ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.01
https://doi.org/10.26493/2590-9770.1254.266
(Also available at http://adam-journal.eu)
Digraphs with small automorphism groups
that are Cayley on two nonisomorphic groups*
Luke Morgan†
Centre for the Mathematics of Symmetry and Computation,
Department of Mathematics and Statistics (M019), The University of Western Australia,
35 Stirling Highway, Crawley, 6009, Australia
Current address: University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia,
and University of Primorska, IAM, Muzejski trg 2, 6000 Koper, Slovenia
Joy Morris‡
Department of Mathematics and Computer Science, University of Lethbridge,
Lethbridge, AB T1K 3M4, Canada
Gabriel Verret
Department of Mathematics, The University of Auckland,
Private Bag 92019, Auckland 1142, New Zealand
Received 5 November 2018, accepted 30 April 2019, published online 9 January 2020
Abstract
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. Specif-
ically, 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 p3, then there is a Cayley digraph Γ on G whose
Cayley index is just p, and whose automorphism group contains a nonabelian regular sub-
group.
Keywords: Cayley digraphs, Cayley index.
Math. Subj. Class.: 05C25, 20B25
*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.
†The first author was supported by the Australian Research Council grant DE160100081.
‡The second author was supported by the Natural Science and Engineering Research Council of Canada grant
RGPIN-2017-04905.
E-mail addresses: luke.morgan@famnit.upr.si (Luke Morgan), joy.morris@uleth.ca (Joy Morris),
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.01
https://doi.org/10.26493/2590-9770.1254.266
(Dostopno tudi na http://adam-journal.eu)
Digrafi z majhnimi grupami avtomorfizmov,
ki so Cayleyjevi na dveh neizomorfnih grupah*
Luke Morgan†
Centre for the Mathematics of Symmetry and Computation,
Department of Mathematics and Statistics (M019), The University of Western Australia,
35 Stirling Highway, Crawley, 6009, Australia
Trenutni naslov: University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia,
in University of Primorska, IAM, Muzejski trg 2, 6000 Koper, Slovenia
Joy Morris‡
Department of Mathematics and Computer Science, University of Lethbridge,
Lethbridge, AB T1K 3M4, Canada
Gabriel Verret
Department of Mathematics, The University of Auckland,
Private Bag 92019, Auckland 1142, New Zealand
Prejeto 5. novembra 2018, sprejeto 30. aprila 2019, objavljeno na spletu 9. januarja 2020
Povzetek
Naj bo Γ = Cay(G,S) Cayleyjev digraf na grupi G in naj bo A = Aut(Γ). Cay-
leyjev indeks grafa Γ je |A : G|. Dokazano je že bilo, da če je p praštevilo, G ciklična
p-grupa, A pa vsebuje neciklično regularno podgrupo, potem je Cayleyjev indeks grafa Γ
nadeksponenten v p.
Predstavimo rezultate, ki kažejo, da so ciklične grupe izjemne v tem pogledu. Konkretno,
pokažemo nasprotni rezultat: če je p liho praštevilo in G abelska, ne pa ciklična, in ima red
potence števila p najmanj p3, potem obstaja Cayleyjev digraf Γ na G, katerega Cayleyjev
indeks je samo p, in katerega grupa avtomorfizmov vsebuje neabelsko regularno podgrupo.
Ključne besede: Cayleyjevi digrafi, Cayleyjev indeks.
Math. Subj. Class.: 05C25, 20B25
g.verret@auckland.ac.nz (Gabriel Verret )
*Zahvaljujemo se recenzentom za njihove pripombe k članku. Prvi in tretji avtor se tudi zahvaljujeta drugemu
avtorju in University of Lethbridge za gostoljubje.
†Prvi avtor je bil podprt s strani Australian Research Council, dotacija DE160100081.
‡Drugi avtor je bil podprt s strani Natural Science and Engineering Research Council of Canada, dotacija
RGPIN-2017-04905.
E-poštni naslovi: luke.morgan@famnit.upr.si (Luke Morgan), joy.morris@uleth.ca (Joy Morris),
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/
g.verret@auckland.ac.nz (Gabriel Verret )