Four-Valent Oriented Graphs of Biquasiprimitive Type

Poznanović, Nemanja; Praeger, Cheryl E.

doi:10.5802/alco.161

Poznanović, Nemanja ¹; Praeger, Cheryl E.²

¹ School of Mathematics and Statistics University of Melbourne Parkville, VIC 3010, Australia
² Department of Mathematics and Statistics University of Western Australia Perth, WA 6009, Australia

Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 409-434.

Abstract

Let $𝒪𝒢 (4)$ denote the family of all graph-group pairs $(Γ, G)$ where $Γ$ is 4-valent, connected and $G$ -oriented ( $G$ -half-arc-transitive). Using a novel application of the structure theorem for biquasiprimitive permutation groups of the second author, we produce a description of all pairs $(Γ, G) \in 𝒪𝒢 (4)$ for which every nontrivial normal subgroup of $G$ has at most two orbits on the vertices of $Γ$ , and at least one normal subgroup has two orbits. In particular we show that $G$ has a unique minimal normal subgroup $N$ and that $N ≅ T^{k}$ for a simple group $T$ and $k \in {1, 2, 4, 8}$ . This provides a crucial step towards a general description of the long-studied family $𝒪𝒢 (4)$ in terms of a normal quotient reduction. We also give several methods for constructing pairs $(Γ, G)$ of this type and provide many new infinite families of examples, covering each of the possible structures of the normal subgroup $N$ .

Received: 2019-04-19
Revised: 2020-08-19
Accepted: 2020-11-25
Published online: 2021-06-22

Zbl

DOI: 10.5802/alco.161

Classification: 05C25, 20B25, 05E18
Keywords: Edge-transitive graphs, automorphism groups, oriented graphs, graph quotients, vertex-transitive graphs, quasiprimitive permutation groups, Cayley graphs.

Author's affiliations:

Poznanović, Nemanja ¹; Praeger, Cheryl E. ²

¹ School of Mathematics and Statistics University of Melbourne Parkville, VIC 3010, Australia
² Department of Mathematics and Statistics University of Western Australia Perth, WA 6009, Australia

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Poznanović, Nemanja; Praeger, Cheryl E. Four-Valent Oriented Graphs of Biquasiprimitive Type. Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 409-434. doi : 10.5802/alco.161. https://alco.centre-mersenne.org/articles/10.5802/alco.161/

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