Commutative Group or Abelian Group

If the commutative law holds in a group, then such a group is called an Abelian group or Commutative group. Thus the group is said to be an Abelian group or commutative group if,      .

A group which is not Abelian is called a non-Abelian group. The group  is called the group under addition while the group  is known as group under multiplication.

Examples:

• The structure  is a group, i.e.,the set of integers with the addition composition is a group. This is so because addition in numbers is associative, the additive identity  belongs to, and the inverse of every element, viz.  belongs to. This is known as additive Abelian group of integers.
• The structures  are all groups, i.e., the sets of rational numbers, real numbers, complex numbers, each with the additive composition, form an Abelian group. But the same sets with the multiplication composition do not form a group, for the multiplicative inverse of the number zero does not exist in any of them.
• The structure  is an Abelian group, where  is the set of non-zero rational numbers. This is so because the operation is associative, the multiplicative identity  belongs to, and the multiplicative inverse of every element  in the set is, which also belongs to. This is known as the multiplicative Abelian group of non-zero rational.
  Obviously  and  are groups, where and are respectively the sets of non-zero real numbers and non-zero complex numbers.