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Math Help - Conjugation a Isomorphism?

  1. #1
    Newbie bakerconspiracy's Avatar
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    Conjugation a Isomorphism?

    Hey,

    I'm having a bit of trouble w/ this one....

    Let G be a group and let a be in G. Define a map Phi: G -> G by Phi(g) = a^-1 g a. Is phi an isomorphism? If so, prove it.

    Thanks in advance
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  2. #2
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    Quote Originally Posted by bakerconspiracy View Post
    Hey,

    I'm having a bit of trouble w/ this one....

    Let G be a group and let a be in G. Define a map Phi: G -> G by Phi(g) = a^-1 g a. Is phi an isomorphism? If so, prove it.

    Thanks in advance

    What's the problem? Indeed, \phi(gh)=\phi(g)\phi(h)\,,\,\,\phi(g)=\phi(h)\Long  leftrightarrow g=h\,,\,\,and\,\,\forall\,g\in G\,\exists\,h\in G\,\,s.t.\,\,\,\phi(h)=g\Longrightarrow conjugation is an automorphism.

    Tonio
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  3. #3
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    Now, what more do you need to show that it is an isomorphism?
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