关于有限秩的可解群的自同构研究

The automorphism groups and automorphisms of groups are the central contents in the theory of groups, and are also difficult to deal with. In this project, we will study two problems on automorphisms of soluble groups of finite rank. . 1. We will study regular automorphisms of prime order and order four of soluble groups of finite rank, on this basis, we will study automorphisms of prime order and order four of soluble groups of finite rank by using regular automorphisms. Give the structure of soluble groups of finite rank which admit automorphisms of prime order and order four, and discuss the property of commutator subgroups and quotient groups.. 2. The nilpotence of automorphism groups of nilpotent groups of finite rank will be studied. Prove the group generated by finitely many automorphisms has some good nilpotence when automorphisms induced by finitely many automorphisms of nilpotent groups of finite rank on the quotient factors of the central series have good properties.

群的自同构和自同构群是群论研究的核心内容之一，也是难以处理的研究对象。本项目拟研究有限秩的可解群的自同构的两个方面的问题：1.在有限秩的可解群的素数阶和4阶正则自同构研究基础上，借助于正则自同构，研究有限秩的可解群的素数阶和4阶自同构，给出具有素数阶和4阶自同构的有限秩的可解群的结构并讨论其换位子群以及商群的性质；2.研究有限秩的幂零群的自同构群的幂零性质，当有限秩的幂零群的有限个自同构在其中心列的商因子上诱导的自同构都具有较好的性质时，证明它们生成的群具有良好的幂零性质。

群的自同构和自同构群是群论研究的核心内容之一，也是难以处理的研究对象。本项目组成员按照项目研究计划，基本完成预期任务，达到了本项目的预期目标。确定了具有素数阶自同构的有限秩的剩余有限可解群和剩余有限minimax可解群的结构以及具有4阶自同构的剩余有限minimax可解群和有限生成无挠幂零群的结构。研究了当有限秩的幂零群的有限个自同构在其上下中心列的商因子上诱导的自同构生成的子环作为Lie环是完全可解的，证明它们生成的群具有良好的幂零性质。定义了多重循环群中的NAFQn-群，并确定了NAFQ1-群和NAFQ2-群的结构。