The Historical Researches Of The Early Developments Of Classical Groups

Classical groups stand for general linear group, symplectic group, unitary groupand orthogonal group. The construction and the representation of Classical groupsplay an significant role in lie groups theory, theory of several complex variables,geometry as well as physics. Classical groups have developed under the cloak ofpermutation groups, and the structure and classification are the primarily problems inthe research, which are also the core problems in structural mathematics. There arethree main methods for the classical groups’ isomorphism and automorphism:involution method, residual space method and matrix method.The present thesis is mainly in chronological order, and the evolvement of theconcept and theoretical development in classical groups are the central clues runningthrough the whole thesis. Based on concept analysis of the school of intellectualhistory, and the study of relative original article and research literature, this thesis triesto analyze comprehensively and study systematically the origin, development andsystematization of the classical groups, then to outline history of the earlydevelopments of the classical groups. The main results are as follows:1. C. Jordan and L. E. Dickson have made outstanding contribution to theclassical groups. This thesis analyzes their research background and methodssystematically. The author compares Jordan’s work with Dickson’s, and reveals theirinheritance relationship in mathematical ideas on the classical groups.2. This thesis analyzes the prominent achievements Otto Schreier, B.E.Van derWaerden and Hermann Weyl have made, and tries to illustrate the development of theclassical groups in close connection with representation theory, invariant theory andtheory of relativity. Furthermore, it is pointed out the promotion of intradisciplinaryand interdisciplinary in the evolution of concepts and the development of theory.3. In this thesis, a comparison between the researches of Jean Dieudonné andLoo-keng Hua on the isomorphism and automorphism of classical groups is drawn.On the basis of this comparison, the present paper tries to contrast two different methods of classical groups: geometric method and matrix method, and finally itcomes to the conclusion that various concrete methods promote the theory forward.4. This thesis penetrates the thinking and methods reflected in achieving aunified structure with classical groups in the study of finite simple group byChevalley Claude, and gives an introduction of unified method of construction aboutclassical groups by others. Accordingly, it reveals that the general and unifiedapproaches play a key role in the process of problem solving and theoreticaldevelopment.5. This thesis analyzes comprehensively the work of three generations ofmathematicians on classical groups, and sets out the evolution of the concepts andterminology in classical groups. Thus, the author gives a comprehensive andsystematic description about the early history of classical groups. In turn, themanifestation of overall macroeconomic of history and logic of mathematics in theresearch of the mathematical has been surveyed briefly.