# Homomorphism & Isomorphism of Group

**Introduction :**

We can say that “o” is the binary operation on set G if : G is an 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^{-1}oa = 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 ∈ G2. f is a one- one mapping3. 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 :

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

**Example** of Isomorphism Group –

If G is the multiplicative group of 3 cube-root units , i.e., (G,o) = ( {1, w, w^{2} } , *) where w^{3} = 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(w^{2}) = 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.

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