Subgroup-closed lattice formations.
type: Item
instance of: scholarly article
title: Subgroup-closed lattice formations.
zbMATH Open document ID: 1346.20017
published in: Journal of Algebra
publication date: 2015-10-13T00:00:00Z
review text: The paper under review deals with finite groups. Therefore, the word ``group'' will mean ``finite group''. Recall that a class of finite groups that is closed under taking quotients and subdirect products is called a \textit{formation}. If every subgroup of a group in a formation \(\mathfrak F\) is also in \(\mathfrak F\), then \(\mathfrak F\) is called subgroup-closed. Moreover, if the formation \(\mathfrak F\) is closed under Frattini extensions, \(\mathfrak F\) is said to be saturated. If \(\mathfrak F\) is a formation, then every group \(G\) has a smallest normal subgroup \(G^{\mathfrak F}\), called the \textit{\(\mathfrak F\)-residual} of \(G\), for which the corresponding quotient is in \(\mathfrak F\). A subgroup \(H\) of a group \(G\) is said to be \textit{\(\mathfrak F\)-subnormal} (respectively, \textit{K-\(\mathfrak F\)-subnormal}) in \(G\) if either \(H=G\) or there exists a maximal chain of subgroups \(H=H_0\leq H_1\leq\cdots\leq H_n=G\) such that \(H_i^{\mathfrak F}\leq H_{i-1}\) (respectively, \(H_{i-1}\) is a normal subgroup of \(H_i\) or \(H_i^{\mathfrak F}\leq H_{i-1}\)) for \(i\in\{1,\ldots,n\}\). In general, for a subgroup-closed formation \(\mathfrak F\), the set of all \(\mathfrak F\)-subnormal (respectively, K-\(\mathfrak F\)-subnormal) subgroups of a group \(G\) is not a lattice of the subgroup lattice of \(G\). The authors obtain that the following statements are equivalent for a subgroup-closed formation \(\mathfrak F\): 1. The set of all K-\(\mathfrak F\)-subnormal subgroups is a sublattice of the subgroup lattice of every group. 2. The set of all \(\mathfrak F\)-subnormal subgroups is a sublattice of the subgroup lattice of every group. 3. \(\mathfrak F=\mathfrak M\times\mathfrak K\times\mathfrak L\) for some subgroup-closed formations \(\mathfrak M\), \(\mathfrak K\), and \(\mathfrak L\) satisfying the following conditions: (a) \(\pi(\mathfrak M)\cap\pi(\mathfrak K)=\emptyset\), \(\pi(\mathfrak K)\cap\pi(\mathfrak L)=\emptyset\), and \(\…
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D35
zbMATH DE Number: 6493622
zbMATH Keywords: formations of finite groups
zbMATH Keywords: saturated formations
zbMATH Keywords: lattice formations
zbMATH Keywords: subgroup-closed formations
zbMATH Keywords: residuals
zbMATH Keywords: chains of subgroups
zbMATH Keywords: lattices of subgroups
zbMATH Keywords: \(\mathfrak F\)-subnormal subgroups
zbMATH Keywords: K-\(\mathfrak F\)-subnormal subgroups
author: Xiaolan Yi
author: S. F. Kamornikov
reviewed by: R. Esteban-Romero
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.07.021
OpenAlex ID: W1777037706
cites work: On the lattice of \(\mathcal F\)-subnormal subgroups
cites work: Classes of Finite Groups
cites work: The \(\mathfrak F\)-normalizers of a finite soluble group
cites work: Finite soluble groups
cites work: Any solubly saturated hereditary lattice formation is saturated.
cites work: Subgroup lattices of finite groups which properly contain the lattice of subnormal subgroups
cites work: Q3084921
cites work: Q3961010
cites work: Q4802810
cites work: Q4344341
DOI: 10.1016/J.JALGEBRA.2015.07.021
description: scientific article; zbMATH DE number 6493622