what is homomorphism?
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a transformation of one set into another that preserves in the second set the relations between elements of the first.
asnamol2005:
a transformation of one set into another that preserves in the second set the relations between elements of the first
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Note :
- Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :
- G is closed under *
- G is associative under *
- G has a unique identity element
- 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 → H is called a group homomorphism if , f(x•y) = f(x)*f(y) ∀ x , y ∈ G . ie. if f preserves the group operation .
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)
Hence , f is a group homomorphism .
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