Finite groups with \(\mathcal H\)-subgroups or strongly closed subgroups.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1256.20015
DOI: 10.1515/JGT.2011.096
published in: Journal of Group Theory
publication date: 2012-05-11T00:00:00Z
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 6032974
zbMATH Keywords: finite groups
zbMATH Keywords: strongly closed subgroups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: minimal non-\(P\mathcal H\)-groups
zbMATH Keywords: minimal non-\(\mathcal T\)-groups
author: Zhencai Shen
author: Shirong Li
MaRDI profile type: Publication
cites work: Some Results onp-Nilpotence and Supersolvability of Finite Groups
cites work: Onp-Nilpotence and Supersolvability of Finite Groups
cites work: On minimal subgroups of finite groups
cites work: On finiteJ-groups
cites work: On finite solvable groups in which normality is a transitive relation
cites work: Finite groups whose minimal subgroups are normal
cites work: On Supersolvable Groups and the Nilpotator
cites work: Strongly closed subgroups of finite groups.
cites work: 2-fusion in finite groups
cites work: Strongly closed 2-subgroups of finite groups
cites work: Endliche Gruppen mit lauter nilpotenten zweitmaximalen Untergruppen
cites work: Finite groups with an automorphism of prime order whose centralizer has small rank.
cites work: Some finite nonsolvable groups characterized by their solvable subgroups
cites work: The influence of SS-quasinormality of some subgroups on the structure of finite groups.
cites work: A Note on Finite Groups in Which Normality is Transitive
cites work: On minimal non PN-groups
cites work: Finite groups with normally embedded subgroups
cites work: Nonsolvable finite groups all of whose local subgroups are solvable
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title: Finite groups with \(\mathcal H\)-subgroups or strongly closed subgroups.
review text: Let \(G\) be a finite group. \textit{M. Bianchi, A. Gillio Berta Mauri, M. Herzog} and \textit{L. Verardi} introduced [in J. Group Theory 3, No. 2, 147-156 (2000; Zbl 0959.20024)] the following notion: A subgroup \(H\) of \(G\) is called an \(\mathcal H\)-subgroup if \(N_G(H)\cap H^g\leq H\) for all \(g\) in \(G\). On the other hand, \textit{D. M. Goldschmidt} [in Ann. Math. (2) 99, 70-117 (1974; Zbl 0276.20011)] and \textit{R. J. Flores} and \textit{R. M. Foote} [in Adv. Math. 222, No. 2, 453-484 (2009; Zbl 1181.20014)] investigated the following concept: Let \(S\) be a subgroup of \(G\); a subgroup \(A\) of \(S\) is said to be strongly closed in \(S\) with respect to \(G\) if for every \(a\in A\) it holds that \(a^G\cap S\leq A\), where \(a^G\) denotes the \(G\)-conjugacy class of \(a\). In particular, when \(A\) is of prime-power order and \(S\) is a Sylow subgroup containing it, \(A\) is simply said to be a strongly closed subgroup. It turns out that \(\mathcal H\)-subgroups of prime-power order are exactly the same as strongly closed subgroups. Note also that Sylow \(p\)-subgroups are \(\mathcal H\)-subgroups.NEWLINENEWLINE In this paper, the authors study groups such that every cyclic subgroup of prime order or of order 4 is an \( \mathcal H\)-subgroup, which are called \(P\mathcal H\)-groups. More concretely, they analyze the structure of minimal non-\(P\mathcal H\)-groups (i.e. non-\(P\mathcal H\)-groups all of whose proper subgroups are \(P\mathcal H\)-groups). Among other results, they show that the structure of minimal non-\(P\mathcal H\)-groups is the same as that of minimal non-\(\mathcal T\)-groups (a \(\mathcal T\)-group is a group for which normality is a transitive relation).
reviewed by: Ana Martínez-Pastor
description: scientific article; zbMATH DE number 6032974