A Research On The History Of The Classification Of Finite Simple Groups

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 thesociety'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).