Question

Two groups G and G' of same order having one to one correspondence between their elements is called:
1. Homomorphism
2. Isomorphism
3. Automorphism
4. Endomorphism

Answer 1

Correct Answer - Option 2 : Isomorphism

Concept:

Group: 

An algebraic set G* (G is set of numbers may be set of integers, * is any mathematical binary operation ) is called a group if 

- It is closed 

- It follows associate property

- It is having an identity in a group

- It is having inverse in group

Homomorphism 

Homomorphism is a mapping f from a group (G* ) and another group (G',o) where f(a*b) = f(a) of f(b).

Example:

We have one Group (R +) which is an additive group of real numbers, and then we have another group (R₀₊ ,× ), which is a group of whole numbers.

We have a mapping f(x) = e ^x

Then f(x₁ + x₂ ) = e (x₁ + x₂) = ex₁ .ex₂

Clearly, this satisfies the condition of the group (R0+,× ), So there is a homomorphism.

Isomorphism

Isomorphism is a type of homomorphism. 

Along with homomorphism, another characteristic is the mapping should be one to one and onto

One One mapping means, if f(a) = f(b), then a = b

It should be onto, which means every number in G, must have a reflection in G'.

In another word, we can say they must have a one-to-one correspondence between their elements. 

An example of Isomorphism mapping is 

f (Z + ) → f (3Z + ); f(x) = 3x

Where Z is a set of integer

If f(x₁) = f (x₂)

3 x₁ = 3 x₂

x₁ = x₂

Also, all elements of (Z + ) for the function will have the reflection in Z + 

Explanation:

From above concepts and given examples, we can conclude that the Two groups G and G' of the same order having one-to-one correspondence between their elements is called Isomorphisam. 

So, Isomorphism is the correct option

• Automorphism: It is the type of Isomorphic mapping done on same group. Here, in the question, it is given the mapping is between two different groups, so this cannot be the answer. 
• Endomorphism: Endomorphism is a type of homomorphism but within the same group.