给定阶特殊子群对群链构造的影响

In this project, we main study the influence of the special generalized supplemented subgroups such as Mp-supplemented subgroup、Mp-embedded subgroup、m-embedded subgroup、nearly m-embedded subgroup with given order on the structure of finite groups. It is aim to characterize the structure of finite groups by applying the method of number theory and the ideal of classification of simple group, though the normalizer of special subgroups with given order, the composition factor and the chief factor... The project is mainly divide into three aspects. On one hand, we will investigate the influence of generalized supplemented subgroups of given order on the p-nilpotency and p-supersolvability of finite group. On the other hand, we will study the influence of special subgroups with given order on the structure of the chain of chief factor. Finally, we will study the influence of special subgroups with given order on the structure of the chain of composition factor and look for the relationship between the generalized supplemented subgroups of given order and the chain of group.

本项目主要研究给定阶特殊子群的广义可补充性（例如Mp-可补充性、Mp-嵌入性、m-嵌入性、几乎m-嵌入性等）对有限群构造的影响, 利用数论的方法和单群分类的思想，通过给定阶特殊子群的正规化子，群G的主因子和合成因子等来寻求有限群结构的刻画。.. 本项目主要从三个方面展开:首先，探讨给定阶特殊子群的广义可补充性对群p-幂零性及p-超可解性的影响；其次，研究给定阶特殊子群的性质对主群链构造的影响；最后，研究给定阶特殊子群的性质对合成群链构造的影响，寻找给定阶特殊子群的广义可补充性与群链之间的关系。

给定阶子群的的性质对有限群的构造有着重要的影响，本项目从单群分类的角度出发，利用分类讨论的思想，揭示子群的数量信息与群结构之间的关系。首先，考虑有限群中Sylow 3-子群、Sylow 5-子群、Sylow 7-子群的给定阶子群的M-可补性对合成因子的影响，得到了有限群中合成因子的结构；其次讨论正规子群E、F(E) 等给定阶子群的Mp-可补性对主因子的影响，得到了主群链中非Frattini G-主因子的结构；最后讨论给定阶子群对可解群、p-超可解群等具体群类构造的影响，得到了给定阶子群与p-拟超可解群的关系。