Abstract

Homological properties of quotient divisible Abelian groups are studied. These groups form an important class of groups, which has been extensively studied in recent years. The first part of the paper is devoted to conditions for the triviality of extension groups in which one of the arguments is a quotient divisible group. Under certain additional assumptions, groups of homomorphisms from quotient divisible groups to reduced Abelian groups are described. Universality properties of quotient divisible Abelian groups are investigated. The second part of the paper considers homological properties of compact Abelian groups dual to quotient divisible groups in the sense of L.S. Pontryagin. Such groups are said to be “quotient toroidal.” Conditions for the triviality of group extensions in which one of the arguments is a quotient toroidal group are studied. Certain groups of continuous homomorphisms in which the second argument is a quotient toroidal group are described. The last part of the paper is devoted to conditions for the triviality of the groups of extensions of quotient divisible groups by compact quotient toroidal ones. The fundamental group of the topological space of a quotient toroidal group is characterized.

中文翻译：

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商对偶阿贝尔群和对偶对偶群的密合性质

摘要

研究了商整除阿贝尔群的同调性质。这些小组构成了重要的小组类别，近年来已经对其进行了广泛的研究。本文的第一部分致力于扩展群的琐碎性条件，其中一个论点是商可整群。在某些其他假设下，描述了从商整除组到简化的Abelian组的同构组。研究了商整除阿贝尔群的普遍性。本文的第二部分从LS Pontryagin的角度考虑了紧凑的Abelian群与商整除群之间的同源性。此类基团被称为“商环”。研究了其中一个参数是商环群的微分群扩展的条件。描述了其中第二自变量是商环组的某些连续同态组。本文的最后一部分专门讨论了由商数紧凑的商圈构成的商可整群扩展的群的平凡性的条件。商环群的拓扑空间的基本群是特征。