Abelian group

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 . For the group to be abelian, the operation and the elements must follow some requirements. These are known as the "abelian group axioms":
Examples.
One example of an abelian group is the set of the integers with the operation of addition. We often write this as formula_1, where formula_2 means the set of all integers. This is an abelian group because formula_1 is a group, and also for any integers formula_4 and formula_5, the equation formula_6 is true. For example, formula_7, because both sides are equal to formula_8.