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Showing Aut(Z) has order 2

  1. Mar 13, 2012 #1
    1. The problem statement, all variables and given/known data
    prove Aut(Z) has order 2.


    2. Relevant equations

    none

    3. The attempt at a solution
    The generators for Z = <-1, 1>.
    if f is a mapping in Aut(Z), f(x)= x or f(x) = -x
     
  2. jcsd
  3. Mar 13, 2012 #2

    Dick

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    Sure, the two elements of Aut(Z) are f(x)=x, and f(x)=(-x). Any question?
     
  4. Mar 13, 2012 #3
    how might I show that Aut(Z) has order 2?
     
  5. Mar 13, 2012 #4

    Dick

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    You just did. Doesn't that show that there are two elements in Aut(Z)? A generator must map to a generator if you are going to get an automorphism.
     
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