Homomorphism & Isomorphism of Group

We can say that  "o" is the binary operation on set G if: G is a non-empty set & G * G = { (a,b) : a , b∈ G } and o : G * G --> G. Here, aob denotes the image of ordered pair (a,b) under the function/operation o.

Example - "+" is called a binary operation on G (any non-empty set ) if & only if: a+b ∈G; ∀ a,b ∈G and a+b give the same result every time when added.

Real example - '+' is a binary operation on the set of natural numbers 'N' because a+b ∈ N; ∀ a,b ∈N and a+b  a+b give the same result every time when added. 

Laws of Binary Operation :

In a binary operation o, such that: o : G * G --> G on the set G is :

1. Commutative -

 aob = boa ; ∀ a,b ∈G

Example:  '+' is a binary operation on the set of natural numbers 'N'. Taking any 2 random natural numbers , say 6 & 70, so here a = 6 & b = 70, 
a+b = 6 + 70 = 76 = 70 + 6 = b + a
This is true for all the numbers that come under the natural number.

2. Associative -

ao(boc) = (aob)oc ; ∀ a,b,c ∈G

Example:  '+' is a binary operation on the set of natural numbers 'N'. Taking any 3 random natural numbers , say 2 , 3 & 7, so here a = 2 & b = 3 and c = 7,
LHS : a+(b+c) = 2 +( 3 +7)  = 2 + 10 = 12
RHS : (a+b)+c = (2 + 3) + 7 = 5 + 7 = 12
This is true for all the numbers that come under the natural number.

3. Left Distributive - 

ao(b*c) = (aob) * (aoc) ; ∀ a,b,c ∈G

4. Right Distributive -

 (b*c) oa = (boa) * (coa)  ; ∀ a,b,c ∈G

5. Left Cancellation -

 aob =aoc  => b = c  ; ∀ a,b,c ∈G

6. Right Cancellation -

 boa = coa  => b = c ; ∀ a,b,c ∈G

Algebraic Structure :

A non-empty set G equipped with 1/more binary operations is called an algebraic structure. 
Example : a. (N,+)  and b. (R, + , .), where N is a set of natural numbers & R is a set of real numbers. Here ' . ' (dot) specifies a multiplication operation. 

GROUP : 

An algebraic structure (G , o) where G is a non-empty set & 'o' is a binary operation defined on G is called a Group if the binary operation "o" satisfies the following properties -

1. Closure - 

a ∈ G ,b ∈ G  => aob ∈ G ;  ∀ a,b ∈ G

2. Associativity -

 (aob)oc = ao(boc) ; ∀ a,b,c ∈ G.

3. Identity Element - 
There exists e in G such that aoe = eoa = a ; ∀ a ∈ G (Example - For addition, identity is 0)

4. Existence of Inverse - 
For each element a ∈ G ; there exists an inverse(a^-1)such that :  ∈ G such that - aoa^-1 = a^-1oa = e

Homomorphism of groups :
Let (G,o) & (G',o') be 2 groups, a mapping "f " from a group (G,o) to a group (G',o') is said to be a homomorphism if -

f(aob) = f(a) o' f(b) ∀ a,b ∈ G

The essential point here is : The mapping f : G --> G' may neither be a one-one nor onto mapping, i.e, 'f' needs not to be bijective.

Example -
If (R,+) is a group of all real numbers under the operation '+' & (R -{0},*) is another group of non-zero real numbers under the operation '*' (Multiplication) &  f is a mapping from (R,+) to (R -{0},*),  defined as : f(a) = 2^a ;  ∀ a ∈ R
Then f is a homomorphism like -  f(a+b) = 2^a+b = 2^a * 2^b = f(a).f(b) . 
So the rule of homomorphism is satisfied & hence f is a homomorphism.

Homomorphism Into - 
A mapping 'f', that is homomorphism & also Into.

Homomorphism Onto - 
A mapping 'f', that is homomorphism & also onto.

Isomorphism of Group :
Let (G,o) & (G',o') be 2 groups, a mapping "f " from a group (G,o) to a group (G',o') is said to be an isomorphism if -

1. f(aob) = f(a) o' f(b) ∀ a,b ∈ G
2. f is a one- one mapping
3. f is an onto mapping.

If 'f' is an isomorphic mapping, (G,o) will be isomorphic to the group (G',o') & we write :

G ≅ G'

Note : A mapping f: X -> Y is called :

1. One - One  - If x₁ ≠x2, then f(x₁) ≠ f(x₂) or if f(x₁) = f(x₂) => x₁ = x_2. Where x₁,x₂ ∈ X
2. Onto           - If every element in the set Y is the f-image of at least one element of set X. 
3. Bijective      - If it is one & Onto.

Example of Isomorphism Group -
If G is the multiplicative group of 3 cube-root units , i.e., (G,o) = ( {1, w, w² } , *) where w³ = 1 &  G' is an additive group of integers modulo 3 -  (G', o') = ( {1,2,3) , +₃). Then : G ≅ G' , we say G is isomorphic to G'.

• The structure & order of both  the tables are same. The mapping 'f' is defined as :
  f : G -> G' in such a way that f(1) = 0 , f(w) = 1 & f(w²) = 2.
• Homomorphism property : f(aob) = f(a) o' f(b) ∀ a,b ∈ G . Let us take a = w & b = 1
  LHS : f(a * b) = f( w * 1 ) = f(w) = 1.
  RHS : f(a) +₃ f(b) = f(w) +₃ f(1) = 1 + 0 = 1
  =>LHS = RHS
• This mapping f is one-one & onto also, therefore, a homomorphism.

Solved Examples

Example 1: Binary Operation Addition on Natural Numbers

Operation: Define o as ++ on N.

Verification:

• Commutative: For any a,b∈N, a+b=b+a.
• Associative: For any a,b,c∈N, a+(b+c)=(a+b)+c.

Example 2: Binary Operation Multiplication on Real Numbers

Operation: Define o as × on R.

Verification:

• Commutative: For any a,b∈R, a×b=b×a.
• Associative: For any a,b,c∈R, a×(b×c)=(a×b)×c.

Example 3: Binary Operation Subtraction on Integers

Operation: Define o as − on Z.

Verification:

• Not Commutative: For any a,b∈Z, a−b≠b−a.
• Not Associative: For any a,b,c∈Z, a−(b−c)≠(a−b)−c.

Example 4: Binary Operation on Matrix Addition

Operation: Define o as matrix addition on M (set of matrices of same dimension).

Verification:

• Commutative: For any A,B∈M A+B=B+A.
• Associative: For any A,B,C∈M, A+(B+C)=(A+B)+C.

Practice Problems on Homomorphism & Isomorphism of Group

1. Prove that the operation oo defined as addition on the set of even integers is a binary operation.
2. Show that multiplication is a binary operation on the set of non-zero rational numbers.
3. Determine if the operation defined as subtraction on the set of natural numbers is commutative.
4. Verify if addition is an associative operation on the set of integers.
5. Prove that the operation defined as multiplication on the set of complex numbers is commutative and associative.
6. Determine if the operation defined as division on the set of positive real numbers is a binary operation.
7. Show that addition is a binary operation on the set of polynomials with real coefficients.
8. Verify if the operation defined as exponentiation on the set of natural numbers is associative.
9. Prove that the operation defined as maximum (max) on the set of integers is associative.
10. Show that the operation defined as bitwise AND on the set of integers is commutative and associative.

Related Articles:

• Homomorphism & Isomorphism of Group
• Group Isomorphisms and Automorphisms