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Group Theory inner automorphism

  1. Oct 31, 2013 #1
    The problem statement, all variables and given/known data

    How do I prove that the inner automorphisms is isomorphic to ##S_3##?


    The attempt at a solution

    I know ##S_3 = \{f: \{ 1,2,3 \}\to\{ 1,2,3 \}\mid f\text{ is a permutation}\}## and I know for every group there is a map whose center is its kernel so the center of of ##S_3## is trivial therefore ##S_3/Z(S_3) = 6##.

    So an inner automorphism of a group ##G## is an automorphism of the form ##ρ_g :x↦gxg^{-1}. ## What I am having trouble with is verifying that distinct elements of ##S_3## give distinct inner automorphism. How can I prove this problem directly?
     
  2. jcsd
  3. Oct 31, 2013 #2

    Office_Shredder

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    The inner automorphisms of what are supposed to be isomorphic to S3?
     
  4. Oct 31, 2013 #3

    pasmith

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    Can you prove that if [itex]\rho_g = \rho_h[/itex] then [itex]g^{-1}h \in Z(G)[/itex]?
     
  5. Oct 31, 2013 #4
    Opps, I forgot to add the inner automorphism of ##S_3## is isomorphic to ##S_3.## Sorry for the confusion!

    Pasmith - Can you elaborate please?
     
  6. Oct 31, 2013 #5

    Dick

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    There's no need to elaborate. pasmith pretty much spelled it out. Try to prove the hint he gave you. Once you've done that figure out what the center of ##S_3## is. It should click easily after that.
     
    Last edited: Oct 31, 2013
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