Direct product of automorphism groups of digraphs

Grech, Mariusz; Imrich, Wilfried; Krystek, Anna Dorota; Wojakowski, Łukasz Jan

doi:10.26493/1855-3974.1498.77b

Cited by 9 publications

(6 citation statements)

References 25 publications

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“…We now present the version of Imrich's result that applies to proper digraphs. During the refereeing process of this article, we were made aware of [5,Theorem 1], which is similar to the theorem below. Theorem 2.2.…”

Section: Cartesian Productsmentioning

confidence: 97%

Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

Morgan

Morris

Verret

2020

Art Discrete Appl. Math.

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Let Γ = Cay(G, S) be a Cayley digraph on a group G and let A = Aut(Γ). The Cayley index of Γ is |A : G|. It has previously been shown that, if p is a prime, G is a cyclic p-group and A contains a noncyclic regular subgroup, then the Cayley index of Γ is superexponential in p. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if p is an odd prime and G is abelian but not cyclic, and has order a power of p at least p 3 , then there is a Cayley digraph Γ on G whose Cayley index is just p, and whose automorphism group contains a nonabelian regular subgroup. * We thank the referee for their comments on the paper. The first and third authors also thank the second author and the University of Lethbridge for hospitality.

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Section: Cartesian Productsmentioning

confidence: 97%

Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

Morgan

Morris

Verret

2020

Art Discrete Appl. Math.

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“…When comparing the results in [7,8,10,14,25] one may observe that usually formulations of theorems concerning graphical representability are more natural and nicer when the problems are considered for edge-colored graphs rather than for simple graphs. In [13] we provide a relatively simple characterization of those cyclic permutation groups that are automorphism groups of edge-colored graphs.…”

Section: B Mariusz Grechmentioning

confidence: 99%

Cyclic Permutation Groups that are Automorphism Groups of Graphs

Grech

Kisielewicz

2019

Graphs and Combinatorics

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“…For example, every abelian group is GRR-detecting (unless it is an elementary abelian 2-group), as was pointed out in Remark 1. 6.…”

Section: Introductionmentioning

confidence: 99%

Groups for which it is easy to detect graphical regular representations

Morris

Verret

2022

Art Discrete Appl. Math.

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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.

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Direct product of automorphism groups of digraphs

Cited by 9 publications

References 25 publications

Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

Cyclic Permutation Groups that are Automorphism Groups of Graphs

Groups for which it is easy to detect graphical regular representations

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