denoted ab. For the group to be abelian, the operation and the elements (A, •) must follow some requirements. These are known as the abelian group axioms:

Closure
For all a, b in A, the result of the operation ab is also in A.
Associativity
For all a, b and c in A, the equation (ab) • c = a • (bc) is true.
Identity element
There exists an element e in A, such that for all elements a in A, the equation ea = ae = a holds.
Inverse element
For each a in A, there exists an element b in A such that ab = ba = e, where e is the identity element.
Commutativity
For all a, b in A, ab = ba.

## Examples

One example of an abelian group is the set of the integers with the operation of addition. We often write this as ${\displaystyle (\mathbb {Z} ,+)}$ , where ${\displaystyle \mathbb {Z} }$  means the set of all integers. This is an abelian group because ${\displaystyle (\mathbb {Z} ,+)}$  is a group, and also for any integers ${\displaystyle a}$  and ${\displaystyle b}$ , the equation ${\displaystyle a+b=b+a}$  is true. For example, ${\displaystyle 3+6=6+3}$ , because both sides are equal to ${\displaystyle 9}$ .