On \(p\)-radical covers of pentavalent arc-transitive graphs

Abstract

Let \(\Gamma\) be a finite connected pentavalent graph admitting a nonabelian simple arc-transitive automorphism group \(T\) and soluble vertex stabilizers. Let \(p>|T|_{2}\) be an odd prime and \((p,|T|)=1\), where \(|T|_{2}\) is the largest power of 2 dividing the order \(|T|\) of \(|T|\). Then we prove that there exists a \(p\)-radical cover \(\widetilde{\Gamma}\) of \(\Gamma\) such that the full automorphism group \(\text{Aut}(\widetilde{\Gamma})\) of \(\widetilde{\Gamma}\) is equal to \(O_{p}(\text{Aut}(\widetilde{\Gamma})).T\) and the covering transformation group is \(O_{p}(\text{Aut}(\widetilde{\Gamma}))\), where \(O_{p}(\text{Aut}(\widetilde{\Gamma}))\) is the \(p\)-radical of \(\text{Aut}(\widetilde{\Gamma})\).