Homomorphism & Isomorphism of Group (original) (raw)

Last Updated : 23 Jul, 2025

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

  1. Associativity -

(aob)oc = ao(boc) ; ∀ a,b,c ∈ G.

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

  2. 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) = 2a ; ∀ a ∈ R
Then f is a homomorphism like - f(a+b) = 2a+b = 2a * 2b = 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 = x2. 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'.

Solved Examples

**Example 1: Binary Operation Addition on Natural Numbers

**Operation: Define o as ++ on N.

**Verification:

**Example 2: Binary Operation Multiplication on Real Numbers

**Operation: Define o as × on R.

**Verification:

**Example 3: Binary Operation Subtraction on Integers

**Operation: Define o as − on Z.

**Verification:

**Example 4: Binary Operation on Matrix Addition

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

**Verification:

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.