Subject :- Mathematics

Content Writer : - Mr. Deepak Gupta

Class & Year :- B.Sc. II Year

Topic :- Group

Key Words :- Binary Operation, Algebraic Structure, Group

Binary Operation on a Set

Let  be any Non-empty set, then a function  is called a binary operation on set G. The image of the ordered pair  under the function  is denoted by .

Thus,  will be a binary operation on set  iff  and  is unique.

Similarly,  will be a binary operation on set  iff  and  is unique.

Example: -  Since the sum to two Natural number is always a natural number, hence Addition is a binary operation on the set of Natural numbers.

Algebraic Structure

A non-empty  together with one or More Binary operation is called Algebraic Structure.

Example:-

GROUP

An algebraic structure , where  is a non empty set and  is binary operation defined on , is called a group if this binary operation satisfies the following postulates :

Closure property: If  and  belong to , then  also belong to , i.e.,

Associativity: The binary operation  is associative i.e.

.  

Existence of identity: There exists an element  in  such that . The element  is called the identity. If  then  is the right identity and when , then  is the left identity in .

Existence of inverse: For each element , there exists an element of  called the inverse of  and denoted by  such that , where  is the identity element for .

Examples of Group

1.  The set  of all Integers is an infinite group with respect to the operation of addition of integers.

2.  The set  of all non-zero rational numbers is an infinite group with respect to the operation of multiplication of rational numbers.