On groups all of whose Haar graphs are Cayley graphs

A Cayley graph of a group H is a finite simple graph Gamma such that 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 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 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.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph.