Let $E$ be a finite set and $G$ a transitive group of permutations of $E$. For $x\in E$, let $S_x$ denote the stabilizer of $x$. Then, for any $x,y\in E$, the number of orbits of $E$ under the action of $S_x$ is equal to the number of orbits of $E$ under the action of $S_y$.
I have seen this mentioned in some books without proof like a triviality, but I can't find a proof. I know Burnside's lemma and I know that $G/S_x$ and $G/S_y$ (sets of left cosets) are isomorphic to $E$, and hence to each other, as $G$-sets. None of this seems to help. I'm probably missing something very obvious. Can you help me?