几类无限幂零群的结构和不变量

Infinite nilpotent groups have very complicated structure. In this project we intend to study some dual nilpotent groups, to obtain their structural theorems and isomorphic invariants, to give their applications..First, by finiteness conditions we study the nilpotent groups with locally cyclic commutator subgroups, including finitely generated nilpotent groups with infinite cyclic commutator subgroup, finitely generated nilpotent groups with commutator subgroup of order p, p-groups with both minimum condition and commutator subgroup of order p, radicable nilpotent groups with locally cyclic commutator subgroups, and so on. .Secondly, by finiteness condition we study nilpotent groups having big center, more precisely, nilpotent groups of central extension by abelian groups of rank two, including the groups with the conditions of maximum, minimum, minimax, finite rank and radicable, etc..Finally, we will apply our results to solve some questions about nilpotent groups, such as automorphism groups, cohomological groups (rings), Schur multiplicators, tensor products, locallizations, completions, etc. .We may obtain some deep structural theorems and isomorphic invariants for these groups, generalize the results for abelian groups and finite p-groups in essential.

无限幂零类具有非常复杂的结构。本项目拟研究几类无限幂零群，得到其结构定理和同构不变量，并给出其应用。.1. 运用有限性条件研究换位子群是局部循环群的幂零群，包括换位子群是无限循环群的有限生成幂零群、换位子群是p阶群的有限生成幂零群、换位子群是p阶群的满足极小条件的p-群、换位子群是局部循环群的可除幂零群等等；.2. 运用有限性条件研究具有大中心的幂零群。即模去其中心后，所得商群是秩2的Abel群的幂零群，包括满足极大条件、极小条件、minimax条件、有限秩条件、可除条件的群；.3. 从所得结构定理和同构不变量出发，推广Abel群和幂零群的相关研究，如自同构群、上同调群(环)等；深化幂零群的相关研究: 如Schur乘子，张量积、局部化、完备化等，得到精确化的结果。.这项研究能够得到关于这些群的深入的结构定理和同构不变量，在本质上推广Abel群和有限p-群的相关结果。

发表26篇群论论文。.主办两次学术研讨会。.本项目运用有限性条件研究了换位子群是局部循环群的有限秩幂零群，得到换位子群是无限循环群的有限生成幂零群的结构定理和同构不变量，把有限生成Abel群的基本定理推广到非交换的情形，这种结构定理在非交换群里是非常稀有的。从所得结构定理和同构不变量出发，得到了几类幂零群的自同构群的结构。.群的自同构群是非常复杂的研究对象。本项目从Abel群的自同态环出发，发现了有限秩幂零群自同构的一种现象，并从Lie环的可解性入手令人满意地解释了这种现象。.把Abel群的一些分解定理推广到主理想整环上的模，并应用于向量空间，深化了已有的结论，可能为进一步的研究提供了一条思路。.Hall的幂零性准则是无限群里的重要定理，Gray把它推广到半Abel范畴，本项目精确化了Gray的结果。.