Let $p$ be an odd prime and denote by $C_{p^n}$ the cyclic group of order $p^n$. We know that $\mathrm{Aut}(C_{p^n})$ is a cyclic group of order $p^{n-1}(p-1)$. In particular, there exists a cyclic subgroup of order $p-1$ inside $\mathrm{Aut}(C_{p^n})$. How can we understand the action of this subgroup on the cyclic group $C_{p^n}$?
In the special case $n = 1$, we can know that $C_p = \mathbb F_p$ is a field and in this case the automorphisms are given by multiplication by non-zero elements of $\mathbb F_p$.