子群的若干算术条件与有限群结构的关系

It is an important subject to study the relationship between some arithemetic conditions of subgroups and the structure of finite groups.This project mainly studies some group preperties such as the p-nilpotecy and hypersolvability of finite groups by using the localization method on finite groups if these subgroups satisfying some fixed ariyhemetic conditions such as the subgroup numbers, the length of the conjugacy classes,the automorphic numbers,etc. We convert the p-nilpotent structure of groups and p-hypersolvability problems to the characteristic questions on some special subgroups with some arithemetic conditions in contact with weakly SS-quasinormal subgroups,D-groups,H-QC-subgroups,classes numbers and automorphic numbers. In the meanwhile, by eliminate the exceptions, some significant inequalities are obtained to represent quantity characteristic of nilpotent groups, and by enumeration method to obtain some classification results on the contrcture of finite groups whose automorphic groups have given orders.The implementation of the project has important significance for enriching and expanding studying on the structure characteristics of finite groups and for bring up young scientific and technological talents.

子群的算术条件与有限群结构的关系是有限群论的一个重要研究课题。本项目主要利用群的局部化方法研究满足一定条件的子群特别是满足某些条件的素数幂阶子群的个数、特殊子群共扼类的长度、群自同构的个数等算术条件对群的p-幂零性、超可解性等性质与群系特征等群结构之间的关系。利用弱SS-拟正规子群、D-群、H-QC-子群、共轭类及自同构算术条件，将有限群p-幂零结构、p-超可解性等问题转化为某些特殊子群的特征问题，同时通过例外群排除得到幂零群数量特征的若干不等式；利用枚举分析方法研究具有给定阶的自同构群的群论结构的分类定理。实施本项目对丰富和拓展有限群的结构特征研究和培养青年科技人才具有重要意义。

子群的算术条件与有限群结构的关系是有限群论的一个重要研究课题。本项目主要利用群的局部化方法研究满足一定条件的子群特别是满足某些条件的素数幂阶子群的个数、特殊子群共轭类的长度、群自同构的个数等算术条件对群的 p-幂零性、超可解性等性质与群系特征等群结构之间的关系。利用弱 SS-拟正规子群、H*-子群、Hp-子群、共轭类及自同构算术条件，将有限群 p-幂零结构、p-超可解性等问题转化为某些特殊子群的特征问题；利用枚举分析方法研究具有给定阶的自同构群的群论结构的分类定理。实施本项目对丰富和拓展有限群的结构特征研究和培养青年科技人才具有重要意义。