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

TL;DR

This paper analyzes the structure of permutation modules for orthogonal groups over finite fields, focusing on their composition factors and submodule lattices in the context of primitive rank 3 actions.

Contribution

It provides a detailed description of the module structure for orthogonal groups acting on nonsingular points, extending previous work to include new cases.

Findings

Detailed module structures for $O_{m}^{\pm}(3)$ groups

Descriptions of composition factors and submodule lattices

Extension of known primitive rank 3 action analyses

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 $m\geq6$ 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.