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Math Help - Group Isomorphism

  1. #1
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    Group Isomorphism

    It is very urgent!!!! need help!!!!

    Let G be any group and let g be an element of G. Define the function f:G-->G by f(a) = g^(-1)*a*g (a is in G)(thus f takes every element to its conjuate by g). Show that f is an isomorphism from G to itself. Show, by example, that f need not be the identity function.
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  2. #2
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    Quote Originally Posted by r7iris View Post
    It is very urgent!!!! need help!!!!

    Let G be any group and let g be an element of G. Define the function f:G-->G by f(a) = g^(-1)*a*g (a is in G)(thus f takes every element to its conjuate by g). Show that f is an isomorphism from G to itself. Show, by example, that f need not be the identity function.
    This is a very special automorphism called "conjugation automorphism".

    It is a homomorphism,
    \phi(x)\phi(y) = gx^{-1}g^{-1}gyg^{-1}=gxyg^{-1}=\phi(xy)

    Is is one-to-one,
    \phi (x) = \phi(y) \implies gxg^{-1} = gyg^{-1} \implies x=y

    Is is onto,
    \forall y\in G \mbox{ choose }x=gyg^{-1}\implies \phi(x) = y
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