A question I'm struggling with is to prove or disprove that the automorphism group of a finite cyclic group must be cyclic. I think the statement is false but have not been able to come up with a counterexample.
Thank you for any assistance you can provide.
For an automorphism, a generator has to go to a unit (a generator). Thus $\operatorname {Aut}(\Bbb Z_n)\cong \Bbb Z_n^×$.
Next, $\Bbb Z_n^×$ is cyclic iff $n$ is $1,2,4,p^k$ or $2p^k$, where $p$ is an odd prime. This is a little less obvious.
But you can check, for instance, that $\Bbb Z_8^×\cong \Bbb Z_2×\Bbb Z_2$ is not cyclic.