I am writing another exam in Algebra this week and this time the main topic is automorphism. I was again going through the example exercises and exams from previous years and this problem is giving me a hard time to understand:
Prove that for group of automorphisms $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})$ holds $\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$ where $\mathbf{Z_{n}^{*}}=\{1 \leq k \leq q | \gcd(k,n)=1\} $.
My main issue with this problem is that it seems very "general" and I don't really know how to address it with any of the techniques we have used.
Could you please show me some ways this could be approached? I appreciate your help.
Answer
Observe that a homomorphism is determined by the image of $\overline{1}$.
Now consider which restrictions you get from the fact that the homomorphism is injective.
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