Application of orbits and Sylow Theorems

Question

Suppose I have a group $G$ such that $n_p \not \equiv 1 \pmod{p^2}$, where $n_p = |\operatorname{Syl}_p(G)|$. Consider a $P \in \operatorname{Syl}_p(G)$ acting on $ \operatorname{Syl}_p(G)$ by conjugation. I want to show that $\exists Q \in \operatorname{Syl}_p(G) : |\mathcal{O} (Q)| = p$. I was considering how to apply the orbit-stabilizer theorem, but I do not know how to apply the fact that $n_p \not \equiv 1 \pmod{p^2}$.

Answer 1

1. For $P$ acting on $Syl_p(G)$, each orbit has size equal to some power of $p$. (Use orbit-stabilizer.)
2. Show that there is only one orbit of size $1$. (Use Sylow's theorems.)
3. The sizes of the orbits sum up to $n_p$, so conclude that there is some orbit of size $p$.