What is Group Theory with an Example

Today in this article i want to demonstrate the definition of Group theory what are the properties that satisfy if the set of elements are in a Group .The article will provide the sample example for Group theory. It is an introduction to GROUP THEORY and with that i will demonstrate what is an abelian Group or Commutative group

Definition of Group :

Let (a,b,c belongs to z ) if a and b are two sets to be in Group (Z,+) Under Addition Operation than it follows some Axioms

1)Closure a+b belongs to Z

2)Associative a+(b+c) = (a+b) + c

3) Identity a+0 = 0 + a

4)Inverse a+(-a) = 0 -a is the additive inverse of a

So let we take two sets A and B for an example

A={1,2} B={3,4}

Closure

We know that 1+2 =3 is again an integer so the first axiom Closure property Satisfy

Associative

a+(b+c) = (a+b) + c

1+(2+3) =(1+2)+3

by looking above we can say the Integers will follow Associative rule hence this axiom is Passed

a+0 =0+a =a there exists 0 where 0 is an Integer and it is an Identity so this Identity rule is Passed.

a+ (-a) = 0 = (-a) + a The additive inverse of a is equal to the identity of an Integer Hence this follows the additive inverse Property

Hence it is called a Group.

If a,b are in Abelian group then it must follow the commutative axiom then a+b = b+a than it is called a Commutative group or an Abelian group.

Similar to here as above (z,*) set of integers also an abelian group or commutative group.