Finite groups admitting an oriented regular representation

In this paper, we investigate finite groups admitting an oriented regular representation and we give a partial answer to a 1980 question of Lazslo 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, apart from C3 2 and C3×C2 3, every finite group, which is neither a generalised dihedral group nor a 2-group, has an ORR. In particular, the classification of the finite groups admitting an ORR is reduced to the class of 2-groups. We also give strong structural conditions on finite 2-groups not admitting an ORR. Finally, based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting an ORR.