Automorphism groups of geometric lattices.
type: Item
instance of: scholarly article
title: Automorphism groups of geometric lattices.
zbMATH Open document ID: 1058.06011
author: Manfred Dugas
author: Rüdiger Göbel
published in: Algebra Universalis
publication date: 2005-04-19T00:00:00Z
review text: In the paper automorphism groups of particular lattices are considered. The main feature of the authors' approach is to apply strong theorems for groups and rings to be realized as automorphism groups and endomorphism groups respectively. The following main theorem is proved: Let \(G\) be a group of cardinality \(| G| \leq \lambda \) such that \(\lambda ^{\aleph _0} = \lambda \) if \(2^{\aleph _0} \leq | G| \) and \(\lambda =| G|^{\aleph _0}\) otherwise. Obviously, there is a commutative group ring \(R = \mathbb Z A\) generated by \(| G| \) elements with free additive structure and \(G\) a subgroup of \(\Aut R\). Then there is an \(R\)-module \(M\) of cardinality \(\lambda \) such that its lattice \(L\) of finitely generated submodels has size \(\lambda \) and automorphism group \(\Aut L \simeq G\).
reviewed by: Radomír Halaš
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 06C10
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 51D25
zbMATH DE Number: 2158724
zbMATH Keywords: automorphism groups
zbMATH Keywords: modular lattices
zbMATH Keywords: projective geometry
MaRDI profile type: Publication
full work available at URL: s000120050223
OpenAlex ID: W2053874028
DOI: 10.1007/S000120050223
description: scientific article; zbMATH DE number 2158724