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Groupg of Automorphisms Aut(G)

  1. Jul 22, 2011 #1
    Wikipedia states:
    when G splits as direct sum of H and K, then

    Aut(H \oplus K) \cong Aut(H) \oplus Aut(K)

    Could someone please help me prove this or perhaps give a reference.
    Thank you
     
  2. jcsd
  3. Jul 22, 2011 #2

    micromass

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    Hi arthurhenry! :smile:

    Where exactly did you see this, what was the context. In general this is false:

    [tex]Aut(\mathbb{Z}_p^2)=GL_2(\mathbb{Z}_p)[/tex]

    but

    [tex]Aut(\mathbb{Z}_p)\times Aut(\mathbb{Z}_p)=\mathbb{Z}_{p-1}^2[/tex]
     
  4. Jul 22, 2011 #3
  5. Jul 22, 2011 #4

    micromass

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    OK, next time, could you please state these things completely?? The wikipedia article says that [itex]Aut(H\times K)\cong Aut(H)\times Aut(K)[/itex] if the groups are finite, abelian and of coprime order!! You need those conditions.

    As for the proof, try to prove it in these steps

    • Given an automorphism [itex]f:H\times K\rightarrow H\times K[/itex], then f(H)=H and f(K)=K.
    • We have a homomorphism

      [tex]\Phi:Aut(H\times K)\rightarrow Aut(H)\times Aut(K):f\rightarrow (f\vert_H, f\vert_K)[/tex]
    • Find an inverse of the homomorphism.
     
  6. Jul 22, 2011 #5
    I am sorry, I realized right after I sent the email; and thank you, now I will work on it.
     
  7. Jul 23, 2011 #6
    This might be bad, but I have had problem finding an inverse. I am afraid I am not suing all of the hypothesis either.
     
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