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Math Help - help with automorphisms

  1. #1
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    help with automorphisms

    I have two problems:

    1) Identify Aut(Aut( \mathbb{Z}_8))
    I know that Aut( \mathbb{Z}_8) is isomorphic to \mathbb{Z}_2 \oplus \mathbb{Z}_2 but what about Aut(Aut( \mathbb{Z}_8 ))?

    2) Identify Aut G, where G=({1, 3, 5, 7}, ._8).
    {1, 3, 5, 7} are generators of \mathbb{Z}_8 so I'm guessing I should head in that direction, but don't know how.

    Please help.
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  2. #2
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    Quote Originally Posted by georgel View Post
    I have two problems:

    1) Identify Aut(Aut( \mathbb{Z}_8))
    I know that Aut( \mathbb{Z}_8) is isomorphic to \mathbb{Z}_2 \oplus \mathbb{Z}_2 but what about Aut(Aut( \mathbb{Z}_8 ))?
    .
    Since \text{Aut}(\mathbb{Z}_2\times \mathbb{Z}_2) \simeq S_3 it means \text{Aut}(\text{Aut}(\mathbb{Z}_8)) \simeq S_3.
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  3. #3
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    Quote Originally Posted by georgel View Post

    2) Identify Aut G, where G=({1, 3, 5, 7}, ._8).

    {1, 3, 5, 7} are generators of \mathbb{Z}_8 so I'm guessing I should head in that direction, but don't know how.
    G is a group of order 4 and every non-identity element of G has order 2. so G \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2 and hence \text{Aut}(G) \cong S_3.
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  4. #4
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    Thank you so much! It all makes sense now.

    I'm just starting to learn Algebra so I'll probably have another question or two in the next few days (weeks? ), I hope that's okay.

    Thanks again!
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