TY - JOUR

T1 - Four-valent oriented graphs of biquasiprimitive type

AU - Poznanović, Nemanja

AU - Praeger, Cheryl E.

PY - 2021/6

Y1 - 2021/6

N2 - 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) ∈ 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 ∼= Tk for a simple group T and k ∈ {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.

AB - 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) ∈ 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 ∼= Tk for a simple group T and k ∈ {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.

KW - Automorphism groups

KW - Cayley graphs

KW - Edge-transitive graphs

KW - Graph quotients

KW - Oriented graphs

KW - Quasiprimitive permutation groups

KW - Vertex-transitive graphs

UR - http://www.scopus.com/inward/record.url?scp=85109303041&partnerID=8YFLogxK

U2 - 10.5802/ALCO.161

DO - 10.5802/ALCO.161

M3 - Article

AN - SCOPUS:85109303041

SN - 2589-5486

VL - 4

SP - 409

EP - 434

JO - Algebraic Combinatorics

JF - Algebraic Combinatorics

IS - 3

ER -