Classification of finite groups that admit an oriented regular representation

TL;DR

This paper completes the classification of finite groups admitting an oriented regular representation, confirming that nearly all non-generalised dihedral groups have an ORR, thus answering a longstanding question from 1980.

Contribution

It provides a complete classification of finite groups with ORRs, resolving a 1980 open problem and identifying 11 small exceptions.

Findings

Almost all non-generalised dihedral groups admit ORRs

11 small exceptions with orders from 8 to 64 do not admit ORRs

Confirms that generalised dihedral groups do not admit ORRs

Abstract

This is the third, and last, of a series of papers dealing with oriented regular representations. Here we complete the classification of finite groups that admit an oriented regular representation (or ORR for short), and give a complete answer to a 1980 question of Laszlo Babai: "Which [finite] groups admit an oriented graph as a DRR?" It is easy to see and well-understood that generalised dihedral groups do not admit ORRs. We prove that, with 11 small exceptions (having orders ranging from 8 to 64), every finite group that is not generalised dihedral has an ORR.