Abelian Group or Commutative Group

If the commutative law holds in a group, then such a group is called an Abelian group or commutative group. Thus the group $$\left( {G, * } \right)$$ is said to be an Abelian group or commutative group if $$a * b = b * a,\forall a,b \in G$$.

A group which is not Abelian is called a non-Abelian group. The group $$\left( {G, + } \right)$$ is called the group under addition while the group $$\left( {G, \times } \right)$$ is known as the group under multiplication.

Examples:

1. • The structure $$\left( {\mathbb{Z}, + } \right)$$ 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 $$0$$ belongs to $$\mathbb{Z}$$, and the inverse of every element $$a$$, viz. $$ – a$$ belongs to $$\mathbb{Z}$$. This is known as additive Abelian group of integers.
   • The structures $$\left( {\mathbb{Z}, + } \right),\left( {\mathbb{R}, + } \right),\left( {\mathbb{C}, + } \right)$$ 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 $$\left( {{\mathbb{Q}_o}, \times } \right)$$ is an Abelian group, where $${\mathbb{Q}_o}$$ is the set of non-zero rational numbers. This is so because the operation is associative. The multiplicative identity $$1$$ belongs to $${\mathbb{Q}_o}$$, and the multiplicative inverse of every element $$a$$ in the set is $$1/a$$, which also belongs to $${\mathbb{Q}_o}$$. This is known as the multiplicative Abelian group of non-zero rational.
     Obviously $$\left( {{\mathbb{R}_o}, \times } \right)$$ and $$\left( {{\mathbb{C}_o}, \times } \right)$$ are groups where $${\mathbb{R}_o}$$ and $${\mathbb{C}_o}$$ are respectively the sets of non-zero real numbers and non-zero complex numbers.