Groups for which it is easy to detect graphical regular representations
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Date
2022
Authors
Morris, Dave W.
Morris, Joy
Verret, Gabriel
Journal Title
Journal ISSN
Volume Title
Publisher
University of Primorska
The Slovenian Discrete and Applied Mathematics Society
The Slovenian Discrete and Applied Mathematics Society
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 Zp wr Zp is not DRR-detecting, for every odd prime p. We also show that if G1 and G2 are nontrivial groups that admit a digraphical regular representation and either gcd(|G1|, |G2|) = 1, or G2 is not DRR-detecting, then the direct product G1 x G2 is not DRR-detecting. Some of these results also have analogues for graphical regular representations.
Description
Open access article. Creative Commons Attribution 4.0 International license (CC BY 4.0) applies
Keywords
Cayley graph , GRR , DRR , Automorphism group , Normalizar
Citation
Morris, D. W., Morris, J., & Verret, G. (2022). Groups for which it is easy to detect graphical regular representations. The Art of Discrete and Applied Mathematics, 5(1), Article #P1.07. https://doi.org/10.26493/2590-9770.1373.60a