Abstract

Let \(\pi\) be a set of primes. A subgroup \(H\) of a group \(G\) is said to be \(\mathbb{P}_{\pi}\)-subnormal in \(G\) if either \(H=G\) or there exists a chain of subgroups beginning with \(H\) and ending with \(G\) such that the index of each subgroup in the chain is either a prime in \(\pi\) or a \(\pi'\)-number. Properties of \(\mathbb{P}_{\pi}\)-subnormal subgroups are studied. In particular, it is proved that the class of all \(\pi\)-closed groups in which all Sylow subgroups are \(\mathbb{P}_{\pi}\)-subnormal is a hereditary saturated formation. Criteria for the \(\pi\)-supersolvability of a \(\pi\)-closed group with given systems of \(\mathbb{P}_{\pi}\)-subnormal subgroups are obtained.

中文翻译：

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关于具有 $$\mathbb{P}_{\pi}$$ 次正规子群的有限群

摘要

令\(\pi\)为一组素数。群\(G\)的子群\(H\)被称为\(\mathbb{P}_{\pi}\)中的次正规群\(G\)，如果\(H=G\)或者存在以\(H\)开头并以\(G\)结尾的子群链，使得链中每个子群的索引要么是\(\pi\)中的素数，要么是\(\pi' \) - 数字。研究了\(\mathbb{P}_{\pi}\)次正规子群的性质。特别地，证明了所有Sylow 子群均为次正规的所有\(\pi\)闭群的类是遗传饱和形成。获得具有给定的次正规子群系统的\ (\pi\ )闭群的\(\pi\) 超可解性准则。