子群的共轭类和自同构导子对有限群结构的影响

In my recent research,the applicant gave the lower bounds on the numbers of conjugacy classes of some non-special subgroups of finite groups in terms of |\pi(G)| and gave some complete classifications of finite groups with certain particular conditions.In addition,the applicant also investigated the structure of finite groups in which the automizers of all cyclic subgroups are small.In the project,we will extend and generalize our privious research and give a complete investigation of the influence of the numbers of conjugacy classes and the automizers of non-special subgroups on the structure of finite groups.The datails are as follows: . (1)To give complete classifications of finite groups in which the numbers of conjugacy classes of some non-specail subgroups satisfy certain particular conditions..(2)To investigate the function relations between the numbers of conjugacy classes of non-special subgroups and other quantitative properties of finite groups and characterize the solvabilities of finite groups. .(3)To investigate the finite groups in which the automizer of every non-nilpotent subgroup(every non-normal non-abelian subgroup,etc.)is large(small)..(4)To give complete characterizations of the finite groups in which the automizer of every abelian subgroup(every non-abelian subgroup,etc.)is either large or small.

在近几年的研究中,申请人通过|\pi(G)|给出了若干非特殊子群共轭类数的下界,并给出了某些特定情形的有限群的完全分类.申请人同时还研究了循环子群具有小的自同构导子的有限群的结构.在本项目中,我们将改进和推广之前的研究工作,完整地研究非特殊子群的共轭类数和自同构导子对有限群结构的影响,具体研究内容包含以下四个方面:.(1)完全分类非特殊子群的共轭类数满足特定条件的有限群;.(2)研究非特殊子群的共轭类数和群的其他数量性质之间的函数关系并刻画群的可解性;.(3)完全刻画非幂零子群、非正规的非交换子群等分别具有大的和小的自同构导子的有限群；.(4)完全刻画交换子群、非交换子群等的自同构导子要么是小的要么是大的有限群.

本项目主要研究非特殊子群的共轭类和自同构导子对有限群结构的影响，主要包括以下内容：.(1)给出了具有较少非循环和非交换子群共轭类的有限群结构..(2)获得了有限群中非交换子群同阶类数以及子群个数的下界. .(3)给出了二极大子群皆交换的有限群的同构分类; 研究了偶阶子群为幂零群以及MS-群的有限群结构..(4)刻画了每个交换子群要么具有小、要么具有大的自同构导子的有限群.