Recently in Dikranjan et al. (Fund Math 249: 185–209, 2020) an uncountable Borel subgroup \(t^s_{(2^n)}({\mathbb T}) \) (called statistically characterized subgroup) was constructed containing the Prüfer group \({\mathbb Z}(2^\infty )\) using the notion of statistical convergence. This note is based on the recent work (Bose et al. in Acta Math Hungar, 2020) which helps us to show that an uncountable chain of distinct Borel subgroups (each of size \(\mathfrak {c}\)) can be generated between \({\mathbb Z}(2^\infty )\) and \(t^s_{(2^n)}({\mathbb T}) \), whereas their intersection actually strictly contains the Prüfer group, with their union being strictly contained in \(t^s_{(2^n)}({\mathbb T})\).

最近在 Dikranjan 等人。(Fund Math 249: 185–209, 2020) 构建了一个不可数的 Borel 子群\(t^s_{(2^n)}({\mathbb T}) \)（称为统计特征子群）包含 Prüfer 群\( {\mathbb Z}(2^\infty )\)使用统计收敛的概念。本笔记基于最近的工作（Bose 等人在 Acta Math Hungar，2020 年），它帮助我们证明可以生成不可数的不同 Borel 子群链（每个子群的大小为\(\mathfrak {c}\)）在\({\mathbb Z}(2^\infty )\)和\(t^s_{(2^n)}({\mathbb T}) \) 之间，而它们的交集实际上严格包含 Prüfer 组，与他们的联合被严格包含在\(t^s_{(2^n)}({\mathbb T})\).