On the permutation modules for orthogonal groups $O_m^{\pm }(3)$ acting on nonsingular points of their standard modules

Abstract

We describe the structure, including composition factors and submodule lattices, of cross-characteristic permutation modules for the natural actions of the orthogonal groups $O_m^{\pm }(3)$ with $m\ge 6$ on nonsingular points of their standard modules. These actions together with those studied in \cite{HN} are all examples of primitive rank 3 actions of finite classical groups on nonsingular points.

19 pages, 3 tables