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Math Help - Automorphism groups of cyclic groups

  1. #1
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    Automorphism groups of cyclic groups

    I was just playing with the automorphism groups of cyclic groups, and I'm finding that perhaps they're not as straightforward as I thought they might be. Does anyone know if there's a formula to determine the automorphism group of \mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z} in general?

    For the prime case, how is this 'conjecture' of mine?
    \mbox{Aut}(\mathbb{Z}_p) \cong \mathbb{Z}_{p-1}, p prime


    Also...I'd like to put a little plug in for my thread on automorphisms of subfields, as I'd really like an answer and it seems to have fallen out of interest...
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    Senior Member abhishekkgp's Avatar
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    Re: Automorphism groups of cyclic groups

    Quote Originally Posted by AlexP View Post
    I was just playing with the automorphism groups of cyclic groups, and I'm finding that perhaps they're not as straightforward as I thought they might be. Does anyone know if there's a formula to determine the automorphism group of \mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z} in general?

    For the prime case, how is this 'conjecture' of mine?
    \mbox{Aut}(\mathbb{Z}_p) \cong \mathbb{Z}_{p-1}, p prime


    Also...I'd like to put a little plug in for my thread on automorphisms of subfields, as I'd really like an answer and it seems to have fallen out of interest...
    Aut(Z_n) \cong (\mathbb{Z}/n \mathbb{Z})^{\times}
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  3. #3
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    Re: Automorphism groups of cyclic groups

    That's what I thought...and it's in the book now that I think about it...so I must have screwed up when I did my calculations. Oh well. Thanks.
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    MHF Contributor Drexel28's Avatar
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    Re: Automorphism groups of cyclic groups

    Quote Originally Posted by AlexP View Post
    That's what I thought...and it's in the book now that I think about it...so I must have screwed up when I did my calculations. Oh well. Thanks.
    Well, \left(\mathbb{Z}/p\mathbb{Z}\right)^\times\cong\mathbb{Z}_{p-1}.
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    Re: Automorphism groups of cyclic groups

    Yes, I know I was right in that case. I worked it all out for \mathbb{Z}_{20} and it did not turn out correctly.
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    MHF Contributor Drexel28's Avatar
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    Re: Automorphism groups of cyclic groups

    Quote Originally Posted by AlexP View Post
    Yes, I know I was right in that case. I worked it all out for \mathbb{Z}_{20} and it did not turn out correctly.
    Well, from the CRT you know that, as rings, \mathbb{Z}/n\mathbb{Z}\cong\left(\mathbb{Z}/p_1^{a_1}\mathbb{Z}\right)\times\cdots\times\left(  \mathbb{Z}/p_m^{a_m}\right) (where n=p_1^{a_1}\cdots p_m^{a_m} is the prime factorization) which then induces a group isomorphism \left(\mathbb{Z}/n\mathbb{Z}\right)^\times\cong\left(\mathbb{Z}/p_1^{a_1}\mathbb{Z}\right)^\times\cdots\times\left  (\mathbb{Z}/\mathbb{Z}p_m}^{a_m\right). Thus, the problem of finding \text{Aut}\left(\mathbb{Z}/n\mathbb{Z}\right)^\times reduces to finding \left(\mathbb{Z}/p^a\mathbb{Z}\right)^\times for primes p. That said, from the above result you have the (interesting) result that if n is square free (i.e. that n=p_1\cdots  p_m where p_k\ne p_j) then \text{Aut}\left(\mathbb{Z}_n\right)\cong\left( \mathbb{Z}/n\mathbb{Z}\right)^\times\cong\mathbb{Z}_{p_1-1}\times\cdots\times\mathbb{Z}_{p_m-1}. For more information on \left(\mathbb{Z}/n\mathbb{Z}\right)^\times you can see the entire chapter devoted to it in this book.
    Last edited by Drexel28; August 15th 2011 at 11:00 AM.
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