Hall-Higman type theorems for semisimple elements of finite classical groups

TL;DR

This paper extends the Hall-Higman theorem to finite classical groups, providing lower bounds on the minimal polynomial degree of semisimple elements of prime power order in cross characteristic representations.

Contribution

It establishes a new analogue of the Hall-Higman theorem for finite classical groups, detailing bounds on minimal polynomial degrees of semisimple elements.

Findings

Lower bounds on minimal polynomial degrees for semisimple elements

Explicit exceptions where bounds do not hold

Generalization of Hall-Higman theorem to classical groups

Abstract

We prove an analogue of the celebrated Hall-Higman theorem, which gives a lower bound for the degree of the minimal polynomial of any semisimple element of prime power order $p^{a}$ of a finite classical group in any nontrivial irreducible cross characteristic representation. With a few explicit exceptions, this degree is at least $p^{a-1}(p-1)$.