I am reading the following theorem:
Let $G$ is a group acting on a set $\Omega$ transitively and let $B\neq\emptyset $ be a block of $G$. Then $|B|$ divides $|\Omega|$.
From the first step till the proof ends, I see the transitively is being used and it is really necessary in this theorem. Is there any counter example showing that omitting transitively doesn't lead us to desire conclusion? Thanks.