Combinatorial structures with regular automorphism groups.

The concept of an automorphism group of a combinatorial structure is a fundamental concept in the cross-section of Combinatorics and Group Theory.
Finding the automorphism group of a specific structure is a notoriously hard problem whose
general complexity has not been resolved but it is believed to be exponential.
In the
talk, I will address the opposite problem of constructing a combinatorial structure
for a given automorphism group.
The left (or right) regular action of a group on itself is one of the most natural group
actions to consider. The focus of our talk will be on combinatorial structures whose
full automorphism groups act regularly on their sets of vertices. Equivalently, we
discuss finite groups whose element sets admit the introduction of a combinatorial
structure whose full automorphism group consists solely of the automorphisms induced by the multiplication by the elements of the underlying group.
Such structures can be thought of as combinatorial representations of the corresponding groups.
Previous results on this topic include the
classification of graphical regular representations (graphs with regular automorphism groups), classification of digraphical regular representations (directed graphs with regular automorphism groups), as well as
the classification of general combinatorial structures (incidence structures) with regular automorphism groups. We generalize these results to the class of k-hypergraphs
which are incidence structures with all blocks of size k.