Cubic graphical regular representations of some classical simple groups

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), PΩ + n (q) (n = 8, 10, 12) and PΩ−n (q) (n = 8, 10, 12), where q = 2 f 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 2 − 1 but not 2 − 1 for each 1 ≤ i < ef together with a random involution y gives rise to a cubic GRR, where e = n − 2 for PΩ+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, 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, 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.

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