Local formations in which every subformation of type \({\mathcal N}_ p\) has a complement
type: Item
instance of: scholarly article
title: Local formations in which every subformation of type \({\mathcal N}_ p\) has a complement
zbMATH Open document ID: 0884.20011
DOI: 10.1007/BF02884221
published in: Chinese Science Bulletin
publication date: 1997-11-25T00:00:00Z
review text: A subformation \({\mathfrak F}_1\) of a formation \({\mathfrak F}\) is said to have a complement in \({\mathfrak F}\), if there is a subformation \({\mathfrak F}_2\) of \({\mathfrak F}\) such that \({\mathfrak F}_1\cap{\mathfrak F}_2=(1)\) and \({\mathfrak F}=\text{form}({\mathfrak F}_1\cup{\mathfrak F}_2)\). The following theorem is a description of local formations in which every local subformation has a complement. Theorem. Let \({\mathfrak F}\neq (1)\) be a local formation. Then the following conditions are equivalent: (i) Every subformation of type \({\mathfrak N}_p\) \((p\in\pi({\mathfrak F}))\) has a complement in \({\mathfrak F}\); (ii) \({\mathfrak F}\) is a nilpotent formation; (iii) The lattice of local subformations of \({\mathfrak F}\) is Boolean.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F17
zbMATH DE Number: 1077943
zbMATH Keywords: complements
zbMATH Keywords: local formations
zbMATH Keywords: nilpotent formations
zbMATH Keywords: lattices of local subformations
author: Wenbin Guo
reviewed by: Nikolay T. Vorob'ev
MaRDI profile type: Publication
cites work: Q3961010
cites work: Q3994689
cites work: Finite soluble groups
cites work: Q3909200
cites work: Q4712395
cites work: Q3934684
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description: scientific article; zbMATH DE number 1077943