1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Find Aut(Z n ) for cases n = 2,3,4, and 5

  1. Feb 18, 2009 #1
    Find Aut(Zn) for cases n = 2,3,4, and 5

    My question is to find Aut(Zn) for cases n = 2,3,4, and 5

    First, I am a little confused about the case Aut(Z), the integers. I know that the only functions that satisfy Aut(Z) are f(x) = x, and f(x) = -x, but what is wrong with f(x) = x+1?

    It would be great if you explain to me how to find automorphic functions, and why the previous example works and doesn't work.

    My attempts at Aut(Z2): I think that f(x) = kx and f(x) = x+k, k [tex]\in[/tex] Z are the only functions that apply in this case.

    The rest, I believe, are similar, but I just need help before I commit to an answer.

    Thank you for your time.
  2. jcsd
  3. Feb 18, 2009 #2
    Re: Automorphisms

    [tex] f \in Aut (Z)[/tex] implies [tex]f:Z \rightarrow Z[/tex] and [tex]f^{-1}:Z \rightarrow Z[/tex] should be injective. Thus, both the kernels of [tex]f, f^{-1}[/tex] should be identity. That implies the identity is mapped to an identity in Z.

    Generally speaking, the set of automorphisms forms a group and the order of [tex]Aut(Z_n)[/tex] is [tex]\phi(n)[/tex] (Euler totient function).
    Once you find an automorphism, you need to make sure that the order of each element is same with [tex]x \in Z_n[/tex] and [tex]f(x) \in Z_n[/tex], where f is an automorphism of [tex]Z_n[/tex].
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook