Graphical Frobenius representations of non-abelian groups

Abstract

A group G has a Frobenius graphical representation (GFR) if there is a simple graph Γ whose full automorphism group is isomorphic to G acting on the vertices as a Frobenius group. In particular, any group G with a GFR is a Frobenius group and Γ is a Cayley graph. By very recent results of Spiga, there exists a function f such that if G is a finite Frobenius group with complement H and |G| > f(|H|) then G admits a GFR. This paper provides an infinite family of graphs that admit GFRs despite not meeting Spiga’s bound. In our construction, the group G is the Higman group A(f, q₀) for an infinite sequence of f and q₀, having a nonabelian kernel and a complement of odd order.