# Abelian Group Example

Last Updated : 15 Mar, 2021

Problem-: Prove that ( I, + ) is an abelian group. i.e. The set of all integers I form an abelian group with respect to binary operation ‘+’.
Solution-:

Set= I ={ ……………..-3, -2 , -1 , 0, 1, 2 , 3……………… }.

Binary Operation= ‘+’

Algebraic Structure= (I ,+)

We have to prove that (I,+) is an abelian group.

To prove that set of integers I is an abelian group we must satisfy the following five properties that is Closure Property, Associative Property, Identity Property, Inverse Property, and Commutative Property.
1) Closure Property

âˆ€ a , b âˆˆ I â‡’ a + b âˆˆ I

2,-3 âˆˆ I â‡’ -1 âˆˆ I

Hence Closure Property is satisfied.

2) Associative Property

( a+ b ) + c = a+( b +c) âˆ€ a , b , c âˆˆ I

2 âˆˆ I, -6 âˆˆ I , 8 âˆˆ I

So, LHS= ( a + b )+c

= (2+ ( -6 ) ) + 8 = 4

RHS= a + ( b + c )

=2 + ( – 6 + 8 ) = 4

Hence RHS = LHS

Associative Property is also Satisfied

3) Identity Property

a + 0 = a âˆ€ a âˆˆ I , 0 âˆˆ I

5 âˆˆ I

5+0 = 5
-17 âˆˆ I
-17 + 0 = – 17

Identity property is also satisfied.

4) Inverse Property

a + ( -a ) = 0 âˆ€ a âˆˆ I , -a âˆˆ I ,0 âˆˆ I

a=18 âˆˆ I then âˆ‹ a number -18 such that 18 + ( -18 ) = 0

So, Inverse property is also satisfied.

5) Commutative Property

a + b = b + a âˆ€ a , b âˆˆ I

Let a=19, b=20
LHS = a + b
= 19+( -20 ) = -1

RHS = b + a
= -20 +19 = -1
LSH=RHS

Commutative Property is also satisfied.

We can see that all five property is satisfied. Hence (I,+) is an Abelian Group.

Note-: (I,+) is also Groupoid, Monoid and Semigroup.