k-Hypergraphs with regular automorphism groups

Abstract

Regular representations of finite groups, as introduced by Cayley,
are among the most natural permutation representations of finite
groups. Thus, the question which regular representations appear as
full automorphism groups of combinatorial structures has been 
addressed and resolved for several classes of structures, notably for
graphs (where they are called Graphical Regular Representations,
GRR's), digraphs (Digraphical Regular Representations, DRR's)
as well as for hypergraphs allowing for hyperedges of 
varying sizes.
In the present paper, we focus on $k$-hypergraphs, which are hypergraphs 
in which all hyperedges are of the same size $k$, and address the question
which $k$-regular hypergraphs possess full automorphism groups
acting regularly on the vertices. We rely on the concept of a Cayley
hypergraph (defined here) and show that all sufficiently large finite
groups admit a regular representation as the full automorphism group
of a $3$-hypergraph.