We show that if all proper subgroups of a locally graded group G are finite-by-abelian-by-finite, then G contains a finite normal subgroup N such that all proper subgroups of G/N are abelian-by-finite. Then we apply this result to the study of groups which are minimal-non-\({\mathcal {P}}\) also for related group properties \({\mathcal {P}}\). Finally we see how for locally (soluble-by-finite) groups of infinite rank, it is enough to restrict attention to the proper subgroups with infinite rank.

中文翻译：

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在具有所有适当子组的组上

我们表明，如果局部分级的组G的所有适当子组都是按逐个阿贝尔有限元划分的，那么G包含有限的正规子组N，这样G / N的所有适当子组都将逐个阿贝尔分布。然后，我们将此结果应用于最小非\（{{mathcal {P}} \）以及相关组属性\（{\ mathcal {P}} \）的研究。最终，我们看到了对于无限等级的局部（有限溶解）组，如何将注意力集中到无限等级的适当子组上就足够了。