The automorphism group of the derangement graph of ${\mathrm{PGL}}_2(q)$ acting on the projective line

Abstract

Given a finite transitive group $G\le \mathrm{Sym}(\Omega )$, the derangement graph ${\Gamma }_G$ is the graph whose vertex set is $G$, and two vertices $g$ and $h$ are adjacent if the ratio $h^{-1}g$ is a fixed-point-free permutation. In this paper, we show that the automorphism group of the derangement graph of the transitive permutation group corresponding to the natural action of ${\mathrm{PGL}}_2(q)$ on the projective line ${\mathrm{PG}}_1(q)$ is \begin{align*} \operatorname{Aut}(Γ_{\operatorname{PGL}_{2}({q})}) = \left(L_{\operatorname{PGL}_2(q)}\times R_{\operatorname{PGL}_2(q)}\right) \rtimes \left(\langle ψ\rangle \times γ_{\operatorname{Aut}(\mathbb{F}_q)}\right), \end{align*} where $L_{{\mathrm{PGL}}_2(q)}$ is the left-regular representation of ${\mathrm{PGL}}_2(q)$, $R_{{\mathrm{PGL}}_2(q)}$ is the right-regular representation of ${\mathrm{PGL}}_2(q)$, ${\gamma }_{\mathrm{Aut}({\mathbb{F}}_q)}$ is the group of conjugation by elements of $\mathrm{Aut}({\mathbb{F}}_q)$, and $\psi :{\mathrm{PGL}}_2(q)\to {\mathrm{PGL}}_2(q)$ such that $\psi (x)=x^{-1}$.