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组合群论是群论的重要分支之一，戴恩是组合群论的创始人。在整理分析文献的基础上，对戴恩组合群论工作的背景、具体贡献和深远影响进行了论述，20世纪初期，戴恩通过基本群的呈示对拓扑学展开研究的过程中，将拓扑学的问题抽象成群论的问题“戴恩问题”,发展了“群的图”研究方法。戴恩在三叶纽结等特殊情况下对戴恩问题的解决，鼓励着数学家继续此方向的工作。戴恩此时期的工作，不仅创立了组合群论这个数学分支，而且影响着其后续发展。
Abstract：Combinatorial group theory is one of the important branches of group theory, and Max Dehn is the founder of combinatorial group theory. Based on the analysis of the relevant literature, the paper discusses the background, specific contributions, and far-reaching implications of Max Dehn's work in combinatorial group theory. The paper reaches the following conclusions: in the early 20th century, in the process of studying topology through the presentation of fundamental groups, Max Dehn abstracted topological problems into group-theoretic problems later known as Dehn's problems and developed a method based on group diagrams. Max Dehn's solution to these problems in special cases, such as the trefoil knot, encouraged later mathematicians to pursue this line of research. Max Dehn's work during this period not only founded the branch of mathematics known as combinatorial group theory, but also influenced its subsequent development.
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(1)“曲面拓扑报告”没有公开发表,直到1987年才在《马克思·戴恩关于群论和拓扑学的论文》中被整理翻译出版,从内容上分析,其手稿完成时间要早于论文中提到的4篇论文。
(2)戴恩当时称字问题为“同一性问题”。
(3)戴恩当时称共轭问题为“转换问题”。
(4)“群论报告”没有公开发表,直到1987年才在《马克思·戴恩关于群论和拓扑学的论文》中被整理翻译出版,从内容上分析,其手稿完成时间要早于论文中提到的4篇论文。
基本信息:
中图分类号:O152
引用信息:
[1]王艳,邓明立.戴恩的组合群论思想研究[J].中国科技史杂志,2026,47(02):223-233.
基金信息:
国家自然科学基金资助项目“结构数学在现代数学中的渗透与应用”(项目编号：12171137)
2026-06-15
2026-06-15