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Math Help - automorphisms of Z_6

  1. #1
    Member ModusPonens's Avatar
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    automorphisms of Z_6

    Hello

    I'm having a problem with exercise II.4.8 of Hungerford's Algebra. It says: find an automorphism of Z_6 that is not an inner automorphism. An inner automorphism is one of the form f(x)=gxg^{-1}.

    The problem is that every automorphism can be written as an inner automorphism because Z_6 is abelian.

    Am I wrong?

    (now i've got the latex)
    Last edited by ModusPonens; August 7th 2011 at 06:27 PM.
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  2. #2
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    Re: automorphisms of Z_6

    Quote Originally Posted by ModusPonens View Post
    Hello

    I'm having a problem with exercise II.4.8 of Hungerford's Algebra. It says: find an automorphism of Z_6 that is not an inner automorphism. An inner automorphism is one of the form f(x)=gxg^{-1}.

    The problem is that every automorphism can be written as an inner automorphism because Z_6 is abelian.
    The fact that Z_6 is abelian does not imply that every automorphism is inner. What it does tell you is that the only inner automorphism is the identity map.

    So any automorphism of Z_6 that is different from the identity must necessarily be outer. Can you find such a map? Hint: an automorphism of a cyclic group must take a generator of the group to a generator.
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  3. #3
    Member ModusPonens's Avatar
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    Re: automorphisms of Z_6

    Of course! What was I thinking? Thank you.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Re: automorphisms of Z_6

    Quote Originally Posted by ModusPonens View Post
    Of course! What was I thinking? Thank you.
    Just a remark, you always know precisely how big \text{Inn}(G) is by the FIT since \text{Inn}(G)\cong G/Z(G).
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