The Historical Research Of Presentation Of Group And Related Fields

The concept of presentation of group appeared for the first time in 1882 in the German mathematician Dyck's “Gruppentheoretische Studien”,which is the first abstract definition of group.Based on the concept of presentation,many mathematical branches have influenced each other in the last hundred years.In this process,Dehn created a new mathematical branch combinatorial group theory,combining the ideas of two great mathematicians Poincaré and Hilbert.More and more seminars and research groups involved this field.The themes of this field often appeared on the International Conference of Mathematicians.This field occupies an important position in modern mathematics.Basing on collecting,reading,organizing and analyzing a lot of the original literature,historical research literature,classical mathematics textbooks,using of historical research methods,with the entry points of important historical turning points,mathematicians and their contributions in the process of the development of the presentation of group,do historical analysis of the different developments of the presentation of group,which was directed respectively by Burnside and Dehn.The former process follows two main lines: time and Burnside-related problem.The latter process follows the two main lines: time and academic heritage.The main contents are as follows:1.The origin of the presentation of group was analyzed from the relations between the presentation and polyhedral group,discrete group.Based on the work of Dyck,the nature and evolution of the concept of the presentation and the relationship between the free group were analyzed.The influences of Cayley and Klein on Dyck were expounded.Through the statistical analysis of the literature and terminology,the development of the presentation of group in China was introduced.2.Burnside's academic background was expounded,focusing on analyzing Burnside's contribution to this field.Burnside's detailed biography was investigated.Burnside's graphical representation of a group was analyzed,comparing with Cayley's.3.The process of Burnside problem,General Burnside problem and Restricted Burnside problem were expounded.The influence of this filed was analyzed,basing on the statistical analysis of the International Conference of Mathematicians which involved the filed.The link among the former Soviet mathematicians who made important contributions to this field was introduced.4.The important ideas and the concept of fundamental group in Poincaré's “Analysis Stius” were analyzed.Tieze's main contribution to the development of the presentation of group in “über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten” was expounded.The brief biography of Tietze was introduced.To Dehn's problem and group diagram as the focus,Dehn's contribution to this field was explored.5.The contributions to this field made by Dehn's students Nielsen and Magnus,and their influences to the development and spread of this filed were analyzed.The biography of Magnus was investigated,who have made important contributions to the development and spread of this filed.6.The appearance of Reidemeister-Schreier method,free products,free products with amalgamations was analyzed,basing on investigating the work of Reidemeister and Schreier.Their influences to the Follow-up development of the field were expounded.7.The fields related the presentation of group were briefly stated,such as,the theory of small cancellation,geometrical group theory,the theory of cryptograph.