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Math Help - automorphism proof

  1. #1
    Senior Member Danneedshelp's Avatar
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    automorphism proof

    Theorem: for any positive integer n, Aut(Z_{n}) is isomorphic to U(n).

    Q: Prove that every element g\in\\Aut(Z_{n}) is determined by g(1); meaning, g(1) results in a well-defined automorphism of Z_{n}.

    A: let g\in\\Aut(Z_{n}) be arbitrary. Then f:Z_{n}\rightarrow\Z_{n} is an isomorphism. So, to show if something is well-defined I need to show if a=c, b=d, then a*b=c*d. I am not sure what to my a,b, and c's are. I know f(k)=kf(1). so, do I let f(m)=mf(1) and f(p)=pf(1). Now, I want to show f(p)*f(q)=mf(1)*pf(1)?

    Thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Danneedshelp View Post
    Theorem: for any positive integer n, Aut(Z_{n}) is isomorphic to U(n).

    Q: Prove that every element g\in\\Aut(Z_{n}) is determined by g(1); meaning, g(1) results in a well-defined automorphism of Z_{n}.

    A: let g\in\\Aut(Z_{n}) be arbitrary. Then f:Z_{n}\rightarrow\Z_{n} is an isomorphism. So, to show if something is well-defined I need to show if a=c, b=d, then a*b=c*d. I am not sure what to my a,b, and c's are. I know f(k)=kf(1). so, do I let f(m)=mf(1) and f(p)=pf(1). Now, I want to show f(p)*f(q)=mf(1)*pf(1)?

    Thanks
    I'm sorry, what is U(n)? That can't be universal notation considering the very common Lie group called the Unitary Group is denoted U(n).
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  3. #3
    Senior Member Danneedshelp's Avatar
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    U(n) represents the group of units modulo n. So, for example, U(10)=\{1,3,7,9\}.

    Sorry for not stating that.
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  4. #4
    Senior Member roninpro's Avatar
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    I think that your analysis of the problem is a little bit too complicated. For an easy example, let us look at the group \mathbb{Z}_{10}. If f:\mathbb{Z}_{10}\to \mathbb{Z}_{10} is an automorphism, it must send a generator to a generator. To be explicit, a number a is a generator of \mathbb{Z}_n if and only if \gcd(a,n)=1. In our case, a is a generator if and only if \gcd(a,10)=1. This means that a=1, 3, 5, 7, 9. So, we have the following possibilities: f(1)=1, f(1)=3, f(1)=5, f(1)=7, or f(1)=9. If you try composing these maps, you will find that you recover the multiplication table for your U_{10}.

    Give it a try. Good luck!
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Drexel28 View Post
    I'm sorry, what is U(n)? That can't be universal notation considering the very common Lie group called the Unitary Group is denoted U(n).
    U(R) is common for denoting the group of units of a ring. I suppose this is just a perversion of that.
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