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When is $Aut\left(\mathbb{Z}_n\right)$ cyclic?

Where $Aut \left( G \right)$ is the group of "Automorphisms of the group $G$" under composition.

I know that $Aut\left(\mathbb{Z}_n\right) \approx U\left(n\right)$ but I don't know when $U\left(n\right)$ is cyclic.

user567182
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    When $n=1,2,4,p^k$ or $2p^k$ for an odd prime $p$. – Derek Holt Jun 10 '18 at 16:03
  • https://math.stackexchange.com/questions/314846/for-what-n-is-u-n-cyclic –  Jun 10 '18 at 16:06
  • @DerekHolt Okay, I am supposed to show that $U\left(8\right) \approx U\left(12\right)$ but $8=2^3$ Hence $ U\left(8\right)$ is cyclic but $U\left(12\right)$ is not cyclic because $12$ is not of the form you mentioned. – user567182 Jun 10 '18 at 17:00
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    $U(8)$ is also not of the form I mentioned, so that is not cyclic either. – Derek Holt Jun 10 '18 at 17:16

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