Direct product and uniqueness of automorphism groups of graphs (Q1817573)

The author considers the problem of representing permutation groups by graphs. If \(\Aut(G)\) denotes the automorphism group of a graph \(G\) and \(A\equiv\Aut(G)\), then \(A\) is a representable permutation group. If \(A\) is represented by exactly one graph \(G\) (up to isomorphism), then \(A\) is called unique. The main result is the following theorem: Let \(A_1\) and \(A_2\) be representable permutation groups. Then the direct product \(A_1\otimes A_2\) is not representable iff \(A_1\) and \(A_2\) are transitive, unique and isomorphic as permutation groups. Moreover all unique transitive permutation groups of prime degree are described.