【摘要】：群的呈示這個概念首次出現(xiàn)于德國數(shù)學(xué)家代克1882年《群論研究》的論文中,是群最早的抽象定義.以群的呈示這個概念為基礎(chǔ),近百年來很多數(shù)學(xué)分支交叉發(fā)展、互相影響.在這個過程中,戴恩結(jié)合兩位偉大的數(shù)學(xué)家龐加萊和希爾伯特的思想,創(chuàng)始了組合群論這個新的數(shù)學(xué)分支.涉及到該領(lǐng)域的研討會議、科研團體越來越多,國際數(shù)學(xué)家大會上也常出現(xiàn)此領(lǐng)域的主題,可見該領(lǐng)域在近現(xiàn)代數(shù)學(xué)中占有重要位置.本論文在搜集、閱讀、整理、分析大量的原始文獻、歷史研究文獻、經(jīng)典數(shù)學(xué)著作的基礎(chǔ)上,綜合運用數(shù)學(xué)史的研究方法,以群的呈示發(fā)展過程中的重要歷史轉(zhuǎn)折點、卓越的數(shù)學(xué)家及其貢獻為切入點,對群的呈示在分別由伯恩賽德與戴恩等人引導(dǎo)的方向上的發(fā)展進行了歷史分析.群的呈示沿前者的發(fā)展過程中,時間與伯恩賽德相關(guān)問題構(gòu)成其兩條主線.群的呈示沿后者的發(fā)展過程中,時間與學(xué)術(shù)傳承構(gòu)成其兩條線索.本論文的主要內(nèi)容如下:1.從群的呈示與多面體群和離散群之間的聯(lián)系,分析了群的呈示的思想起源.以代克的工作為基礎(chǔ),從群的呈示這個概念的本質(zhì)、演變發(fā)展、與自由群之間的關(guān)系等方面,深入細(xì)致地分析了群的呈示的概念.闡述了凱萊和克萊因?qū)Υ说挠绊?通過對文獻、術(shù)語的統(tǒng)計分析,介紹了群的呈示在中國的發(fā)展情況.2.以對伯恩賽德在此領(lǐng)域的貢獻進行深入細(xì)致的分析為中心,對伯恩賽德工作的背景進行了闡述.對伯恩賽德的生平進行了詳細(xì)考察.在與凱萊的群的圖表示對比分析的基礎(chǔ)上,分析了伯恩賽德的群的圖表示.3.闡述了伯恩賽德問題、廣義伯恩賽德問題、限制性伯恩賽德問題的發(fā)展.統(tǒng)計涉及該領(lǐng)域的國際數(shù)學(xué)家大會,分析了該領(lǐng)域產(chǎn)生的影響.介紹了在該領(lǐng)域做出重要貢獻的前蘇聯(lián)數(shù)學(xué)家之間的聯(lián)系.4.分析了龐加萊的《位置分析》中的重要思想和基本群的概念.闡述了梯采的《多維流形的拓?fù)洳蛔兞俊分袑θ旱某适镜陌l(fā)展的主要貢獻.簡單介紹了梯采的生平.以戴恩問題和群的圖為重點,對戴恩在此領(lǐng)域的貢獻進行了深入細(xì)致的探究.5.分析了戴恩的學(xué)生尼爾森與馬格努斯對此領(lǐng)域的貢獻,以及他們對此領(lǐng)域的傳播發(fā)展的影響.馬格努斯對組合群論的發(fā)展和傳播都有突出貢獻,本論文對他的生平進行了詳細(xì)考察.6.在對瑞德邁斯特和施萊爾的工作進行考察的基礎(chǔ)之上,分析了瑞德邁斯特-施萊爾方法、自由積、融合自由積的提出.闡述了他們在此領(lǐng)域后續(xù)發(fā)展中的影響.7.通過小消去理論、幾何群論、編碼理論的例證,簡要闡述了與群的呈示相關(guān)的其他領(lǐng)域的情況.
[Abstract]:The presentation of group is the earliest abstract definition of group, which first appeared in the paper "study of Group Theory" by German mathematician Dek in 1882. Based on the concept of group presentation, many branches of mathematics have developed and interacted with each other in the last hundred years. In the process, Dayne combined the ideas of two great mathematicians, Poincare and Hilbert, to create the new branch of combinatorial group theory. There are more and more scientific research organizations involved in this field, and the subject of this field is often appeared in the International Congress of mathematicians, which shows that this field plays an important role in modern mathematics. On the basis of collecting, reading, sorting out and analyzing a large number of original documents, historical research documents and classical mathematical works, this paper synthetically applies the research methods of mathematical history to show an important historical turning point in the course of the development of group presentation. Excellent mathematicians and their contributions as the starting point, the presentation of the group in the direction led by Burnside and Dayne, etc., respectively, the historical analysis of the development. In the course of the development of group presentation, the related problems of time and Burnside form its two main lines. In the course of the development of the group, time and academic heritage constitute its two clues. The main contents of this thesis are as follows: 1. From the relation between the presentation of groups and polyhedron groups and discrete groups, the origin of the ideas of presentation of groups is analyzed. Based on Deke's work, the concept of group presentation is analyzed in detail from the aspects of the essence, evolution and development of the concept, and the relationship between the group and the free group. The influence of Kelley and Klein on Deke is expounded. Through the statistical analysis of literature and terminology, this paper introduces the development of group presentation in China. 2. Based on the detailed analysis of Burnside's contribution in this field, the background of Burnside's work is expounded. The life of Burnside was examined in detail. On the basis of comparative analysis of graph representation of group with Kelley, the graph representation of group of Burnside is analyzed. 3. 3. The development of Burnside problem, generalized Burnside problem and restricted Burnside problem are expounded. The international Congress of mathematicians involved in this field analyzes the impact of the field. This paper introduces the relationship between the mathematicians of the former Soviet Union who have made important contributions in this field. 4. The important ideas and the concept of basic group in Poincare's position Analysis are analyzed. The main contributions to the development of the presentation of groups in the topological invariants of Multidimensional Manifolds are described. The life history of ladder mining is introduced briefly. With the focus on Dayne problem and group map, Dayne's contribution in this field has been explored in detail. 5. The contribution of Dayne's students Nielsen and Magnus to this field and their influence on the communication and development of this field are analyzed. Magnus has made outstanding contributions to the development and dissemination of combinatorial group theory. Based on the investigation of the work of Redmeister and Schlyle, this paper analyzes the presentation of the Redmeister Schlyle method, the free product and the fusion free product. This paper expounds their influence in the follow-up development in this field. 7. 7. Through the examples of small elimination theory, geometric group theory and coding theory, this paper briefly describes other fields related to the presentation of groups.
【學(xué)位授予單位】：河北師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O152

【共引文獻】

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