Homomorphism & Isomorphism of Group

In group theory, homomorphisms and isomorphisms are mappings between groups that preserve algebraic structure. A homomorphism preserves the group operation, while an isomorphism is a bijective homomorphism showing that two groups are structurally identical.

Homomorphism of Groups

Let (G,*) and (H, o) be two groups, a mapping "f " from a group (G,*) to a group (H ,o) is said to be a homomorphism if -

f(a * b) = f(a) o f(b), ∀ a,b ∈ G

A homomorphism need not be one-one, onto. It only needs to preserve the group operation.

Properties

Let f: G→H , be a homomorphism.

Identity Preservation: A homomorphism maps the identity element of one group to the identity element of the other group.

f(e_G) = e_H where e_G​ and e_H​ are identity elements of groups G and H.

Inverse Preservation: The image of the inverse of an element is equal to the inverse of its image.

f(a⁻¹) = (f(a))^-1, for every a ∈ G

Kernel of Homomorphism: The kernel of a homomorphism is the set of all elements of G whose image is the identity element of H.

ker⁡(f) = {g ∈ G: f(g) = e_H}

If ker⁡(f) = {e_G} then the homomorphism is one-one.

Example: Consider the mapping f(x) = e^x from the group (R,+) to the group (R⁺,×)

Verification: f(x+y) = e^x+y = e^xe^y= f(x)f(y)

Hence, f is a homomorphism.

Types of Homomorphism

• Homomorphism Into : A mapping 'f', that is homomorphism and also Into.
• Homomorphism Onto : A mapping 'f', that is homomorphism and also Onto.

Isomorphism of Group

Let (G , *) and (H ,o) be two groups, a mapping "f " from a group (G,*) to a group (H ,o) is said to be an isomorphism if -

1. f is homomorphism i.e., f(a*b) = f(a) o f(b) ∀ a,b ∈ G
2. f is a one- one mapping
3. f is an onto mapping.

If there exists an isomorphism between groups G and H, then the groups are called isomorphic.

It is written as G ≅ H

Properties

• Structure Preservation: An isomorphism preserves the group operation.
• Equality of Order: Isomorphic finite groups have the same number of elements.
• Abelian Property Preservation: If one group is Abelian, the other is also Abelian.

Example: If G is the multiplicative group of 3 cube-root units , i.e., (G,o) = ( {1, w, w² } , *) where w³ = 1 and 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 mapping 'f' is defined as :
  f : G -> G' in such a way that f(1) = 0 , f(w) = 1 and f(w²) = 2.
• Homomorphism property : f(a o b) = 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.

Homomorphism vs Isomorphism

➢Practice: Solved Examples