ON THE ORDER OF ARC-STABILIZERS IN ARC-TRANSITIVE GRAPHS

摘要

Let p be a prime. We say that a transitive action of a group L on a set  is p-sub-regular if there existnx; y 2  such that hLx ; Ly i D L and LYnxnu0018 D Zp, where Y D yLx is the orbit of y under Lx . Our main resultnis that if 0 is a G-arc-transitive graph and the permutation group induced by the action of Gv on 0.vis p-sub-regular, then the order of a G-arc-stabilizer is equal to ps 1 where s u0014 7, s 6D 6, and moreover,nif p D 2, then s u0014 5. This generalizes a classical result of Tutte on cubic arc-transitive graphs as well asnsome more recent results. We also give a characterization of p-sub-regular actions in terms of arc-regularnactions on digraphs and discuss some interesting examples of small degree.