On the existence of Frobenius digraphical representations
Summary: A Frobenius group is a transitive permutation group that is not regular and such that only the identity fixes more than one point. A digraphical, respectively graphical, Frobenius representation, DFR and GFR for short, of a Frobenius group \(F\) is a digraph, respectively graph, whose automorphism group as a group of permutations of the vertex set is \(F\). The problem of classifying which Frobenius groups admit a DFR and GFR has been proposed by Mark Watkins and Thomas Tucker and is a natural extension of the problem of classifying which groups that have a digraphical, respectively graphical, regular representation. In this paper, we give a partial answer to a question of Mark Watkins and Thomas Tucker concerning Frobenius representations [\textit{J. K. Doyle} et al., ``Graphical Frobenius representations'', J. Algebr. Comb. (to appear; \url{doi:10.1007/s10801-018-0814-6})]: ``All but finitely many Frobenius groups with a given Frobenius complement have a DFR''.
scientific article; zbMATH DE number 6873175
scientific article