On Automorphisms of Haar graphs of Abelian Groups

Let G be a group and S\subseteq G. A Haar graph of G with connection set S has vertex set \Z_2\times G and edge set \{(0,g)(1,gs):g\in G{\rm\ and\ }s\in S\}. Haar graphs are then natural bipartite analogues of Cayley digraphs. We first examine the relationship between the automorphism group of a Cayley digraph of G with connection set S and a Haar graph of G with connection set S. We establish that the automorphism group of a Haar graph contains a natural subgroup isomorphic to the automorphism group of the corresponding Cayley digraph. In the case where G is abelian, we then give four situations in which the automorphism group of the Haar graph can be larger than the natural subgroup corresponding to the automorphism group of the Cayley digraph together with a specific involution, and analyze the full automorphism group in each of these cases. As an application, we show that all s-transitive Cayley graphs of generalized dihedral groups have a quasiprimitive automorphism group, can be “reduced” to s-arc-transitive graphs of smaller order, or are Haar graphs of abelian groups whose automorphism groups have a particular permutation group theoretic property.