On lattice-isomorphic Abelian groups
type: Item
instance of: scholarly article
title: On lattice-isomorphic Abelian groups
zbMATH Open document ID: 0761.20009
DOI: 10.1007/BF01292920
author: Kazem Mahdavi
author: John C. Poland
published in: Archiv der Mathematik
publication date: 1992-09-27T00:00:00Z
review text: The authors consider the question of when two Abelian groups of torsion-free rank one can have the same lattice of subgroups. They review the history of published, sometimes incorrect, results on lattice-isomorphic Abelian groups, and carefully derive when two splitting Abelian groups of torsion-free rank one can be lattice-isomorphic. This occurs precisely when their torsion subgroups are isomorphic and the type of one of the groups can be obtained from the type of the other by some permutation of the primes which leaves fixed all primes occurring as orders of elements. The method of proof involves a direct construction of the correspondence between cyclic subgroups and hence all subgroups.
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20K21
Mathematics Subject Classification ID: 20K27
zbMATH DE Number: 62260
zbMATH Keywords: Abelian groups of torsion-free rank one
zbMATH Keywords: lattices of subgroups
zbMATH Keywords: lattice-isomorphic Abelian groups
zbMATH Keywords: splitting Abelian groups
zbMATH Keywords: types
MaRDI profile type: Publication
cites work: Crossed Isomorphisms
cites work: Q3281123
cites work: Q5673086
cites work: On mixed groups of torsion-free rank one
cites work: Torsion-free and mixed abelian groups
cites work: Q5840056
cites work: Q3232305
full work available at URL: https://doi.org/10.1007/bf01292920
OpenAlex ID: W1996037238
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description: scientific article; zbMATH DE number 62260