On \(\mathfrak F_{hq}\)-supplemented subgroups of a finite group
type: Item
instance of: scholarly article
title: On \(\mathfrak F_{hq}\)-supplemented subgroups of a finite group
zbMATH Open document ID: 1399.20018
author: M. Ezzat Mohamed
author: M. I. Elashiry
author: M. M. Al-Shomrani
published in: Periodica Mathematica Hungarica
publication date: 2018-07-23T00:00:00Z
review text: Let \(G\) be a finite group, and \(H\) a subgroup of \(G\). Suppose there is a quasinormal subgroup \(N\) of \(G\) such that the subgroup \(H N\) of \(G\) is a Hall subgroup of \(G\), and \((H \cap N) H_{G} / H_{G}\) is contained in \(Z_{\mathfrak F}(G / H_{G})\). Then \(H\) is said to be \({\mathfrak F}_{hq}\)-supplemented in \(G\). Here, a subgroup \(N\) of \(G\) is said to be quasinormal in \(G\) if \(N K = K N\) for all subgroups \(K\) of \(G\); a Hall subgroup is a subgroup whose order is coprime to its index; and \(Z_{\mathfrak F}(L)\) is the \({\mathfrak F}\)-hypercenter of the group \(L\), where \(\mathfrak F\) is a formation. The main results of the paper under review are the following. Let \(G\) be the smallest prime divisor of the order of \(G\), and \(G_{p}\) be a Sylow \(p\)-subgroup of \(G\). If all of the maximal subgroups of \(G_{p}\) are \({\mathfrak U}_{h q}\)-supplemented in \(G\), then \(G\) is \(p\)-nilpotent. (Lemma~3.1) Here, \(\mathfrak U\) is the formation of supersolvable groups. Suppose \(G\) has a normal \(p\)-subgroup \(P\) such that \(G/P\) is supersolvable. If the maximal subgroups of \(P\) are \({\mathfrak U}_{hq}\)-supplemented in \(G\), then \(G\) itself is supersolvable (Theorem~3.3).
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 6909491
zbMATH Keywords: finite groups
zbMATH Keywords: saturated formation
zbMATH Keywords: supplemented subgroup
zbMATH Keywords: Sylow subgroup
zbMATH Keywords: supersolvable group
reviewed by: Andrea Caranti
MaRDI profile type: Publication
full work available at URL: s10998-017-0234-8
OpenAlex ID: W2775058349
cites work: Q4114014
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DOI: 10.1007/S10998-017-0234-8
description: scientific article; zbMATH DE number 6909491