Arithmetic results on orbits of linear groups

TL;DR

This paper classifies certain linear groups over finite fields that have order divisible by a prime but act with orbits of size coprime to that prime, impacting representation theory and transitive group actions.

Contribution

It provides a classification of p-exceptional linear groups, revealing new structural insights and implications for longstanding conjectures in representation theory and group actions.

Findings

Classification of p-exceptional linear groups

Implications for a well-known conjecture in representation theory

Results on 1/2-transitive linear groups with order divisible by p

Abstract

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups. This has consequences for a well known conjecture in representation theory, and also for a longstanding question concerning 1/2-transitive linear groups (i.e. those having all orbits on nonzero vectors of equal length), classifying those of order divisible by $p$.