Let act transitively on a finite set . A 'block' in is a non-empty subset of such that for all either or (where ).

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

Prove that:

The action(transitive) of on is primitive if and only if for each is a maximal subgroup of . ( stabilizer of in )

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

Define . Its easy to see that . Moreover if then .

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