. What is abelian group ? Show that (z6, +) is an abelian group ?
Answers
A group which holds commutative property with respect to binary operation is called abelian group.
Note :
- Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :
- G is closed under *
- G is associative under *
- G has a unique identity element
- 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 .