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Math Help - Automorphism

  1. #1
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    Automorphism

    Aut(Aut \mathbb{Z}_{8}) is isomorphic to which group? I came out with \mathbb{Z} but I'm not sure.
    Thanks.
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  2. #2
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    Re: Automorphism

    Hi,
    Aut(\mathbb{Z}_n)\cong\mathbb{Z}_n^\ast where \mathbb{Z}_n^\ast is the multiplicative group of integers relatively prime to n.

    So Aut(\mathbb{Z}_8)\cong\mathbb{Z}_2\oplus\mathbb{Z}  _2

    Thus Aut(Aut(\mathbb{Z}_8))\cong\text{GL}(2,2) where \text{GL}(2,2) is the group of non-singular 2 by 2 matrices over the field with 2 elements; \text{GL}(2,2) has order 6. This is definitely not the group of integers.
    Thanks from Haven and kicma
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