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Math Help - Automorphism

  1. #1
    Junior Member hercules's Avatar
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    Automorphism

    I need help with the following proof:
    Prove that Aut( \mathbb{Z}_{4}) \approx \mathbb{Z}_{2}

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  2. #2
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    Note, \mathbb{Z}_4 = \{ [0],[1],[2],[3] \} = \left< [1]\right>. Any homomorphism \phi: \mathbb{Z}_4 \mapsto \mathbb{Z}_4 is completely determined by [1] (since it is the generator). Thus, there are four possibilities, \phi([1]) = [0],\phi([1]) = [1], \phi([1]) = [2],\phi([1])= [3]. Which of these give rise to automorphisms?
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