If $G$ is a group of order $p + 1$, then $p$ does not divide $|\text{Aut}(G)|$ ($p$ is prime and $p + 1$ isn't a prime power)

Question

If $G$ is a group of order $p + 1$, then $p$ does not divide $|\text{Aut}(G)|$ ($p$ is prime and $p + 1$ isn't a prime power).

Here is how far I got: Assume $p$ divides $|\text{Aut}(G)|$. By Cauchy's theorem, $\text{Aut}(G)$ has an element of order $p$. I am not sure how to progress from here. How do I prove this?

Answer 1

Hint: If $\varphi$ is an automorphism of order $p$, what are the possible sizes of orbits of $\varphi$? What are the orders of their elements?