广义可补群的研究

The existence of complements for certain subgroups of a finite group provides a lot of information about its structure. For example, P.Hall proved that a finite group G is solvable if and only if every Sylow subgroup of G is complemented. Since the complementarity of subgroups play an important role in group theory, a topic of some interest is to investigate the finite groups in which certain subgroups are complemented. In this project we study finite groups with some complemented subgroups. In detail, we study the following: . (1) The influence of generalized supplemented subgroups on the structure of finite groups;. (2) The arithmetic contitions of conjugacy classes of non-complemented subgroups and the structure of finite groups;. (3) Finite groups in which non-complemented subgroups contaied in the Fitting subgroups.

子群的可补性是有限群研究中最常用最基本的性质之一。自从群论大家P.Hall给出著名的可解群判定定理起，子群的可补性与有限群的结构及相关问题的研究一直是有限群研究的热点课题之一。本项目将系统研究广义可补群的性质和结构分类问题，主要包括以下3个方面：. （1）非可补子群的共轭类等算术条件与有限群的结构；. （2）非可补子群含于Fitting子群的有限群；. （3）子群的广义可补性对有限群结构的影响。. 本项目的研究成果将丰富可解群理论，也促进子群的性质等相关问题的研究。

本项目利用局部化分析方法和技巧，结合非可补子群以的有关性质，研究广义可补群的性质与结构。经过项目组成员的共同努力，目前已完成学术论文6篇。得到了可解群的若干判定准则，及其他若干有意义的结果。例如，如果有限群G的每个奇素数阶子群在G中SS-可补，那么G可解；如果有限群G的非正规子群共轭类个数不超过|G|的素因子个数的2倍时，那么G可解。另有一些结果在撰写整理中，研究工作按计划顺利进行并基本达到预期目标。