This article is cited in 2 papers

On Finite Groups with $\mathbb{P}_{\pi}$-Subnormal Subgroups

T. I. Vasilyeva^a, A. G. Koranchuk<sup>b

a</sup> Belarusian State University of Transport, Gomel'
^b Gomel State University named after Francisk Skorina

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.

Keywords: $\mathbb{P}_{\pi}$-subnormal subgroup, ${\pi}$-solvable group, ${\pi}$-supersolvable group, Sylow subgroup, hereditary saturated formation.

UDC: 512.542

MSC: 20D20, 20F16, 20F17, 20F22

Received: 02.04.2023
Revised: 08.05.2023

DOI: 10.4213/mzm13973

 English version: Mathematical Notes, 2023, 114:4, 421–432

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