Groups for which it is easy to detect graphical regular representations

TL;DR

This paper investigates which finite groups are easily identifiable as automorphism groups of Cayley digraphs, introducing the concept of DRR-detecting groups and exploring their properties and limitations.

Contribution

It characterizes nilpotent DRR-detecting groups as p-groups and identifies specific group constructions that are not DRR-detecting, advancing understanding of automorphism groups of Cayley digraphs.

Findings

Nilpotent DRR-detecting groups are p-groups.

Wreath product of two cyclic groups of order p is not DRR-detecting for odd p.

Direct product G x H is not DRR-detecting under certain conditions.

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 phi of G such that phi(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product of two cyclic groups of order p is not DRR-detecting, for every odd prime p. We also show that if G and H are nontrivial groups that admit a digraphical regular representation and either gcd(|G|,|H|) = 1, or H is not DRR-detecting, then the direct product G x H is not DRR-detecting. Some of these results also have analogues for graphical regular representations.