有限单群分类的历史研究
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摘要
群是近代数学的基本结构之一,而有限群则是整个群论的核心。正如素数是正整数乘法的“原子”或“积木块”一样,有限单群宛如有限群的“原子”或“积木块”。这样一来,对有限群的研究便分为两大部分:一是确定出所有有限单群;二是探索有限群如何由这些单群结合而成。前一部分乃是数学中绝无仅有的一项大工程——有限单群分类。
本文在阅读大量专业文献与历史文献的基础上,以有限单群的类型为纲,以时间为轴,将对比法、关键点剖析法、分类法以及列表法相综合,对有限单群分类的历史进行了系统的梳理与总结。主要结果如下:
1.考察了有限单群的产生背景以及它们在有限群论中占有至关重要地位的原因,揭示出对有限单群分类的研究是历史的产物。
2.系统分析了若尔当和迪克森研究典型群的背景与方法,在详细考察原始文献的基础上,对二人工作进行纵横对比,揭示出他们的思想传承关系。
3.在对谢瓦莱关于谢瓦莱群以及后人关于扭型谢瓦莱群的工作进行研究的基础上,揭示出一般的、统一的方法才是从本质上解决问题的关键,在一定程度上反衬出19世纪末霍尔德、米勒等人所采用的列举法的局限性。
4.在查阅大量原始文献的基础上,系统总结了26个散单群的发现与确定过程,揭示了它们彼此之间的密切联系,以期读者能够从中借鉴到研究问题的思路与途径。
5.深入研究了最早的散单群——马蒂厄群的历史,考察了它们的产生背景、构造方法及重大意义,强调了在一段时期内问题意识所产生的深刻影响。
6.作为有限单群中最大的散单群,大魔群自20世纪70年代就以其复杂的结构和迷人的性质受到广泛关注。本文考察了人们对大魔群求索与认知的过程,分析了它与数学其他学科乃至物理学的关系,展现了它的广阔发展前景和重要意义。
7.介绍了高林斯坦为分类所有有限单群而制定的16步计划,阐述了它的主要内容及进展情况,揭示出团结协作已成为当今社会进步的强劲推动力,成为顺利解决问题的一大方法。
8.挖掘了分类定理最初证明中存在的弊端,提炼并概括出对分类定理进行二代证明的必要性与可行性,通过查阅最新研究资料,详细介绍了目前正在进行的GLS计划。
9.概述了有限单群分类在数学内外所产生的重大影响,希望从事群论应用研究的读者能够有所借鉴。
10.鉴于布饶尔在有限单群分类中做出的重大贡献,全面系统地评述了布饶尔的生平、思想、工作以及对我国数学产生的影响(附录1)。
Groups are one of the fundamental structures in modern mathematics, and finite groups are the core of the whole group theory. Along from being analogous with prime numbers for the multiplication of positive integers, finite simple groups serve as the“atoms”or“building blocks”for finite groups. As a result, the study on finite groups falls naturally into two parts: to determine all the finite simple groups and to find how finite groups are constructed by simple groups. In fact, the first part is a peerless and magnificent project in mathematics, i.e. the classification of finite simple groups.
According to the chronological order and the types of finite simple groups, this dissertation tries to summarize and study the history of the classification of finite simple groups systematically, and in the process of which the methods of the comparison, analysis to the key points, classification and tabulation play an important role. The main results are as follows:
1. This dissertation reviews the background of the appearance of finite simple groups, examines the reasons why finite simple groups are very important for finite groups, and points out it is the product of the history to study the classification of finite simple groups.
2. C.Jordan and L.E.Dickson made outstanding contributions to classical groups. This dissertation analyzes their research background and methods systematically. On the basis of original literatures, the author compares Jordan’s work with Dickson’s, and reveals their inheritance relationship in mathematical ideas.
3. Based on the study to the prominent achievements on Chevalley groups and twisted Chevalley groups acquired by Chevalley and others respectively, this dissertation points out that the general and unified approaches are the key to finish the classification of finite simple groups. Accordingly, we can realize the limits of the enumeration used by O.H?lder, G.A.Miller and others.
4. There are close connections among 26 sporadic simple groups. Based on abundance of original literatures, this dissertation summarizes the process of the detection and determination of these sporadic groups. I hope that it can provide an effective way for the readers to consider and solve the general problems.
5. This dissertation studies the history of the oldest sporadic groups, Mathieu groups, including their background, construction, significance and so on. The author emphasizes that problem awareness can influence our study profoundly in some period. 6. Since 1970s,the monster, which is the largest sporadic simple group, has drawn the
society’s great attention because of its intricate structure and charming properties. This dissertation examines the process of the pursuit of it and the comprehension to it, analyzes the relations between it and other disciplines in mathematics and physics, and points out its broad prospects and significance.
7. This dissertation formulates the 16-step program made by D.Gorenstein to classify all the finite simple groups, and examines its main contents and development. It indicates that the collaboration has become a strong driving force for the whole society to go ahead, and has been an important method to solve the problems successfully.
8. In view of the disadvantages existing in the proof of the theorem of the classification, this dissertation analyzes the necessity and feasibility to give a second proof, and elaborates on the GLS Project according to the latest research information.
9. The finish of the classification of finite simple groups has profound influence on mathematics and other fields. The author hopes that this dissertation can give some helpful hints for the readers who study the applications of groups.
10. Given the great achievements acquired by R.D.Brauer in the process of classifying finite simple groups, this dissertation reviews Brauer’s life, ideas, work, as well as the impact on Chinese mathematics comprehensively and systematically (appendix 1).
本文在阅读大量专业文献与历史文献的基础上,以有限单群的类型为纲,以时间为轴,将对比法、关键点剖析法、分类法以及列表法相综合,对有限单群分类的历史进行了系统的梳理与总结。主要结果如下:
1.考察了有限单群的产生背景以及它们在有限群论中占有至关重要地位的原因,揭示出对有限单群分类的研究是历史的产物。
2.系统分析了若尔当和迪克森研究典型群的背景与方法,在详细考察原始文献的基础上,对二人工作进行纵横对比,揭示出他们的思想传承关系。
3.在对谢瓦莱关于谢瓦莱群以及后人关于扭型谢瓦莱群的工作进行研究的基础上,揭示出一般的、统一的方法才是从本质上解决问题的关键,在一定程度上反衬出19世纪末霍尔德、米勒等人所采用的列举法的局限性。
4.在查阅大量原始文献的基础上,系统总结了26个散单群的发现与确定过程,揭示了它们彼此之间的密切联系,以期读者能够从中借鉴到研究问题的思路与途径。
5.深入研究了最早的散单群——马蒂厄群的历史,考察了它们的产生背景、构造方法及重大意义,强调了在一段时期内问题意识所产生的深刻影响。
6.作为有限单群中最大的散单群,大魔群自20世纪70年代就以其复杂的结构和迷人的性质受到广泛关注。本文考察了人们对大魔群求索与认知的过程,分析了它与数学其他学科乃至物理学的关系,展现了它的广阔发展前景和重要意义。
7.介绍了高林斯坦为分类所有有限单群而制定的16步计划,阐述了它的主要内容及进展情况,揭示出团结协作已成为当今社会进步的强劲推动力,成为顺利解决问题的一大方法。
8.挖掘了分类定理最初证明中存在的弊端,提炼并概括出对分类定理进行二代证明的必要性与可行性,通过查阅最新研究资料,详细介绍了目前正在进行的GLS计划。
9.概述了有限单群分类在数学内外所产生的重大影响,希望从事群论应用研究的读者能够有所借鉴。
10.鉴于布饶尔在有限单群分类中做出的重大贡献,全面系统地评述了布饶尔的生平、思想、工作以及对我国数学产生的影响(附录1)。
Groups are one of the fundamental structures in modern mathematics, and finite groups are the core of the whole group theory. Along from being analogous with prime numbers for the multiplication of positive integers, finite simple groups serve as the“atoms”or“building blocks”for finite groups. As a result, the study on finite groups falls naturally into two parts: to determine all the finite simple groups and to find how finite groups are constructed by simple groups. In fact, the first part is a peerless and magnificent project in mathematics, i.e. the classification of finite simple groups.
According to the chronological order and the types of finite simple groups, this dissertation tries to summarize and study the history of the classification of finite simple groups systematically, and in the process of which the methods of the comparison, analysis to the key points, classification and tabulation play an important role. The main results are as follows:
1. This dissertation reviews the background of the appearance of finite simple groups, examines the reasons why finite simple groups are very important for finite groups, and points out it is the product of the history to study the classification of finite simple groups.
2. C.Jordan and L.E.Dickson made outstanding contributions to classical groups. This dissertation analyzes their research background and methods systematically. On the basis of original literatures, the author compares Jordan’s work with Dickson’s, and reveals their inheritance relationship in mathematical ideas.
3. Based on the study to the prominent achievements on Chevalley groups and twisted Chevalley groups acquired by Chevalley and others respectively, this dissertation points out that the general and unified approaches are the key to finish the classification of finite simple groups. Accordingly, we can realize the limits of the enumeration used by O.H?lder, G.A.Miller and others.
4. There are close connections among 26 sporadic simple groups. Based on abundance of original literatures, this dissertation summarizes the process of the detection and determination of these sporadic groups. I hope that it can provide an effective way for the readers to consider and solve the general problems.
5. This dissertation studies the history of the oldest sporadic groups, Mathieu groups, including their background, construction, significance and so on. The author emphasizes that problem awareness can influence our study profoundly in some period. 6. Since 1970s,the monster, which is the largest sporadic simple group, has drawn the
society’s great attention because of its intricate structure and charming properties. This dissertation examines the process of the pursuit of it and the comprehension to it, analyzes the relations between it and other disciplines in mathematics and physics, and points out its broad prospects and significance.
7. This dissertation formulates the 16-step program made by D.Gorenstein to classify all the finite simple groups, and examines its main contents and development. It indicates that the collaboration has become a strong driving force for the whole society to go ahead, and has been an important method to solve the problems successfully.
8. In view of the disadvantages existing in the proof of the theorem of the classification, this dissertation analyzes the necessity and feasibility to give a second proof, and elaborates on the GLS Project according to the latest research information.
9. The finish of the classification of finite simple groups has profound influence on mathematics and other fields. The author hopes that this dissertation can give some helpful hints for the readers who study the applications of groups.
10. Given the great achievements acquired by R.D.Brauer in the process of classifying finite simple groups, this dissertation reviews Brauer’s life, ideas, work, as well as the impact on Chinese mathematics comprehensively and systematically (appendix 1).
引文
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