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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 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:
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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. -
The structure  is a 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 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. |
