Desarguesian spreads and field reduction for elements of the semilinear group

TL;DR

This paper develops a unified framework connecting field reduction, semilinear groups, and projective groups to simplify proofs and extend results on Desarguesian spreads, stabilizers, and related structures in finite projective spaces.

Contribution

It introduces an elementary approach to analyze Desarguesian spreads and stabilizers, extending existing results to broader groups and providing new insights into linear sets.

Findings

Reproved Dye's stabilizer result in PGL more simply.

Extended results to PΓL(n, q) for Desarguesian spreads.

Connected Singer cycles with Desarguesian spreads and analyzed subspreads.

Abstract

The goal of this note is to create a sound framework for the interplay between field reduction for finite projective spaces, the general semilinear groups acting on the defining vector spaces and the projective semilinear groups. This approach makes it possible to reprove a result of Dye on the stabiliser in PGL of a Desarguesian spread in a more elementary way, and extend it to P{\Gamma}L(n, q). Moreover a result of Drudge [5] relating Singer cycles with Desarguesian spreads, as well as a result on subspreads (by Sheekey, Rottey and Van de Voorde [19]) are reproven in a similar elementary way. Finally, we try to use this approach to shed a light on Condition (A) of Csajbok and Zanella, introduced in the study of linear sets [4].