Prove: Let H be a finite group of permutations of the set Y. Suppose Y acts transitivity on Y. Then Y is a finite set, and |Y|divide |H|
It's a standard theorem that $\displaystyle H/St_H(x) \equiv orb(x)$ as H-sets where $\displaystyle St_H(x):= \{ h \in H : hx=x \}$ and $\displaystyle orb(x)= \{ hx \in Y : h\in H \}$ and since $\displaystyle H$ acts transitively $\displaystyle orb(x)=Y$ so we have $\displaystyle \vert H \vert = \vert orb(x) \vert \vert St_H(x) \vert = \vert Y \vert \vert St_H(x) \vert$ and the result is immediate.