Subgroups of Classical Groups that are Transitive on Subspaces

TL;DR

This paper classifies subgroups of finite classical groups that act transitively on specific sets of subspaces, and applies these results to classify point-transitive automorphism groups of finite generalized quadrangles.

Contribution

It provides a comprehensive classification of subgroups acting transitively on certain subspace sets in classical groups, using maximal factorisations of almost simple groups.

Findings

Classified subgroups transitive on totally isotropic or nondegenerate subspaces.

Applied results to classify point-transitive automorphism groups of finite generalized quadrangles.

Utilized the classification of maximal factorisations in the proofs.

Abstract

For each finite classical group $G$, we classify the subgroups of $G$ which act transitively on a $G$-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the classification of the maximal factorisations of almost simple groups. As a first application of these results we classify all point-transitive subgroups of automorphisms of finite thick generalised quadrangles.