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Math Help - normal subgroups proofs please help

  1. #1
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    normal subgroups proofs please help

    Let f: G -> K be an epimorphism of groups. If N is a normal
    subgroup of G, show that f(N) := {f(n) | n e N} is a normal subgroup of K.

    please note in this proof "e" is the id for G
    and "i" is the id for K
    My proof:
    Let g, h be an elements of G
    Suppose N is a normal subgroup of G,

    then f(e)=i is element of f(N)
    f(g^(-1))= f(g)^(-1) is element of f(N)
    f(g)f(h) = f(gh)=f(N)
    therefore N is subgroup of K

    Now to proof that it is a normal subgroup of K

    f(ghg(-1))=f(g)f(h)f(g)^(-1)

    Now im stuck...
    Can some one please show me how to proof this , and if my proof that i gave so far
    is incorrect plz show me how in the simplest way ...

    Thank you in advance.
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  2. #2
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    Quote Originally Posted by Dreamer78692 View Post
    Let f: G -> K be an epimorphism of groups. If N is a normal
    subgroup of G, show that f(N) := {f(n) | n e N} is a normal subgroup of K.

    please note in this proof "e" is the id for G
    and "i" is the id for K
    My proof:
    Let g, h be an elements of G
    Suppose N is a normal subgroup of G,

    then f(e)=i is element of f(N)
    f(g^(-1))= f(g)^(-1) is element of f(N)
    f(g)f(h) = f(gh)=f(N)
    therefore N is subgroup of K

    Now to proof that it is a normal subgroup of K

    f(ghg(-1))=f(g)f(h)f(g)^(-1)

    Now im stuck...
    Can some one please show me how to proof this , and if my proof that i gave so far
    is incorrect plz show me how in the simplest way ...

    Thank you in advance.

    \forall k_1\in K\,\exists g_1\in G\,\,s.t.\,\,f(g_1)=k_1 , so that if k\in f(N) then also k=f(g) for some g/in G, and now:

    For any elements k_1\in K\,,\,k\in f(N) , k_1^{-1}kk_1=f(g_1)^{-1}f(g)f(g_1)=f(g_1^{-1}gg_1) , and since g_1^{-1}gg_1\in N then k_1^{-1}kk_1\in f(N) and we're done.

    Tonio
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