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Homomorphism

 Homomorphism:

Let (𝐺 , ∙) and (𝐻 ,∗) be two groups. A mapping 𝜑 ∶ 𝐺 ⟶ 𝐻 is said to be homomorphism if 

𝜑 (𝑥 ∙ 𝑦)= 𝜑( 𝑥)∗ 𝜑(𝑦)for 𝑥, 𝑦 ∈ 𝐺. The range of 𝜑 in 𝐻 is called the homomorphic image of 𝜑.

Endomorphism:

 Let (𝐺 ,∗) be a group. A homomorphism 𝜑 ∶ 𝐺 ⟶ 𝐺 is called endomorphism.

Monomorphism:

Let (𝐺 , ∙) and (𝐻 ,∗) be two groups. A mapping 𝜑 ∶ 𝐺 ⟶ 𝐻 is said to be monomorphism if 

a) 𝜑 is homomorphism.

b) 𝜑 is injective.

Epimorphism:

Let (𝐺 , ∙) and (𝐻 ,∗) be two groups. A mapping 𝜑 ∶ 𝐺 ⟶ 𝐻 is said to be 

epimorphism if 

a) 𝜑 is homomorphism.

b) 𝜑 is surjective. i.e., for all 𝑏 ∈ 𝐻, there is an element 𝑎 ∈ 𝐺 such that 𝜑 (𝑎) = 𝑏.

Endomorphism

Let (𝐺 ,∗) be a group. A homomorphism 𝜑 ∶ 𝐺 ⟶ 𝐺 is called endomorphism.

 Isomorphism

Let (𝐺 , ∙) and (𝐻 ,∗) be two groups. A mapping 𝜑 ∶ 𝐺 ⟶ 𝐻 is said to be 

isomorphism if 

a) 𝜑 is homomorphism.

b) 𝜑 is injective.

c) 𝜑 is surjective.

The isomorphism between two groups is denoted by " ≅ ".i.e., the isomorphism between 𝐺 and 𝐻 is denoted by 𝐺 ≅ 𝐻.


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