Properties, General Results on of Aut(G) ?

  • Thread starter WWGD
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  • #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?
 

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  • #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.
 
  • #3
lavinia
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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.

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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