# permutation/transitivity

It's a standard theorem that $H/St_H(x) \equiv orb(x)$ as H-sets where $St_H(x):= \{ h \in H : hx=x \}$ and $orb(x)= \{ hx \in Y : h\in H \}$ and since $H$ acts transitively $orb(x)=Y$ so we have $\vert H \vert = \vert orb(x) \vert \vert St_H(x) \vert = \vert Y \vert \vert St_H(x) \vert$ and the result is immediate.