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.
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