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Properties, General Results on of Aut(G) ?

  1. Jan 18, 2015 #1

    WWGD

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    Hi, just curious: I saw a result that for the multiplicative group G:= {1,-1, i, -i } , "every

    homomorphism ## h: G \rightarrow G ## is of the form ## z \rightarrow z^k ## for some

    ##k \in \mathbb Z ##. I can show this is true by considering the image of a generator, but

    I was wondering if there are general results about the automorphism groups that may also

    apply. Any ideas?
     
  2. jcsd
  3. Jan 23, 2015 #2

    WWGD

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    I think this has to see with the fact that this G is cyclic. Then once we know the value of ## z^k## for some generator, we know the
    whole homomorphism. But that is not a proof.
     
  4. Jan 23, 2015 #3

    lavinia

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    I think it is a proof. What do you think is missing? For a cyclic group you are correct that the automorphism is determined by the image of a generator.
     
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