Abelian Group: Group whose group operation 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’.

An abelian group is a set, A, together with an operation "•". It combines any two elements a and b to form another element denoted a • b. 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 a • b is also in A.
Associativity
For all a, b and c in A, the equation (a • b) • c = a • (b • c) is true.
Identity element
There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = a holds.
Inverse element
For each a in A, there exists an element b in A such that a • b = b • a = e, where e is the identity element.
Commutativity
For all a, b in A, a • b = b • a.

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 ${\displaystyle 9}$ .

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