Abstract If 𝔛 {\mathfrak{X}} is a class of groups, define a sequence of group classes 𝔛 1 , 𝔛 2 , … , 𝔛 k , … {\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots,\mathfrak{X}_{k},\ldots} by putting 𝔛 1 = 𝔛 {\mathfrak{X}_{1}=\mathfrak{X}} and choosing 𝔛 k + 1 {\mathfrak{X}_{k+1}} as the class of all groups whose non-permutable subgroups belong to 𝔛 k {\mathfrak{X}_{k}} . In particular, if 𝔄 {\mathfrak{A}} is the class of abelian groups, 𝔄 2 {\mathfrak{A}_{2}} is the class of quasimetahamiltonian groups, i.e. groups whose non-permutable subgroups are abelian. The aim of this paper is to study the structure of 𝔛 k {\mathfrak{X}_{k}} -groups, with special emphasis on the case 𝔛 = 𝔄 {\mathfrak{X}=\mathfrak{A}} . Among other results, it will also be proved that a group has a finite normal subgroup with quasihamiltonian quotient if and only if it is locally graded and belongs to 𝔄 k {\mathfrak{A}_{k}} for some positive integer k.

中文翻译：

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不可置换子群是 metaquasihamiltonian 的群

摘要 如果 𝔛 {\mathfrak{X}} 是一个群类，定义一个群类序列 𝔛 1 , 𝔛 2 , ... , 𝔛 k , ... {\mathfrak{X}_{1},\mathfrak{X} _{2},\ldots,\mathfrak{X}_{k},\ldots} 通过放置 𝔛 1 = 𝔛 {\mathfrak{X}_{1}=\mathfrak{X}} 并选择 𝔛 k + 1 {\mathfrak{X}_{k+1}} 作为其不可置换子群属于 𝔛 k {\mathfrak{X}_{k}} 的所有群的类。特别地，如果 𝔄 {\mathfrak{A}} 是阿贝尔群的类，那么 𝔄 2 {\mathfrak{A}_{2}} 是拟元哈密尔顿群的类，即不可置换子群是阿贝尔群的群。本文的目的是研究𝔛 k {\mathfrak{X}_{k}} -groups 的结构，特别强调情况 𝔛 = 𝔄 {\mathfrak{X}=\mathfrak{A}} 。在其他结果中，