On the existence of graphical Frobenius representations and their asymptotic enumeration

摘要

We give a complete answer to the GFR conjecture, proposed by Conder, Doyle, Tucker and Watkins: "All but finitely many Frobenius groups F=N (sic) H with a given complement H have a GFR, with the exception when vertical bar H vertical bar is odd and N is Abelian but not an elementary 2-group". Actually, we prove something stronger, we enumerate asymptotically GFRs; we show that, besides the exceptions listed above, as vertical bar N vertical bar tends to infinity, the proportion of GFRs among all Cayley graphs over N containing F in their automorphism group tends to 1. (C) 2019 Elsevier Inc. All rights reserved.