群论思想在代数组合中的渗透与应用  

The Penetration and Application of Group Theory Thought in Algebraic Combinatatorics

作　　者：闫焱[1,2] 邓明立 YAN Yan;DENG Mingli(School of Mathematical Sciences,Hebei Normal University,Shijiazhuang 050024,China;College of Science,North China University of Science and Technology,Tangshan 063210,China)

机构地区：[1]河北师范大学数学科学学院,石家庄050024 [2]华北理工大学理学院,唐山063210

出　　处：《自然科学史研究》2023年第2期249-263,共15页Studies in The History of Natural Sciences

基　　金：国家自然科学基金资助项目“结构数学在现代代数学中的渗透与应用”(项目编号:12171137)、“代数几何及其相关领域的历史研究”(项目编号:12271138)。

摘　　要：代数组合是组合数学与抽象代数的交叉学科,被国际数学界视为“没有群的群论”。本文从结构数学的视角,选取了结合方案、本原结合方案、西罗定理、特征标理论、对偶性等代数组合中的代表性问题进行剖析,阐释了群论思想与方法对代数组合的渗透与应用。通过探寻代数组合和群论的共同逻辑基础、共同概念基础和共同方法基础,呈现出群论在代数组合发展史中的重要地位与促进作用。algebra,is regarded as“group theory without the group”in the international mathematics world.In this paper some representative problems in algebraic combinatorics,such as association schemes,primitive association schemes,Sylow theorems,character theory and duality are selected for analysis from the perspective of structural mathematics.The penetration and application of the theory and methods of group theory to algebraic combinatorics are explained.By finding the common logic basis,common concept basis and common method basis of algebraic combinatorics and group theory,the important position and promoting role of group theory in the history of algebraic combinatorics is presented.

关 键 词：代数组合 群论 结合方案 有限群 结构数学 

分 类 号：N09[自然科学总论—科学技术哲学]