子群完备码  

Subgroup Perfect Codes

作　　者：张星 王燕[1] 曲海鹏[2] ZHANG Xing;WANG Yan;QU Hai-peng(School of Mathematics and Information Sciences,Yantai University,Yantai 264005,China;School of Mathematics and Computer Science,Shanxi Normal University,Shanxi 041000,China)

机构地区：[1]烟台大学数学与信息科学学院,山东烟台264005 [2]山西师范大学数学与计算机科学学院,山西临汾041000

出　　处：《烟台大学学报（自然科学与工程版）》2020年第2期127-130,211,共5页Journal of Yantai University(Natural Science and Engineering Edition)

基　　金：国家自然科学基金资助项目(11671347,61771019);山东省自然科学基金资助项目(ZR2017MAO22).

摘　　要：有限群G的子群H可作为群G凯莱图的完备码当且仅当H存在一个右(左)陪集代表系可作为群G的广义凯莱子集.从这一基本理论出发,本文从群论角度给出了子群可作为完备码的充要条件,得出了奇数阶群的正规子群,奇数阶的正规子群以及指数为奇数的正规子群均可作为凯莱图的完备码.对于偶数阶群的子群完备码,本文分别给出了广义四元数群和广义二面体群的子群可作为完备码的充要条件.A subgroup H of a finite group G is a perfect code of some Cayley graph of G if and only if there is a right(left)transversal of G that can be a generalized Cayley subset of G.Starting from this basic fact,we give necessary and sufficient conditions for subgroups to be perfect codes.We reprove the known results that the normal subgroups of odd-order groups,the normal subgroups of odd orders and the normal subgroups of odd indices can be perfect codes of Cayley graphs.For subgroup perfect codes of even-order groups,we give necessary and sufficient conditions for a subgroup to be a perfect code in some Cayley graph of generalized quaternion groups and generalized dihedral groups,respectively.

关 键 词：完备码 凯莱图 陪集代表系 子群 

分 类 号：O157.6[理学—数学]