Definition of Group
An algebraic structure , where is a nonempty set with a binary operation “” defined on it is said to be a group; if the binary operation satisfies the following axioms (called group axioms).
(G1) Closure Axiom. is closed under the operation , i.e., , for all .
(G2) Associative Axiom. The binary operation is associative. i.e., .
(G3) Identity Axiom. There exists an element such that . The element is called the identity of “” in .
(G4) Inverse Axiom. Each element of possesses inverse, i.e., for each element , there exists an element such that
The element is then called the inverse of a with respect to “” and we write . Thus is an element of such that
Examples:

The structures and are not groups i.e., the set of natural numbers considered with the addition composition or the multiplication composition, does not form a group. For, the postulate (G3) and (G4) in the former case, and (G4) in the latter case, are not satisfied.

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 group of integers.
The structure , i.e., the set of integers with the multiplication composition does not form a group, as the axiom (G4) is not satisfied. 
The structures are all groups, i.e., the sets of rational numbers, real numbers, complex numbers, each with the additive composition, form a 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.