Question

. What is abelian group ? Show that (z6, +) is an abelian group ?

Answer 1

A group which holds commutative property with respect to binary operation is called abelian group.

Answer 2

Note :

• Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

• Moreover , if a group (G,*) also holds commutative property , then it is called commutative group or abelian group .

Solution :

Given :

Z₆ = { 0 , 1 , 2 , 3 , 4 , 5 }

To prove :

Z₆ is an abelian group .

Proof :

For Cayley's table (composition table) please refer to the attachment .

1) Closure property :

All the elements of the composition table are the elements of Z₆ . ie. a +₆ b ∈ Z₆ ∀ a , b ∈ Z₆ .

2) Associative property :

We know that , the addition of integers is associative , thus a +₆ (b +₆ c) = (a +₆ b) +₆ c ∀ a , b , c ∈ Z₆ .

3) Existence of identity :

We have 0 ∈ Z₆ such that 0 +₆ a = a +₆ 0 = a ∀ a ∈ Z₆ .

Thus , 0 is the identity element in Z₆ .

4) Existence of inverse element :

∀ a ∈ Z₆ , there exists a⁻¹ ∈ Z₆ such that a +₆ a⁻¹ = a⁻¹ +₆ a = 0 where a⁻¹ is called the inverse of a .

Here ,

0⁻¹ = 0

1⁻¹ = 5

2⁻¹ = 4

3⁻¹ = 3

4⁻¹ = 2

5⁻¹ = 1

5) Commutative property :

The Cayley's table is symmetrical about the principal diagonal , thus Z₆ is commutative , ie. a +₆ b = a +₆ b ∀ a , b ∈ Z₆ .

Hence , Z₆ is an abelian group .