On groups that are dominated by countably many proper subgroups (Q1643200)

In the article under review, the authors study the structure of a group having a countably set of proper subgroups with the property that every proper subgroup is contained in some member of the set (such a group is called \(\mathbf{CD}\)-\textit{group}). In the article, the authors first give various examples of \(\mathbf{CD}\)-groups. Further, some classes of groups with \(\mathbf{CD}\) property are described. Among them, all virtually nilpotent \(\mathbf{CD}\)-groups are characterized. Some aspects of soluble \(\mathbf{CD}\)-groups is also studied. If every subgroup of a \(\mathbf{CD}\)-group \(G\) is a \(\mathbf{CD}\)-group, then \(G\) is called \(\mathbf{SCD}\)-\textit{group}. In the article, examples of \(\mathbf{CD}\)-groups are given that are not \(\mathbf{SCD}\). A theorem describing all soluble \(\mathbf{SCD}\)-groups is also given.