Properties, General Results on of Aut(G) ?

[WWGD]

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?

 

[WWGD]

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.

 

[lavinia]

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.