结合方案上Sylow p-子集的若干性质研究
Research on Some Properties of Sylow p-Subsets on Association Schemes
DOI: 10.12677/PM.2023.135136, PDF,    国家自然科学基金支持
作者: 张晓婷：华北理工大学理学院，河北 唐山；河北省数据科学与应用重点实验室，河北 唐山 ；闫 焱*：华北理工大学理学院，河北 唐山；河北省数据科学与应用重点实验室，河北 唐山 ；河北师范大学数学科学学院，河北 石家庄；阎 熠, 王 悦：华北理工大学理学院，河北 唐山
关键词: 结合方案；Sylow定理；本原结合方案；Sylow p-子集；Association Scheme； Sylow Theorem； Primitive Association Scheme； Sylow p-Subset
摘要: 本文主要研究结合方案上Sylow p-子集(p是素数)的若干性质，并利用Sylow p-子集的存在性判断结合方案是非本原的。此外，利用Sylow定理分析价为6和200的结合方案的数学结构，给出价为6的结合方案至少存在一个价为2和3的闭子集，价为200的结合方案至少存在一个价为2、4、5、8和25的闭子集。
Abstract: In this paper, we study some properties of Sylow p-subset (p is a prime number) on association schemes, and the existence of Sylow p-subset is used to judge that the association scheme is imprimitive. Sylow theorem is used to analyze the mathematical structure of the association schemes with valency 6 and 200. The association scheme of valency 6 has at least one closed subset with valency 2 and 3. And the association scheme of valency 200 has at least one closed subset with valency 2, 4, 5, 8 and 25.
文章引用：张晓婷, 闫焱, 阎熠, 王悦. 结合方案上Sylow p-子集的若干性质研究[J]. 理论数学, 2023, 13(5): 1333-1340. https://doi.org/10.12677/PM.2023.135136

参考文献

[1] Bose, R.C. and Shimamoto, T. (1952) Classification and Analysis of Partially Balanced Incomplete Block Designs with Two Association Classes. Journal of the American Statistical Association, 47, 151-184. [Google Scholar] [CrossRef] 
[2] Bose, R.C. and Meaner, D.M. (1959) On Linear Associa-tive Algebras Corresponding to Association Schemes of Partially Balanced Designs. The Annals of Mathematical Sta-tistics, 30, 21-38. [Google Scholar] [CrossRef] 
[3] Bannai, E. and Ito, T. (1984) Algebraic Combinatorics I (Association Schemes). Benjinamin/Cummings, Menlo Park.
[4] Zieschang, P.H. (2005) Theory of Association Schemes. Springer Monographs in Mathematics. Springer-Verlag, Berlin.
[5] Zieschang, P.H. (1996) An Algebraic Approach to Association Schemes. In: Lecture Notes in Mathematics, Vol. 1628, Springer-Verlag, Berlin. [Google Scholar] [CrossRef] 
[6] Bannai, E. (2010) Combinatorics Regarded as Pure Mathematics: The Aims of Algebraic Combinatorics. Sugaku, 62, 433-452.
[7] Rassy, M. and Zieschang, P.H. (1998) Basic Structure Theory of Association Schemes. Mathematische Zeitschrift, 227, 391-402. [Google Scholar] [CrossRef] 
[8] Hirasaka, M., Muzychuk, M. and Zieschang, P.H. (2002) A Generali-zation of Sylow’s Theorems on Finite Groups to Association Schemes. Mathematische Zeitschrift, 241, 665-672. [Google Scholar] [CrossRef] 
[9] Blau, H.I. and Zieschang, P.H. (2004) Sylow Theory for Table Algebras, Fusion Rule Algebras, and Hypergroups. Journal of Algebra, 273, 551-570. [Google Scholar] [CrossRef] 
[10] Hanaki, A. (2008) Nilpotent Schemes and Group-Like Schemes. Journal of Combinatorial Theory, 115, 226-236. [Google Scholar] [CrossRef] 
[11] French, C. and Zieschang, P.H. (2015) A Schur-Zassenhaus The-orem for Association Schemes. Journal of Algebra, 435, 88-123. [Google Scholar] [CrossRef] 
[12] Andrey, V. and Zieschang, P.H. (2022) Solvable Hypergroups and a Generalization of Hall’s Theorems on Finite Solvable Groups to Association Schemes. Journal of Algebra, 594, 733-750. [Google Scholar] [CrossRef] 
[13] 韩士安, 林磊. 近世代数[M]. 第2版. 北京: 科学出版社, 2009.
[14] 徐明曜, 曲海鹏. 有限p群[M]. 北京: 北京大学出版社, 2010.

为你推荐