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Math Help - Automorphisms

  1. #1
    Forum Admin topsquark's Avatar
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    Automorphisms

    This one is mostly a problem with the definitions, I think.

    I am asked to prove that \text{Aut} \mathbb{Z} is isomorphic to \mathbb{Z}_2, where \text{Aut} \mathbb{Z} is the group of all automorphisms of \mathbb{Z}.

    Obviously the identity function I: \mathbb{Z} \to \mathbb{Z}: x \mapsto x is in \text{Aut} \mathbb{Z}. The only other automorphism I can come up with is f: \mathbb{Z} \to \mathbb{Z}: x \mapsto -x. etc, etc. to finish showing the isomorphism between \text{Aut} \mathbb{Z} and \mathbb{Z}_2.

    Are there truly only two members in \text{Aut} \mathbb{Z}? It seems there should be more, but I can't find any.

    -Dan
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    You need to show that these are the only automorphisms of \mathbb{Z}. Use the fact that automorphisms map generators to generators. Now, what are the generators of \mathbb{Z}?
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ojones View Post
    You need to show that these are the only automorphisms of \mathbb{Z}. Use the fact that automorphisms map generators to generators. Now, what are the generators of \mathbb{Z}?
    Thank you. I was unaware of that fact. (Edit: It's pretty obvious, actually, now that I've had some time to think about it.) To finish the argument then, since there are only two generators of \mathbb{Z} ( -1 and 1) there are only two automorphisms that are in \text{Aut} \mathbb{Z}. Since \text{Aut} \mathbb{Z} is a group of two members it must be isomorphic to \mathbb{Z}_2.

    -Dan
    Last edited by topsquark; March 29th 2011 at 06:04 AM.
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  4. #4
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    Yes, this will do it provided you know how to fill in the details.
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