An algebraic structure where is a non-empty 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 an 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
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 the 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.