Representing Finite Groups As Regular Automorphism Groups Of Combinatorial Structures.
Robert Jajcay · 2002
Given a regular action of a finite group G on a set V , we ask (and answer) the question of the existence of an incidence structure I = (V; B) on the set V whose full automorphism group Aut(I) is the group G in its regular action. Additional conditions on I also allow us to refine the original problem to the class of hypergraphs. Using results on graphical and digraphical regular representations ([4], [1]), we show the existence of a desired combinatorial structure (incidence structure or hypergraph) for all but a finite list of finite groups. 1 Introduction In our paper, we take up the problem of classifying finite groups G for which there exists an incidence structure I or a hypergraph H such that the full automorphism group of the combinatorial structure is the group G acting regularly on the vertices of the structure. We show the existence of such structures for all but very few finite groups of small orders. This problem is closely related to the so called GRR-problem -- the p...