广义Norm和可解融合系

On the one hand ，we will give the system thoery of generelized Norm. 1. We will give the generation Theorem and the equivalent conditions of D-groups and S-groups and solve two conjectures which are posed by the referees of our two papers having published in Journal of Algebra. 2. By the definition of the center series of nilpotency class of nilpotent group，we give the concepts of D-length of a group G with G' nilpotent and S-length of a meta-nilpotent group. In order to investigate the structure of D-groups and S-groups, we will give the classification of minimal non-D-group and minimal non-S-groups, in fact, they are interesting independently. We will give the new concepts of the Norms of the general F-residuals of subgroups; for example, let F be a saturated fomation containing all nilpotent fomation or all supersolvable fomation. 3. By the suggestion of Prof. Jiping Zhang(For the concepts of a few generalized Norms being given by us, ones can only consider some respresentation subgroups and need not consider all subgroups), we can give the concepts of the more and new generalized Norms. On the other hand，for Fusion System. First, we investigate Zhang Jiping's visualizer (decide when the results from solvable group theory will hold for the theory of fusion system). Second, we investigate Solomen and Stancu' conjecture (Is there a weakly normal subsystem over every strongly closed subgroup of a saturated fusion system).

一、拟给出广义Norm的系统理论。1.给出D-群和S-群的生成定理和等价条件以及解决申请人近期发表在Journal of Algebra上两篇文章中审稿人提出的两个猜想。2.根据幂零群的幂零类的上中心链定义提出导子群幂零的群的D-长、亚幂零群的S-长的概念；为了更好的研究D-群和S-群的结构，给出极小非D-群和S-群的分类，事实上它们也有独立的意义；给出子群的F-剩余的Norm的新概念，例如考虑F是包含幂零群系和超可解群系的饱和群系。3.根据张继平教授的建议（对于我们提出的几类广义Norm的概念不需要考虑所有的子群，只需要找出有代表的某类子群）能够给出更多新的广义Norm的概念。二、关于融合系，首先是对张继平教授提出的构想（根据可解群的理论给出可解融合系的系统理论）进行研究；然后对Solomon和Stancu猜想（是否饱和融合系的每个强闭子群上存在一个弱正规子系）进行研究。

一、给出了广义Norm的系统理论；1、给出了D-群和S-群的生成定理和等价条件以及解决了申请.人近期发表在Journal of Algebra上两篇文章中审稿人提出的一个猜想。2、根据幂零群的.幂零类的上中心链定义提出导子群幂零的群的D-长、亚幂零群的S-长的概念；给出极小非D-群和S-群的分类，它们有着独立的意义；给.出了子群的F-剩余的Norm的新概念，例如考虑F是包含幂零群系和超可解群系的饱和群系。3、根据张继平教授的建议.（对于我们提出的几类广义Norm的概念不需要考虑所有的子群，只需要找出有代表的某类子群）给出了更多新的广义Norm的概念。.二、关于融合系，首先是对张继平教授提出的构想（根据可解群的理论给出可解融合系的系统理论）进行了系统的研究.；然后给出了Solomon和Stancu猜想p-可解情形的证明，构造了一般融合系的反例（是否饱和融合系的每个强闭子群上存在一个弱正规子系.）进行研究。