If G is a primitive group action on A, then $G_a$ is a maximal subgroup of G
A block is a subset $B \subseteq A$ such that for any $\sigma \in G$, either $\sigma (B) = B$ or $\sigma(B) \cap B = \emptyset$. A transitive group action is called primitive if the only blocks in $A$ are single elements $a \in A$ or $A$ itself.
I'm having trouble showing that the stabilizer of an element $G_a$ is maximal in $G$. Its fairly easy to show that $G_a \le G_B$, but moving beyond that point is hard. I googled and found a site saying that there is a one to one correspendance between blocks countaining $a$ and subgroups containing $G_a$, but I don't see why this is true.