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 PSLn(q) (n=4,6,8), PSpn(q) (n=6,8), PΩn+(q) (n=8,10,12) and PΩn−(q) (n=8,10,12), where q=2f with f≥1. We prove that for each of these groups, with probability tending to 1 as q→∞, any element x of odd prime order dividing 2ef−1 but not 2i−1 for each 1≤i<ef together with a random involution y gives rise to a cubic GRR, where e=n−2 for PΩn+(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 PSL2(q), PSL3(q), PSU3(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.