关于有限p-群的自同构和上同调的研究

In the project, automorphisms and cohomologies of finite p-groups are studied. By use of general linear groups and symplectic groups, automorphism groups of finite p-groups with derived subgroup of p order are determined from different respects, and further automorphism groups of some special types of finite p-groups with a cyclic derived subgroup and finite p-groups with a cyclic center are studied. With the help of a representing set, Frobenius category, Quillen complex, etc, the mod-p cohomologies of finite p-groups with a cyclic derived subgroup, finite p-groups with a cyclic center and finite p-groups with a cyclic Frattini subgroup are studied. On one hand, by studying the cardinality of a maximal non-commuting set in these finite p-groups, the mod-p cohomology lengths of these finite p-groups are determined; On the other hand, the mod-p cohomologies of finite groups with these finite p-groups as Sylow p-subgroups are studied, in these finite p-groups which types are Swan groups or resistant groups are determined.

本课题研究有限p-群的自同构和上同调。借助于一般线性群，辛群，从不同方面刻画导群为p阶的有限p-群的自同构群，进一步，研究导群为循环的有限p-群和中心为循环的有限p-群中一些特殊类型的自同构群。借助于表示集，Frobenius范畴，Quillen 复形等理论，研究导群为循环的有限p-群，中心为循环的有限p-群，Frattini子群为循环的有限p-群的模p上同调。一方面，通过研究这些有限p-群的极大非交换集的势，刻画它们的模p上同调长；另一方面，研究以这些有限p-群为Sylow p-子群的有限群的模p-上同调，确定这些有限p-群中哪些是Swan群或者resistant群。

课题组成员按照项目研究计划, 基本完成预期任务，达到了本课题的预期目标。主要研究了导群是p阶群的有限p-群， Frattini子群是循环的有限p-群，中心是循环的有限p-群的极大非交换集的势; 确定了导群是p阶群，循环群被初等Abel群中心扩张的有限p-群的自同构群和一类中心是循环的，中心商群是齐次循环的有限p-群的自同构群; 确定了几乎所有导群是p阶群的，循环群被初等Abel群中心扩张的有限p-群是Swan群。研究成果对一些自同构的提升问题提供了有效的方法，对非交换集势的确定提供了有效的思路，对Swan群的判断提供了不同的计算方法。