无限可解群的自同构问题

Studying automorphism groups and automorphisms of groups is one of the most central contents in the theory of groups, and is also difficult to deal with. In this project, three problems on automorphisms of infinite soluble groups will be studied..1. The influence of almost regular automorphisms of order p² of infinite soluble groups upon the structure will be studied. Applying regular automorphisms, we will give the structure of infinite soluble groups which admit almost regular automorphisms of order p²..2. The normal automorphisms of infinite soluble groups will be studied. The structure of normal automorphism groups of infinite soluble groups will be given, which solves Endimioni´s problem..3. The structure of automorphism subgroups generated by finitely many automorphisms of infinite soluble groups will be studied. When the image of the commutator of automorphisms induced by finitely many automorphisms of infinite soluble groups on every finite quotient is a finite cyclic group, finite abelian group or finite nilpotent group, the structure of the group generated by finitely many automorphisms will be determined, which generalizes Hirsch´s theorem.

群的自同构和自同构群是群论里最核心的研究内容之一，但也是难以处理的问题。本项目拟研究无限可解群的自同构的三个问题。1.研究p²阶几乎正则自同构对无限可解群的结构的影响。借助于正则自同构，确定具有p²阶几乎正则自同构的无限可解群的结构；2.研究无限可解群的正规自同构，确定无限可解群的正规自同构群的结构，解决Endimioni问题；3.研究无限可解群的有限个自同构生成的自同构子群的结构。当无限可解群的有限个自同构在每一个有限特征商群上诱导的自同构的换位子的像是有限循环群、有限Abel群、有限幂零群时，确定它们生成的群的结构，推广Hirsch定理。

项目主要研究了无限可解群的自同构问题。项目总体上按计划进行，获得的主要成果如下：确定了具有p²阶正则自同构的多重循环群的结构；给出了宽度为1的可解群的正规自同构群和类保持自同构群的结构；当有限秩的幂零群的有限个自同构在中心列的每一个商因子上诱导的自同构具有较好的性质时，确定了它们生成的群的结构，推广了Hirsch定理。