On Haar digraphical representations of groups

Abstract

In this paper we extend the notion of digraphical regular representations in the context of Haar digraphs. Given a group $G$, a {\em Haar digraph} $\Gamma$ over $G$ is a bipartite digraph having a bipartition $\{X,Y\}$ such that $G$ is a group of automorphisms of $\Gamma$ acting regularly on $X$ and on $Y$. We say that $G$ admits a {\em Haar digraphical representation} (HDR for short), if there exists a Haar digraph over $G$ such that its automorphism group is isomorphic to $G$. In this paper, we classify finite groups admitting a HDR.