Abstract

We distinguish 2-maximal and strictly 2-maximal subgroups of a finite group and give examples of soluble and simple groups in which every 2-maximal subgroup is strictly 2-maximal. Let M be a maximal subgroup of a group G. We prove that every maximal subgroup of M is strictly 2-maximal in G if M is normal in G or if G is p-soluble and $|G:M|=p.$ We describe the structure of a finite group in which all 2-maximal subgroups are Hall subgroups. In particular, it has a Sylow tower and all its Sylow subgroups are elementary abelian.

中文翻译：

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关于有限群的 2-极大子群

摘要

我们区分有限群的 2-极大子群和严格 2-极大子群，并给出每个 2-极大子群都是严格 2-极大的可溶群和简单群的例子。令M为组G的最大子组。我们证明如果M在G中是正规的或如果G是p可溶的，则M的每个最大子群在G中是严格 2-极大的$|G：米|=p.$我们描述了一个有限群的结构，其中所有 2-极大子群都是霍尔子群。特别是，它有一个 Sylow 塔，它的所有 Sylow 子群都是基本的阿贝尔。