On groups all of whose Haar graphs are Cayley graphs

Abstract

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple bipartite graph $\Sigma$ such that ${\rm Aut}(\Sigma)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(\Sigma)$ and the $H$-orbits are equal to the bipartite sets of $\Sigma$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs (a group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian). As an application, it is also shown that every non-solvable group has a Haar graph which is not a Cayley graph.