Transitive action of a $p$-group on minimal block systems

Question

I have trouble proving the following theorem:

Let $P$ be a transitive $p$-subgroup of ${\rm Sym}(A)$ with $|A| > 1$. Then any minimal $P$-block system consists of exactly $p$ blocks. Furthermore, the subgroup $P'$ which stabilizes all of the blocks has index $p$ in $P$.

Thank you.