On strongly \(\Pi \)-permutable subgroups of a finite group
type: Item
instance of: scholarly article
title: On strongly \(\Pi \)-permutable subgroups of a finite group
zbMATH Open document ID: 1457.20020
DOI: 10.1134/S0037446619040177
author: Yanyan Li
published in: Siberian Mathematical Journal
publication date: 2019-11-27T00:00:00Z
review text: All groups in this report are finite. Consider a partition \(\sigma=\{\sigma_i\mid i\in I\}\) of the set \(\mathbb{P}\) of all primes and a subset \(\emptyset\ne \Pi\subseteq \sigma\) and a group~\(G\). A set \(\mathcal{H}\) of subgroups of \(G\) is a complete Hall \(\Pi\)-set of \(G\) when every non-trivial member of \(\mathcal{H}\) is a Hall \(\sigma_i\)-subgroup of \(G\) for some \(\sigma_i\in \Pi\) and \(\mathcal{H}\) has exactly one Hall \(\sigma_i\)-subgroup of \(G\) for every \(\sigma_i\in \pi\) such that \(\sigma_i\cap \pi(G)\ne \emptyset\). A group \(G\) with a complete Hall \(\Pi\)-set is called \(\Pi\)-full. A subgroup \(A\) of \(G\) is called \(\Pi\)-permutable in \(G\) if \(G\) has a complete Hall \(\Pi\)-set \(\mathcal{H}\) such that \(AH^x=H^xA\) for all \(H\in\mathcal{H}\) and all \(x\in G\); \(\sigma\)-subnormal in \(G\) if \(G\) hs a chain of subgroups \(A=A_0\le A_1\le\dots\le A_t=G\) such that either \(A_{i-1}\trianglelefteq A_i\) or \(A_i/(A_{i-1})_{A_i}\) is a \(\sigma_k\)-group for some \(k=k(i)\) for each \(i\in\{1,\dots, t\}\); and strongly \(\Pi\)-permutable in \(G\) if \(A\) is \(\Pi\)-permutable and \(\sigma\)-subnormal in~\(G\). A group \(G\) is said to be \(\sigma\)-nilpotent if \(G\) is the direct product of the members of a complete Hall \(\sigma\)-set of~\(G\). A subgroup \(A\) of \(G\) is said to be \(\Pi\)-modular when \(G\) is \(\Pi\)-full and \(\langle A, H\cap C\rangle=\langle A, H\rangle \cap C\) for every Hall \(\sigma_i\)-subgroup \(H\) of \(G\) and all \(\sigma_i\in \Pi\) and \(A\le C\le G\). The group \(G\) is said to be a \(\Pi\)-full group of Sylow type if every subgroup of \(G\) has the dominance property Hall \(\sigma_i\)-subgroups for all \(\sigma_i\in \Pi\). The first main theorem of this paper (Theorem~A) shows, for a \(\Pi\)-full group~\(G\) and a \(\sigma\)-nilpotent subgroup \(A\) of \(G\), the equivalence between the following facts: (1) \(A\) is strongly \(\Pi\)-permutable in~\(G\); (2) every characteristic subg…
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 7137344
zbMATH Keywords: finite group
zbMATH Keywords: subgroup lattice
zbMATH Keywords: \(\sigma\)-subnormal subgroup
zbMATH Keywords: strongly \(\Pi\)-permutable subgroup
reviewed by: R. Esteban-Romero
MaRDI profile type: Publication
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full work available at URL: s0037446619040177
OpenAlex ID: W2967639597
description: scientific article; zbMATH DE number 7137344