Groups for which it is easy to detect graphical regular representations

Abstract

We say that a finite group G is DRR-detecting if, for every subset S of G, either the Cayley digraph Cay(G, S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism ϕ of G such that ϕ(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product Z p Z p is not DRR-detecting, for every odd prime p. We also show that if G 1 and G 2 are nontrivial groups that admit a digraphical regular representation and either gcd |G 1 |, |G 2 | = 1, or G 2 is not DRR-detecting, then the direct product G 1 × G 2 is not DRR-detecting. Some of these results also have analogues for graphical regular representations.

The core of a vertex transitive complementary prism of a lexicographic product

“…Here we follow the notation in [12]. The same product appears in the literature also as Γ 1 • Γ 2 (see [2,3,13,29,31,32]) and as Γ 1 ≀ Γ 2 in which case it is referred to as the wreath product [6,20]…”