Direct product and uniqueness of automorphism groups of graphs

We consider the direct product of permutation groups in which both factors are automorphism groups of graphs and ask when the resulting permutation group is again an automorphism group of a graph. We prove that this is always the case except for when both the factors are isomorphic as permutation groups, transitive and unique in the following sense. A permutation group A is called unique (as an automorphism group of a graph) if up to graph isomorphism there is exactly one graph whose automorphism group is A. In the second part of the paper we describe all unique transitive permutation groups of prime degree and prove some other results for composite degree. (C) 1999 Elsevier Science B.V. All rights reserved.