有限群论及其相关领域的历史研究

Structural mathematics is the mainstream of the mathematics in 20th century. As an important part of it, the theory of finite groups has become one of the most popular and active branches in the mathematics in recent years. On the basis of excavating, selecting and studying a large number of original documents, the project tries to make a research on the history of the theory of finite groups and its related areas from three aspects: fundamental concepts, relative theory and their intersectant applications, as well as crucial mathematicians. Specifically, it mainly includes the following substantial contents: dissecting the main thread of thought throughout the finite Abel group, quotient group, Mathieu group and other fundamental concepts, tracing their origin and evolution, and giving some helpful hints for the historical research on the other mathematical concepts; exploring the history and current situation of the classical group, the representation theory of finite groups, and the classification of finite simple groups, which have significant influence on the finite group theory, and analyzing the meeting point of the finite group theory and other subjects; writing biographies for the mathematicians who had made outstanding contributions in the development of finite group theory, and exploring their background and motivation, such as W. Burnside and J.G.Thompson. Finite group theory is the best-developed theory in abstract algebra. If the theory of finite groups and its related areas can be studied and summarized from a historical perspective, it will provide a good case for the research on the structural mathematics in 20th century, and have significant theoretical and practical value.

20世纪数学的主流是结构数学，有限群论作为结构数学的一个重要组成部分，是近年来研究最多、最活跃的数学分支之一。本项目在深入挖掘、整理和研读大量原始文献的基础上，从基本概念、相关理论与交叉应用、重要人物三个方面对有限群论及其相关领域的历史展开研究。剖析有限Abel群、商群、Mathieu群等基本概念的思想发展主线，追溯其产生与演化过程，为其他数学概念的历史考察提供借鉴；探讨有限群中具有重大影响的典型群理论、有限群表示论、有限单群分类的历史与现状，并在此基础上分析有限群论与其他学科的契合点；对W.Burnside、J.G.Thompson等在有限群论中做出突出贡献的数学家树碑立传，探究其创造的背景与动因，启发教育后人。有限群论是抽象代数学中发展得最为完整的一门理论，从历史角度对它及相关领域进行系统的梳理与总结，必将对20世纪结构数学的研究提供一个很好的案例，具有重大理论价值和现实意义。

有限群论是从实践中发展出来的一门比较抽象的学科，是代数学最基本的内容之一。对有限群论的历史研究是认识结构数学历史发展的一个案例。本项目围绕有限群论，从基本概念、相关理论与交叉应用、重要人物三个方面进行了研究，顺利完成研究计划任务，共发表论文9篇。主要研究内容和结果为：（1）在对数论中二元二次型早期发展进行研究的基础上，特别分析了C. F. Gauss的型的合成理论，揭示了有限Abel群在数论中产生的历史根源，它虽然没有直接催生群论产生，但对抽象群概念的形成起到巨大推动作用。（2）以有限单群为核心，以美国代数学发展为主线，探讨了美国代数学崛起的主要因素，着重分析了群论大家F. N. Cole、G. A. Miller和L. E. Dickson在有限单群和典型群等方面所做的工作，对有限单群分类的早期列举研究具有重要意义。（3）考察了E. T. Bell在两次世界大战期间对加州理工学院数学的影响，他在教师队伍建设和人才培养方面做出重大贡献。此外还对W. Burnside的传记进行了分析和总结，研究了他的生平以及在有限单群分类和有限群表示论方面的贡献。（4）将对有限群论与域论的历史研究结果应用到高校近世代数教学中，揭示数学概念的发现、发展过程，推展教学改革，实现数学史的教育功能。