Prove that there does not exist a group $G$ with a group of automorphisms isomorphic to $\mathbb{Z}$ or $\mathbb{Z}/n\mathbb{Z}$ with odd $n \gt 2$.
I assumed the opposite, let there exist a group $G$ such that $\mathrm{Aut} (G) \cong \mathbb{Z}$ or $\mathrm{Aut} (G) \cong \mathbb{Z} / n \mathbb {Z}$. Since the subgroup of the cyclic group is also cyclic, $\mathrm{Inn}(G)$ is cyclic. $\operatorname{Inn} (G) \cong G / Z (G) $.
So $G$ is Abelian.
If $\mathrm{Aut}(G)$ has an element whose order is not equal to $2$, then we take the automorphism $p (x) = x^{-1}$. This is an automorphism of order $2$. There remains a case where the order of each automorphism is $2$. Help with this case.
