The modulo 2 structure of rank 3 permutation modules for odd characteristic symplectic groups. (Q1415348)

Let \(G=\text{Sp}(2n,q)\) with \(q\) odd and \(n\geq 2\). Then \(G\) has a rank \(3\) permutation representation on the set of \(1\) dimensional subspaces of the natural module. Let \(V\) be the corresponding permutation module over a field \(F\) of characteristic coprime to \(q\). If this characteristic is not \(2\), then \textit{M. W. Liebeck} [J. Algebra 92, 9-15 (1985; Zbl 0559.20027)] determined the submodule structure of \(V\). In this paper, the authors handle the case of characteristic \(2\). The idea is to first consider the composition factors restricted to a maximal parabolic and then use a recent result of \textit{R. M. Guralnick, K. Magaard, J. Saxl} and \textit{Pham Huu Tiep} [J. Algebra 257, No. 2, 291-347 (2002; Zbl 1025.20002)]. The description of doubly transitive and rank three permutation modules has many applications and this paper is an important contribution to the area.