Definition of Group

An algebraic structure \left( {G, * } \right) where G 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: G is closed under the operation  * , i.e., a * b \in G, for all a,b \in G.

(G2) Associative Axiom: The binary operation  * is associative, i.e., \left( {a * b} \right) * c = a * \left( {b * c} \right)   \forall a,b \in G.

(G3) Identity Axiom: There exists an element e \in G such that e * a = a * e = a   \forall a \in G. The element e is called the identity of “ * ” in G.

(G4) Inverse Axiom: Each element of G possesses an inverse, i.e., for each element a \in G, there exists an element b \in G such that b * a = a * b = e
The element b is then called the inverse of a with respect to “ * ” and we write b = {a^{ - 1}}. Thus {a^{ - 1}} is an element of G such that {a^{ - 1}} * a = a * {a^{ - 1}} = e

 

Examples:

  1. The structures \left( {\mathbb{N}, + } \right) and \left( {\mathbb{N}, \times } \right) 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.
  2. 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 the additive group of integers.
    The structure \left( {\mathbb{Z}, \times } \right), i.e., the set of integers with the multiplication composition, does not form a group, as the axiom (G4) is not satisfied.
  3. 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 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 \left( {{\mathbb{Q}_o}, \times } \right) is a 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