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 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.
Cite
@article{arxiv.2005.09798,
title = {Groups for which it is easy to detect graphical regular representations},
author = {Dave Witte Morris and Joy Morris and Gabriel Verret},
journal= {arXiv preprint arXiv:2005.09798},
year = {2020}
}
Comments
11 pages. v2: added acknowledgments and author address