Group Identity Theory

Introduction about group theory

Group:            

    If a set is defined collectively with the binary o[operation  * then it is called a group. It is defined as the symbol G.If  x and y are the member of a set G and it related to  the binary operation * means then the g is said to be a Group. Here the binary operation should satisfy the certain Axioms.

Axioms of Group theory:

Closure
Associativity of an element
Identity of an element
Inverse of an element.

Properties of group theory


Closure property in group theory:

    If any two elements x ,y  in the group G, then the outcome of the product operation   (x.y) is also present in the group G.

Associativity property in Group theory:

        If any set of elements present in the group G then it satisfies the following condition

                      (x .y).z= x. (y .z).

 Inverse property in group theory:

   For each x is present in the group G , then there exist an element y is also present in  the group  G such that       x . y = y .x =e. here e is defined as an identity element. The outcome of combining element x with y   are not necessitate to acquiesce the same outcome as combining element y with x.

               x. y = y. as It is not always be true.

Identity of an element in the group theory:

If an element x is present in the group G and there exist an another element e in the group G then it satisfies the following,

                 e. x= x . e =x.Here we say that e is an identity element.

Identity of an element is the important axiom of group theory.


Applications of group identity theory:


It is used in the Galois Theory for describe the polynomial roots of symmetries.
The connection between the algebraic field extension and the group theory derived fro m the Galois fundamental theorem.
Another use of group theory is the Algebraic topology. Some of the topological invariants are defined by group.
Group theory is primarily used in the Cryptography and the Algebraic geometry.