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Math Help - Normal endomorphism of a group

  1. #1
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    Normal endomorphism of a group

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    (1) An endomorphism f of a group G is called normal endomorphism if af(b)a^{-1} = f(aba^{-1}) for all a,b \in G.
    (2) f + g denotes the function G->G given by a \mapsto f(a)g(a) .
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    [Hungerford, p88, Q3.8]

    Let f and g be normal endomorphisms of a group G. Prove that if f + g is an endomorphism, then it is normal.
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  2. #2
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    Quote Originally Posted by aliceinwonderland View Post
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    (1) An endomorphism f of a group G is called normal endomorphism if af(b)a^{-1} = f(aba^{-1}) for all a,b \in G.
    (2) f + g denotes the function G->G given by a \mapsto f(a)g(a) .
    --------------------------------------------------------------------
    [Hungerford, p88, Q3.8]

    Let f and g be normal endomorphisms of a group G. Prove that if f + g is an endomorphism, then it is normal.
    This is just a straightforward computation. Let h=f+g be the function a\mapsto f(a)g(a).
    If h is an endomorphism then h(aba^{-1}) = f(aba^{-1})g(aba^{-1}) = (af(b)a^{-1} )(ag(b)a^{-1}) so h(aba^{-1}) = af(b)g(b)a^{-1} = ah(b)a^{-1}.
    This shows that h is a normal endomorphism.
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  3. #3
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    [Edited] I removed another question, coz it turned out trivial.
    Thanks for your help.
    Last edited by aliceinwonderland; December 9th 2008 at 07:25 PM.
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