group in which the operation is commutative.

An abelian group is a group whose operation is commutative, ie *=*. An example of an abelian group is the integers with the usual addition operation. An example of a group which is not abelian is the rotations of a cube (try it out). When dealing with an abelian group is it conventional to denote the group operation as addition (+), rather than multiplication (*) and to denote the unit element as 0 rather than 1.

a group that satisfies the commutative law

a group whose group operation is commutative

a set together with an operation on the elements of the set, where the operation has all the properties of addition of the integers

In mathematics, an abelian group, also called a commutative group, is a group (G, * ) such that a * b = b * a for all a and b in G. In other words, the order in which the binary operation is performed doesn't matter. Such groups are generally easier to understand, although infinite abelian groups remain a subject of current research.