Transitive sets of derangements in primitive actions of PSL_2(q)

TL;DR

This paper investigates a problem about the existence of derangements in primitive permutation groups, showing that groups with socle PSL_2(q) do not provide counterexamples to a specific permutation property.

Contribution

It proves that almost simple primitive groups with socle PSL_2(q) always contain derangements that map one point to another without fixing any point.

Findings

Groups with socle PSL_2(q) do not produce negative examples.

The result narrows down the search for counterexamples in permutation group theory.

Abstract

Problem 8.75 of the Kourovka Notebook [10], attributed to John G. Thompson, asks the following: Suppose $G$ is a finite primitive permutation group on $\Omega$, and $\alpha$, $\beta$ are distinct points of $\Omega$. Does there exist an element $g\in G$ such that $\alpha^g=\beta$ and $g$ fixes no point of $\Omega$? A recent negative example is given in [12], where $G$ is the Steinberg triality group ${}^{3}D_{4}(2)$ acting primitively on 4,064,256 points. At present this is the only negative example known. In this note we show that almost simple primitive permutation groups with socle isomorphic to PSL_2(q) do not give negative examples.