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Asked Oct 25, 2019
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Step 1

Let G and G’ are any groups and f is a homomorphism from G to G’ then

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G =f(G) kerf

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Step 2

Image and kernel of a group both are the subgroup of the group,

  • The kernel of f, written ker f, is the set of elements g G such that f(g) = 1.
  • The image of f, written im f, is the set of elements of H of the form f(g) for some g G.

Now if f is a homomorphism of a group G to G’ then the image of f is a subgroup of H, and the kernel of f is a normal subgroup of G.

 

Now let’s show that ker f is a normal subgroup of G. Given an element x ∈ G and an element g ∈ ker f , we have

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f (x gx) f(x)f(g) g(x) =f(x)l(x) =f (x)f(x) =1

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Step 3

The normal subgroup of the S3 and co...

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normal subgroup of f(G) ker f S e} e} S Z2 {4} 3 {4} S e S

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