Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture

TL;DR

This paper proves a conjecture that finite simple groups acting transitively have a uniform positive proportion of derangements, with most cases exceeding 1.6%, and shows this proportion tends to 1 in many actions.

Contribution

It confirms the Boston-Shalev conjecture and establishes that most actions of finite simple groups have a high proportion of derangements, advancing understanding of permutation group behavior.

Findings

Proved the Boston-Shalev conjecture with delta = 0.016 for most cases.

Showed the proportion of derangements approaches 1 in many actions.

Identified finitely many exceptions to the derangement proportion bound.

Abstract

This is the fourth paper in a series. We prove a conjecture made independently by Boston et al and Shalev. The conjecture asserts that there is an absolute positive constant delta such that if G is a finite simple group acting transitively on a set of size n > 1, then the proportion of derangements in G is greater than delta. We show that with possibly finitely many exceptions, one can take delta = .016. Indeed, we prove much stronger results showing that for many actions, the proportion of derangements goes to 1 as n increases and prove similar results for families of permutation representations.