Given a group G and a subgroup H, we let $\mathcal {O}_G(H)$ denote the lattice of subgroups of G containing H. This article provides a classification of the subgroups H of G such that $\mathcal {O}_{G}(H)$ is Boolean of rank at least $3$ when G is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilisers of chains of regular partitions, and the other arises by taking stabilisers of chains of regular product structures. As an application, we prove in this case a conjecture on Boolean overgroup lattices related to the dual Ore’s theorem and to a problem of Kenneth Brown.

中文翻译：

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有限交替和对称群中的布尔格

给定一组G和一个子群H，我们让 $\mathcal {O}_G(H)$ 表示子群的格子G包含H. 本文提供了亚组的分类H的G这样 $\mathcal {O}_{G}(H)$ 至少是等级的布尔值 $3$ 什么时候G是有限交替或对称群。除了一些零星的例子和一些扭曲的版本外，还有两种不同类型的这种格子。一种是采用规则分区链的稳定剂，另一种是采用规则产品结构的链的稳定剂。作为一个应用，我们在这种情况下证明了一个关于布尔超群格的猜想，该猜想与对偶 Ore 定理和 Kenneth Brown 的问题有关。