The paper studies the permutation representations of a finite general linear group, first on finite
projective space and then on the set of vectors of its standard module. In both cases the submodule lattices
of the permutation modules are determined. In the case of projective space, the result leads to the solution
of certain incidence problems in finite projective geometry, generalizing the rank formula of Hamada. In
the other case, the results yield as a corollary the submodule structure of certain symmetric powers of the
standard module for the finite general linear group, from which one obtains the submodule structure of all
symmetric powers of the standard module of the ambient algebraic group.