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Math Help - automorphism group of a cyclic group

  1. #1
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    automorphism group of a cyclic group

    Hi guys,

    Is the automorphism group of a cyclic group always abelian?

    I believe so but I need to prove it. I don't really know where to start...any ideas?
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  2. #2
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    Hi

    Let (G,.) be a cyclic group, a one of its generators. Two automorphisms \phi and \psi are determined by \phi(a)=a^k and \psi(a)=a^l for some k,l \in\mathbb{Z}.

    Since \psi\circ\phi and \phi\circ\psi are automorphisms, they're also determined by \psi\circ\phi(a) and \phi\circ\psi(a), therefore if \psi\circ\phi(a)=\phi\circ\psi(a) then \psi\circ\phi=\phi\circ\psi and the automorphism group is abelian.

    So is \psi\circ\phi(a)=\phi\circ\psi(a) true?
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  3. #3
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    Quote Originally Posted by ziggychick View Post
    Hi guys,

    Is the automorphism group of a cyclic group always abelian?

    I believe so but I need to prove it. I don't really know where to start...any ideas?
    You can do better that what clic-clac said if you want to try and additional exercise.

    If |G|=n is a cyclic group then \text{Aut}(G) \simeq \mathbb{Z}_n^{\times}.
    If |G|=\infty then \text{Aut}(G) \simeq \mathbb{Z}_2.
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  4. #4
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    oh i see now. i can prove those two things.


    thanks!
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