陪集图的同构与自同构

Isomorphisms and Automorphisms of Coset Graphs

令G是一个有限图,H是G的无核子群,D是形如HgH(gH)的一些双陪集的并,且满足D=D-1。记(Cos(G,H,D)表示G关于H和D的陪集图,A=Aut(Cos(G,H,D))。用RH(G)表示G在H的全体右陪集所在的集合Ω=[G:H]上的右乘置换表示,σ(g)表示g∈G通过共轭作用诱导在G上的自同构。本文不但证明了NA(RH(G))=RH(G)Aut(G,H,D)且RH(G)∩Aut(G,H,D)=I(H),其中Aut(G,H,D)={α∈Aut(G)|Hα=H,Dα=D},I(H)={σ(h)|h∈H},而且证明了Cos(G,H,D)是一个CI-图当且仅当对任意的σ∈SΩ,满足RH(G)σ≤A,必存在a∈A使得RH(G)a=RH(G)σ。作为对本文两个定理的应用,本文考虑了一类线性群上陪集图的CI-性问题及其在同构意义下的计数问题。

Let Gbe a finite group,Ha core-free subgroup of Gand Da union of several double-cosets of the form HgH with gH such that D=D-1.Let Cos（G,H,D）be the coset graph of G with respect to Hand D,and let A=Aut（Cos（G,H,D））.Denote the right multiplication action of GonΩ=[G：H]by RH（G）,the set of right cosets of H in G,and denote the automorphism of Ginduced by the conjugate of g∈Gon Gbyσ（g）.In this paper,it is shown that NA（RH（G））=RH（G）Aut（G,H,D）and RH（G）∩ Aut（G,H,D）=I（H）,where Aut（G,H,D）= {α∈Aut（G）｜Hα=H,Dα=D}and I（H）={σ（h）｜h∈H},and it is also shown that Cos（G,H,D）is a CI-graph if and only if for anyσ∈SΩwith RH（G）σ≤A,and that there exists a∈A such that RH（G）a=RH（G）σ.The CI-propety problem and isomorphism count problem of a class coset graphs on linear groups are considered in application.