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Isomorphism with z sub prime#

  1. Mar 6, 2012 #1
    1. The problem statement, all variables and given/known data
    Let P be a prime integer, prove that Aut(Z sub P) ≈ Z sub p-1


    2. Relevant equations

    none

    3. The attempt at a solution
    groups must preserve the operation, be 1-1, and be onto and they can be called an isomorphism. Z sub p-1 has one less element in it so and all the elements in them are the same except for the one less element. Not sure what this tells me though. HELP ME FOR LINEARRRRR
     
  2. jcsd
  3. Mar 6, 2012 #2

    UD1

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    Let f be an automorphism of Z_p, and x a generator for Z_p, so that <x>=Z_p. Explain why f is determined by what it does to x, i.e. why knowing f(x) suffices to to know where f sends any other element of the group.

    Now think about whether (x is a generator) => (f(x) is a generator) is true. You will see that Aut(Z_p) is isomorphic to the group of units of Z_p -- there is a natural isomorphism. Think how you can prove that the latter is cyclic.
     
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