Representing Finite Groups as Regular Automorphism Groups of Combinatorial Structures

Abstract

Given a regular action of a finite group \(G\) on a set \(V\), we consider the problem of the existence of an incidence structure \(\mathcal{I} = (V, \mathcal{B})\) on the set \(V\) whose full automorphism group \(Aut(\mathcal{I})\) is the group \(G\) in its regular action. Using results on graphical and digraphical regular representations \(([2,7], [1])\), we show the existence of such an incidence structure for all but four small finite groups.