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Inverse of Automorphism is an Automorphism

  1. Aug 7, 2011 #1
    1. The problem statement, all variables and given/known data
    Verify that the inverse of an automorphism is an automorphism.


    2. Relevant equations


    3. The attempt at a solution

    Let [itex]f:G\to G[/itex] be an automorphism. Then, [itex]f(xy)=f(x)f(y)[/itex] [itex]\forall x,y\in G[/itex].
    Then, we define the inverse [itex]f^{-1}:G\to G[/itex] by [itex]f^{-1}(f(x)) = f(f^{-1}(x)) = x[/itex] [itex]\;\;\forall x\in G[/itex]. We get [itex]f^{-1}(f(x)f(y))=f^{-1}(f(xy))=xy=f^{-1}(f(x))f^{-1}(f(y))[/itex]. Since [itex]f^{-1}(f(x)f(y))=f^{-1}(f(x))f^{-1}(f(y))[/itex], [itex]f^{-1}[/itex] is an automorphism.

    I was watching http://www.extension.harvard.edu/openlearning/math222/" [Broken], and this came up as an exercise in lecture 3. I was wondering if I did this problem correctly.
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Aug 7, 2011 #2

    micromass

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    Seems good!! :smile:
    As an exercise however, it would be benificial if you indicated where exactly you use injectivity and surjectivity...
     
    Last edited by a moderator: May 5, 2017
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