Graphical Frobenius representations of non-abelian groups

TL;DR

This paper demonstrates that an infinite family of non-abelian groups, specifically Higman groups, possess graphical Frobenius representations, expanding understanding of symmetries in graph automorphisms.

Contribution

It proves that Higman groups have graphical Frobenius representations for infinitely many parameters, addressing an open question about non-abelian Frobenius kernels.

Findings

Higman groups have GFR for infinitely many parameters

Existence of GFR for non-abelian 2-groups confirmed

Expands class of groups known to have Frobenius graphical representations

Abstract

A group $G$ has a Frobenius graphical representation (GFR) if there is a simple graph $\varGamma$ whose full automorphism group is isomorphic to $G$ and it acts on vertices as a Frobenius group. In particular, any group $G$ with GFR is a Frobenius group and $\varGamma$ is a Cayley graph. The existence of an infinite family of groups with GFR whose Frobenius kernel is a non-abelian $2$-group has been an open question. In this paper, we give a positive answer by showing that the Higman group $A(f,q_0)$ has a GFR for an infinite sequence of $f$ and $q_0$.