Question

Difference between group homomorphism and isomorphism

Answer 1

In the category of topological spaces, morphisms are continuous functions, andisomorphisms are homeomorphisms. Extra remark: A  fundamental difference between algebra and topology is that in algebra any morphism (homomorphism) which is 1-1 and onto is an isomorphism - i.e., its inverse is a morphism.

Answer 2

Note :

• Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

• Abelian group : If a group (G,*) also holds commutative property , then it is called commutative group or abelian group , ie . if x*y = y*x ∀ x , y ∈ (G,*) , then the group G is said to be abelian .

Answer :

Group homomorphism : Let (G,•) and (G',*) be any two groups , then the mapping f : G → G' is called a group homomorphism if , f(x•y) = f(x)*f(y) ∀ x , y ∈ G . ie. if f preserves the group operation .

Group isomomorphism : Let (G,•) and (G',*) be any two groups , then the mapping f : G → G' is called a group isomomorphism if , it is an one-to-one homomorphism , ie.

• f(x•y) = f(x)*f(y) ∀ x , y ∈ G
• f(x) = f(y) → x = y ∀ x , y ∈ G

Examples :

We know that , the set of all integers Z and the set of all even integers E are group with with respect to addition .

Now ,

We define a function f : Z → E , such that

f(x) = 2x ∀ x ∈ Z

Now let x , y ∈ Z , then

→ f(x + y) = 2(x + y)

→ f(x + y) = 2x + 2y

→ f(x + y) = f(x) + f(y)

→ f is a group homomorphism .

Moreover , if f(x) = f(y) , then

→ 2x = 2y

→ x = y

→ f is one-to-one .

Hence , f is a group isomomorphism .