Upper semimodularity of finite subgroup lattices
type: Item
instance of: scholarly article
title: Upper semimodularity of finite subgroup lattices
zbMATH Open document ID: 0853.20013
author: Francesco Regonati
published in: European Journal of Combinatorics
publication date: 1997-01-06T00:00:00Z
review text: The paper gives a very nice combinatorial characterization of upper semimodular subgroup lattices. On one hand, using the known description of finite groups with upper semimodular subgroup lattices, the author shows that in such a lattice every interval sublattice has the property that the number of atoms is less than or equal to the number of coatoms in it. On the other hand, even a stronger converse is true: by inspecting the minimal sections with non-upper-semimodular subgroup lattice it is obtained that if the subgroup lattice of a finite group is not upper semimodular, then it contains an interval sublattice of height 3 in which the number of atoms exceeds the number of coatoms.
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 06C10
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 895877
zbMATH Keywords: upper semimodular subgroup lattices
zbMATH Keywords: finite groups
zbMATH Keywords: number of atoms
zbMATH Keywords: number of coatoms
zbMATH Keywords: minimal sections
zbMATH Keywords: interval sublattice
reviewed by: Péter P. Pálfy
MaRDI profile type: Publication
full work available at URL: eujc.1996.0034
OpenAlex ID: W2019918622
DOI: 10.1006/EUJC.1996.0034
description: scientific article; zbMATH DE number 895877