Let $\displaystyle G$ act transitively on a finite set $\displaystyle A$. A 'block' in $\displaystyle A$ is a non-empty subset $\displaystyle B$ of $\displaystyle A$ such that for all $\displaystyle \sigma \in G$ either $\displaystyle \sigma(B)=B$ or $\displaystyle \sigma(B) \cap B= \phi$ (where $\displaystyle \sigma(B)=\{ \sigma(b)|b \in B \}$).

This action is called 'primitive' if the only blocks in $\displaystyle A$ are trivial ones: the sets of size $\displaystyle 1$ and $\displaystyle A$ itself.

Prove that:

The action(transitive) of $\displaystyle G$ on $\displaystyle A$ is primitive if and only if for each $\displaystyle a \in A,$ $\displaystyle G_a$ is a maximal subgroup of $\displaystyle G$. ($\displaystyle G_a=\{g \in G| g \cdot a=a \}=$ stabilizer of $\displaystyle a$ in $\displaystyle G$)

Here is what i have(with a little help from my friend):

Define $\displaystyle G_B= \{ \sigma \in G| \sigma(B)=B \}$. Its easy to see that $\displaystyle G_B \leq G$. Moreover if $\displaystyle a \in B$ then $\displaystyle G_a \leq G_B$.

I came to know that there exists a bijection between the blocks containing $\displaystyle a$ and the subgroups of $\displaystyle G$ containing $\displaystyle G_a$.I couldn't prove this. Help needed.