Let $G$ be a finite group acting transitively on the set $A$. Let $p$ be a prime and $S \in Syl_{p}(G)$. Show that $N_{G}(S)$ acts transitively on $Fix_{A}(S)$(the set of fixed-points of S in A).
I tried to take an element $s$ in $S$, and it fixes $a, a' \in Fix_{A}(S)$. Then there exists a element $g \in G$ such that $ga = a'$. I noted that we also have $s^{-1}gsa = a'$, and want to show that $s^{-1}gs \in N_{G}(S)$, but don't know how to do. Any hint is appreciated, thank you!