# Four-Valent Oriented Graphs of Biquasiprimitive Type

## TL;DR

This paper classifies 4-valent oriented graphs with specific symmetry properties using biquasiprimitive groups, revealing the structure of their automorphism groups and providing methods to construct new examples.

## Contribution

It offers a complete description of certain 4-valent oriented graphs with biquasiprimitive automorphism groups, including the structure of their minimal normal subgroups.

## Key findings

- Every such group has a unique minimal normal subgroup isomorphic to a power of a simple group.
- The minimal normal subgroup has order 1, 2, 4, or 8 copies of a simple group.
- Many new infinite families of examples are constructed.

## Abstract

Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(\Gamma,G)$ where $\Gamma$ 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 $(\Gamma, G) \in\mathcal{OG}(4)$ for which every nontrivial normal subgroup of $G$ has at most two orbits on the vertices of $\Gamma$. In particular we show that $G$ has a unique minimal normal subgroup $N$ and that $N \cong T^k$ 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 $\mathcal{OG}(4)$ in terms of a normal quotient reduction. We also give several methods for constructing pairs $(\Gamma, G)$ of this type and provide many new infinite families of examples, covering each of the possible structures of the normal subgroup $N$.