Equivalence theorems
type: Item
instance of: scholarly article
title: Equivalence theorems
zbMATH Open document ID: 0781.20032
author: P. Hill
published in: Rocky Mountain Journal of Mathematics
publication date: 1994-02-21T00:00:00Z
review text: Two subgroups of an abelian group are equivalent if there is an automorphism of the group mapping one onto the other. The equivalence problem is to find useful necessary and sufficient conditions on the subgroups and their factor groups for equivalence. This paper is a survey of the known results on the equivalence problem, written by the leading contributor to the area. Even for finite \(p\)- groups the problem is not solved, being equivalent to the classification of finite valuated \(p\)-groups. However, solutions are known for several special cases, and these results are what the author calls Equivalence Theorems. The special cases include arbitrary subgroups of a free group, basic subgroups of a countable \(p\)-group, almost balanced subgroups of a simply presented group, isotype subgroups of a simply presented \(p\)-group and various classes of subgroups of a completely decomposable group.
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20K25
zbMATH DE Number: 218639
zbMATH Keywords: Axiom 3 groups
zbMATH Keywords: abelian group
zbMATH Keywords: equivalence problem
zbMATH Keywords: subgroups
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: finite valuated \(p\)-groups
zbMATH Keywords: equivalence theorems
zbMATH Keywords: basic subgroups
zbMATH Keywords: almost balanced subgroups
zbMATH Keywords: isotype subgroups
zbMATH Keywords: simply presented \(p\)-group
zbMATH Keywords: completely decomposable group
reviewed by: Phillip Schultz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072617
OpenAlex ID: W4247309077
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DOI: 10.1216/RMJM/1181072617
description: scientific article; zbMATH DE number 218639