\(M_ 7\) as an interval in a subgroup lattice
type: Item
instance of: scholarly article
title: \(M_ 7\) as an interval in a subgroup lattice
zbMATH Open document ID: 0567.20011
DOI: 10.1007/BF01194535
author: Peter Köhler
published in: Algebra Universalis
publication date: 1983-01-01T00:00:00Z
review text: This note is a contribution to the problem, whether every finite lattice is isomorphic to an interval in the subgroup lattice of a finite group. The problem originates from universal algebra [\textit{P. P. Pálfy} and \textit{P. Pudlák}, Algebra Univers. 11, 22-27 (1980; Zbl 0386.06002)]. Let \(M_ n\) be the lattice of length two with n atoms, G a finite group and H a subgroup of G. Suppose that the interval between H and G is the subgroup lattice of G is isomorphic to \(M_ n\) and H contains no nontrivial normal subgroup of G. The author proves that if n-1 is not a prime power then G is subdirectly irreducible with nonabelian minimal normal subgroup. Examples are known only for \(n=7\) and \(n=11\) [\textit{W. Feit}, ibid. 17, 220-221 (1983)].
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 06B15
zbMATH DE Number: 3904821
zbMATH Keywords: finite lattice
zbMATH Keywords: interval in the subgroup lattice of a finite group
zbMATH Keywords: lattice of length two
zbMATH Keywords: subdirectly irreducible
zbMATH Keywords: nonabelian minimal normal subgroup
reviewed by: Péter P. Pálfy
MaRDI profile type: Publication
cites work: Q5727672
cites work: Endliche Gruppen I
full work available at URL: https://doi.org/10.1007/bf01194535
OpenAlex ID: W2022701405
description: scientific article; zbMATH DE number 3904821