Combinatorial structures with regular automorphism groups

The topic of our talk is a refinement of the classical result of Frucht, who has shown that for any finite group G, there exists a graph whose full automorphism group is isomorphic to G. Instead of seeking a graph whose automorphism group is isomorphic to G, we choose a specific permutation representation of G on a set V, and ask about the existence of a combinatorial structure on V whose full automorphism group is equal to G in its particular representation. If the permutation representation considered is the regular permutation representation and the combinatorial structures we consider are graphs, the question becomes the well-known questions of which finite groups are the full automorphism groups of some Cayley graph (in case of existence, such Cayley graph is called a Graphical Regular Representation for G). In our talk, we extend the question of regular representations to other classes of combinatorial structures such as incidence structures, directed graphs, designs, maps, or hypergraphs. Our presentation is based on a collaboration with Tatiana Jajcayova.