Finite groups with nilpotent commutator subgroups, having a distributive lattice of normal subgroups
type: Item
instance of: scholarly article
title: Finite groups with nilpotent commutator subgroups, having a distributive lattice of normal subgroups
zbMATH Open document ID: 0588.20019
DOI: 10.1016/0021-8693(86)90109-2
author: P. Longobardi
author: M. Maj
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: The authors are concerned with a finite group G whose lattice, n(G), of normal subgroups is distributive. First they show that if G' is nilpotent then n(G) is distributive if and only if every epimorph of G has a faithful irreducible character; in fact, if G is solvable and n(G) is distributive then every epimorph of G has a faithful irreducible character. An example is given of a group G with G' supersolvable for which every epimorph has a faithful irreducible character but n(G) is not distributive. Then the distributivity of n(G) when G' is nilpotent is reduced to the distributivity of n(H) for certain subgroups H of G. This means that the distributivity of n(G) for G supersolvable reduces to the strictly p-closed case; distributivity in this case is characterized in Section 2. Then in Section 3 a nonsupersolvable group G is constructed, using the wreath product, with G' nilpotent and n(G) distributive.
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 3944135
zbMATH Keywords: distributive lattice of normal subgroups
zbMATH Keywords: lattice of normal subgroups
zbMATH Keywords: finite group
zbMATH Keywords: nilpotent
zbMATH Keywords: faithful irreducible character
zbMATH Keywords: supersolvable
zbMATH Keywords: distributivity
zbMATH Keywords: strictly p-closed
reviewed by: W. E. Deskins
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90109-2
OpenAlex ID: W2024136399
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description: scientific article; zbMATH DE number 3944135