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:
- For all a, b in A, the result of the operation a • b is also in A.
- 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.
- 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 , where means the set of all integers. This is an abelian group because is a group, and also for any integers and , the equation is true. For example, , because both sides are equal to .