The modulo 2 structure of rank 3 permutation modules for odd characteristic symplectic groups

This paper studies the permutation representation of the symplectic group where q is odd, on the 1-spaces of its natural module. The complete submodule lattice for the modulo ℓ reduction of this permutation module is known for all odd primes ℓ not dividing q. In this paper we determine the complete submodule lattice for the mod 2 reduction. Similar results are then obtained for the orthogonal group .