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Math Help - Please help with group automorphism

  1. #1
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    Please help with group automorphism

    Let G be an infinite cyclic group. Prove that Aut(G)≅(Z₂,⊕).
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  2. #2
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    Hint1: G\simeq \mathbb{Z}
    Hint2: If G_1\simeq G_2 then \mbox{Aut}(G_1)\simeq \mbox{Aut}(G_2).
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  3. #3
    Senior Member JaneBennet's Avatar
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    Quote Originally Posted by hzhang610 View Post
    Let G be an infinite cyclic group. Prove that Aut(G)≅(Z₂,⊕).
    Any homomorphism from \mathbb{Z} to itself is of the form n\to kn for some k\in\mathbb{Z}. This is a bijection only if k=\pm1. Hence there are only two automorphisms of \mathbb{Z}.
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