幂零群及其自同构的研究

Nilpotent groups and their automorphisms are profound, complex and active research topics and play fundamental roles in group theory. The proposer has made substantial progress on the study of residual finiteness of infinite groups, Wells maps of automorphism groups of group extensions and automorphism groups of groups, etc. These works win high praise and are intrinsically quoted. For instance, Robinson has said that our work on Wells maps for automorphism groups of a group extension is important and used this result to study the automorphism pairs of group extensions;Recently Robinson's work on "extended residually finite property" is based on our ideas.This project will study residually finite properties and pro-p topologies of nilpotent groups with finite rank and discuss the lifting problems and linear representations for automorphism groups of this class of groups；We will plan to generalize both Hall's theorem and Stewart's theorem to weakly supersoluble groups and generalize both Hirsch's theorem and Endimioni's theorem to minimax soluble groups with residually finite property; Finally we are going to determine the automorphism groups of several classes of finite p-groups, and figure out Wells maps for the automorphism groups of some specific group extensions and give some applications on Wells maps.

幂零群及其自同构是群论里非常深刻、复杂、活跃的研究对象，具有基本的重要性。申请人已经在无限群的剩余有限性、群扩张的自同构群的Wells映射、群的自同构群等方面取得了实质性的进展。申请人明确定义了群的"强剩余有限性"，这是群论大家Robinson近来两篇文章的研究对象；Robinson评价申请人关于群扩张的自同构群Wells映射的工作是"important"，并运用这项工作去研究群扩张的"自同构对"。本项目拟研究无限幂零群的剩余有限性和pro-p拓扑，研究这类群的自同构群的"提升"问题和线性表示；以幂自同构为工具给出Hall-Stewart幂零性定理在弱超可解群等群类上的推广；把Hirsch和Endimioni关于多重循环群的幂零性定理进一步推广到剩余有限的可解minimax群；确定几类有限p-群的自同构群；计算某些具体的群扩张的自同构群的Wells映射，给出 Wells映射的一些应用。

本项目完成论文29篇，在无限群及其自同构（群）、有限p-群的非交换集和自同构群、有限群的融合系理论、矩阵分析等方面取得了比较系统深入的研究成果。