Absolutely irreducible quasisimple linear groups containing elements of order a specified Zsigmondy prime

Abstract

This paper is concerned with absolutely irreducible quasisimple subgroups $G$ of a finite general linear group $G{L}_d({\mathbb{F}}_q)$ for which some element $g\in G$ of prime order $r$, in its action on the natural module $V=({\mathbb{F}}_q{)}^d$, is irreducible on a subspace of the form $V(1-g)$ of dimension $d/2$. We classify $G,d,r$, the characteristic $p$ of the field ${\mathbb{F}}_q$, and we identify those examples where the element $g$ has a fixed point subspace of dimension $d/2$. Our proof relies on representation theory, in particular, the multiplicities of eigenvalues of $g$, and builds on earlier results of DiMuro.

34 pages, 9 tables, dedicated to the memory of Richard A. Parker