广西师范大学学报(自然科学版) ›› 2015, Vol. 33 ›› Issue (4): 68-72.doi: 10.16088/j.issn.1001-6600.2015.04.012

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陪集图的同构与自同构

化小会, 陈利   

  1. 河南师范大学数学与信息科学学院,河南新乡453007
  • 收稿日期:2015-07-04 出版日期:2015-12-25 发布日期:2018-09-21
  • 通讯作者: 化小会(1979—),女,河南许昌人,河南师范大学副教授,博士。E-mail: xhhua@outlook.com
  • 基金资助:
    国家自然科学基金资助项目(11301159);河南省教育厅科学与技术重点项目(13A110543);河南师范大学青年教师基金项目(2012QK01)

Isomorphisms and Automorphisms of Coset Graphs

HUA Xiao-hui, CHEN Li   

  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang Henan 453007, China
  • Received:2015-07-04 Online:2015-12-25 Published:2018-09-21

摘要: 令G是一个有限图,H是G的无核子群,D是形如HgH(g$\notin$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。

关键词: 弧传递图, 陪集图, Cayley图

Abstract: Let G be a finite group, H a core-free subgroup of G and D a union of several double-cosets of the form HgH with g$\notin$H such that D=D-1. Let Cos(G, H, D) be the coset graph of G with respect to H and D, and let A=Aut(Cos(G, H, D)). Denote the right multiplication action of G on Ω=[G:H] by RH (G), the set of right cosets of H in G, and denote the automorphism of G induced by the conjugate of g∈G on G by σ(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 any σ∈SΩ with 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.

Key words: rc-transitive graph, coset graph, Cayley graph

中图分类号: 

  • O157.5
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