Every finite non-solvable group admits an Oriented Regular Representation

TL;DR

This paper proves that all finite non-solvable groups have an oriented regular representation (ORR), advancing understanding of group actions on oriented graphs and partially addressing a question from 1980.

Contribution

It establishes that every finite non-solvable group admits an ORR and characterizes finite groups with up to three generators regarding their ORR admitance.

Findings

All finite non-solvable groups admit ORRs.

Characterization of groups with ≤3 generators and their ORR status.

Introduction of a new tool for analyzing ORRs in solvable groups.

Abstract

In this paper we give a partial answer to a 1980 question of Lazslo Babai: "Which [finite] groups admit an oriented graph as a DRR?" That is, which finite groups admit an oriented regular representation (ORR)? We show that every finite non-solvable group admits an ORR, and provide a tool that may prove useful in showing that some families of finite solvable groups admit ORRs. We also completely characterize all finite groups that can be generated by at most three elements, according to whether or not they admit ORRs.