有限（非交换）群的整群环、表示环和Burnside环的增广商群

Structure of augmentation ideals and their consecutive quotients for integral group rings of finite groups is a very important problem in group ring theory. They are closely linked with the problem of dimension subgroup, which is defined by augmentation ideal. People focus on augmentation ideals and augmentation quotients for integral group ring at first. Furthermore, augmentation quotients for Burnside ring and representation rings are tackled. First of all, we will give a basis or a generating set for the powers of augmentation ideals for complex representation rings of some finite groups, such as point group and nonabelian group 2-gorups. Then we determine the structure of their augmentation quotients. Secondly, we study augmentation ideals and augmentation quotients for real representation rings of some finite groups by comparing them with corresponding complex representation rings. Thirdly, we classifying subgroups of some non-abelian groups (like Dihedral groups), which help us determine the multiple structure of their Burnside rings. Then we give a basis of augmentation ideals (and their powers) for Burnside rings of Dihedral groups. Finally, we want to give some recursive formulas for Ulm invariants of augmentation quotients for integral group ring of finite abelian groups. Then we try to compute them by computer programming.

增广理想和增广商群的结构是群环理论中的重要问题之一，它们与经典的维数子群问题有紧密的联系---整群环的维数子群就是由其增广理想确定的。最初人们研究整群环的增广理想和增广商群，后来又进一步讨论Burnside环以及表示环的增广理想和增广商群。本课题拟从以下几个方面展开研究：首先，利用某些性质较好的非交换群（如和非交换2-群）的表示论结果，对其表示环的增广理想的幂的基底或生成元进行讨论，进而确定相应增广商群的结构；其次，考虑某些有限群的实表示环的增广理想和增广商群的结构，与复表示环的相关结果进行比较；再次，对某些性质较好的非交换群（如二面体群）的子群进行分类，讨论其Burnside 环的乘法规则，确定其增广理想的幂作为自由交换群的基底，进而给出增广商群的结构；最后，建立有限齐次交换p-群的整群环的各阶增广商群的Ulm不变量之间的递推关系，给出一个计数算法并上机实现。

本项目针对以下问题展开了研究. 首先, 对广义四元数群, 具体构造了其复表示环的增广理想各次幂的基, 并确定了相应增广商群的结构. 其次, 对下列有限群: 奇数阶交换群的广义二面体群, 具有循环Sylow 2子群的有限交换群的广义二面体群, 具有循环极大子群的p^3阶非交换群, 具体构造了它们的广义二面体群的Burnside环的增广理想各次幂的基, 并确定了相应增广商群的结构, 其中p是奇素数. 最后, 对有限循环群, 具体构造了其实表示环的增广理想各次幂的基, 并确定了相应增广商群的结构.