On \(\mathfrak F_n\)-normal subgroups of finite groups.
type: Item
instance of: scholarly article
title: On \(\mathfrak F_n\)-normal subgroups of finite groups.
zbMATH Open document ID: 1230.20015
DOI: 10.1134/S0037446611020029
published in: Siberian Mathematical Journal
publication date: 2011-06-24T00:00:00Z
review text: All groups considered are finite. For a class \(\mathfrak F\) of groups, in the paper under review a subgroup \(H\) of a group \(G\) is defined to be \(\mathfrak F_n\)-normal in \(G\) if there exists a normal subgroup \(T\) of \(G\) such that \(HT\) is a normal subgroup of \(G\) and \((H\cap T)H_G/H_G\) is contained in the \(\mathfrak F\)-hypercenter \(Z_\infty^{\mathfrak F}(G/H_G)\) of \(G/H_G\), where \(H_G\) denotes the core of \(H\) in \(G\). We recall that the \(\mathfrak F\)-hypercenter is the largest normal subgroup of \(G\) whose \(G\)-chief factors are \(\mathfrak F\)-central (a chief factor \(H/K\) of \(G\) is \(\mathfrak F\)-central if \([H/K](G/C_G(H/K))\in\mathfrak F\)). This concept is a generalization of \(c\)-normality (introduced by \textit{Y. Wang} [in J. Pure Appl. Algebra 110, No. 3, 315-320 (1996; Zbl 0853.20015)]), \(\mathfrak F_n\)-supplementation (or \(\mathfrak F_c\)-normality) (considered by \textit{N. Yang} and the first author [in Asian-Eur. J. Math. 1, No. 4, 619-629 (2008; Zbl 1176.20018)] and by \textit{A. Y. Alsheik Ahmad, J. J. Jaraden} and \textit{A. N. Skiba} [in Algebra Colloq. 14, No. 1, 25-36 ( 2007; Zbl 1126.20012)]) and \(\mathfrak F_h\)-normality (defined by \textit{X. Feng} and the first author [in Front. Math. China 5, No. 4, 653-664 (2010; Zbl 1226.20011)]). In this paper the authors use this new embedding property to obtain some criteria for supersolubility and \(p\)-nilpotency (\(p\) a prime) of groups, taking further some previous developments.
reviewed by: Ana Martínez-Pastor
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 5912826
zbMATH Keywords: finite groups
zbMATH Keywords: supersoluble groups
zbMATH Keywords: saturated formations
zbMATH Keywords: \(p\)-nilpotent groups
zbMATH Keywords: supersolubility
zbMATH Keywords: \(p\)-nilpotency
zbMATH Keywords: supplemented subgroups
zbMATH Keywords: c-normal subgroups
zbMATH Keywords: hypercenter
author: Wenbin Guo
author: Xiaolong Yu
MaRDI profile type: Publication
cites work: Q3941589
cites work: Finite groups whose minimal subgroups are normal
cites work: Two sufficient conditions for supersolvability of finite groups
cites work: Influence of normality on maximal subgroups of Sylow subgroups of a finite group
cites work: \(C\)-normality of groups and its properties
cites work: On $\mathfrak{F}_n$-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS
cites work: Q3994689
cites work: On \(\mathcal F\)-supplemented subgroups of finite groups.
cites work: On weakly \(s\)-permutable subgroups of finite groups.
cites work: Q3961010
cites work: Q3629861
cites work: Q3913574
cites work: The classification of (J, σ)-irreducible monoids of typeA4
cites work: Q4676829
cites work: \(G\)-covering subgroup systems for the classes of supersoluble and nilpotent groups.
cites work: On minimal subgroups of finite groups
cites work: ON c-NORMAL MAXIMAL AND MINIMAL SUBGROUPS OF SYLOW SUBGROUPS OF FINITE GROUPS
cites work: On \(c\)-normal maximal and minimal subgroups of Sylow \(p\)-subgroups of finite groups.
full work available at URL: s0037446611020029
OpenAlex ID: W1972985361
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description: scientific article; zbMATH DE number 5912826