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Abelian Group Example

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  • Difficulty Level : Expert
  • 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

∀ a , b ∈ I ⇒ a + b ∈ I

2,-3 ∈ I ⇒ -1 ∈ I

Hence Closure Property is satisfied.

2) Associative Property

( a+ b ) + c = a+( b +c) ∀ a , b , c ∈ I

2 ∈ I, -6 ∈ I , 8 ∈ 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 ∀ a ∈ I , 0 ∈ I

5 ∈ I

5+0 = 5
-17 ∈ I
-17 + 0 = – 17

Identity property is also satisfied.

4) Inverse Property

a + ( -a ) = 0 ∀ a ∈ I , -a ∈ I ,0 ∈ I

a=18 ∈ I then ∋ a number -18 such that 18 + ( -18 ) = 0

So, Inverse property is also satisfied.

5) Commutative Property

a + b = b + a ∀ a , b ∈ 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.

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