Cubic graphical regular representations of some classical simple groups

Abstract

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. In this paper we study cubic GRRs of ${\mathrm{PSL}}_n(q)$ ($n=4,6,8$), ${\mathrm{PSp}}_n(q)$ ($n=6,8$), $\mathrm{P}{\Omega }_n^{+}(q)$ ($n=8,10,12$) and $\mathrm{P}{\Omega }_n^{-}(q)$ ($n=8,10,12$), where $q=2^f$ with $f\ge 1$. We prove that for each of these groups, with probability tending to $1$ as $q\to \infty$, any element $x$ of odd prime order dividing $2^{ef}-1$ but not $2^i-1$ for each $1\le i<ef$ together with a random involution $y$ gives rise to a cubic GRR, where $e=n-2$ for $\mathrm{P}{\Omega }_n^{+}(q)$ and $e=n$ for other groups. Moreover, for sufficiently large $q$, there are elements $x$ satisfying these conditions, and for each of them there exists an involution $y$ such that $\{x,x^{-1},y\}$ produces a cubic GRR. This result together with certain known results in the literature implies that except for ${\mathrm{PSL}}_2(q)$, ${\mathrm{PSL}}_3(q)$, ${\mathrm{PSU}}_3(q)$ and a finite number of other cases, every finite non-abelian simple group contains an element $x$ and an involution $y$ such that $\{x,x^{-1},y\}$ produces a GRR, showing that a modified version of a conjecture by Spiga is true. Our results and several known results together also confirm a conjecture by Fang and Xia which asserts that except for a finite number of cases every finite non-abelian simple group has a cubic GRR.