The transitivity of local Warfield groups
type: Item
instance of: scholarly article
title: The transitivity of local Warfield groups
zbMATH Open document ID: 0916.20039
author: P. Hill
author: W. Ullery
published in: Journal of Algebra
publication date: 1999-05-25T00:00:00Z
review text: In any algebraic system two elements are indistinguishable in some sense if one can be mapped onto the other by an automorphism. This is a global property and one may ask how this relation between elements can be detected locally by inspecting the elements. Let \(G\) be an abelian group and \(a\) and \(b\) two elements of \(G\). If \(a\) can be mapped onto \(b\) by an automorphism, the \(a\) and \(b\) must have the same \(p\)-height for every \(p\), and as Kaplansky observed, their height sequences \((\text{height}_p(p^na):n=0,1,\ldots)\) and \((\text{height}_p(p^nb):n=0,1,\ldots)\) must be the same. As there was no other local property in sight, Kaplansky defined a \(p\)-group to be transitive if any two elements with identical height sequences could be mapped onto one another by an automorphism. It turned out that not all \(p\)-groups are transitive but the well-behaved \(p\)-groups, such as totally projective groups, were transitive. In the present paper the question of transitivity is studied in local \(p\)-groups, i.e., modules over the integers localized at \(p\), that are mixed. In this generality transitivity is rare. The authors succeed (Theorem 8) in giving a local characterization of elements that can be mapped onto one another by an automorphism in a local Warfield group. The characterization adds to the height sequences another invariant called type vector. Concrete examples are given of Warfield groups of rank \(3\) that are not transitive. The proofs engage the large tool box developed for Warfield groups and there are many technical but interesting new concepts. The paper may serve as a model of exposition with very careful and complete explanations for the reader.
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K21
Mathematics Subject Classification ID: 20K30
zbMATH DE Number: 1230356
zbMATH Keywords: Warfield groups
zbMATH Keywords: transitive groups
zbMATH Keywords: automorphisms
zbMATH Keywords: height sequences
zbMATH Keywords: local \(p\)-groups
zbMATH Keywords: mixed groups
zbMATH Keywords: type vectors
reviewed by: Adolf Mader
MaRDI profile type: Publication
full work available at URL: jabr.1998.7539
OpenAlex ID: W2083336670
cites work: Q5688707
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cites work: Axiom 3 Modules
cites work: Q5589390
cites work: Isotype Submodules of p-Local Balanced Projective Groups
DOI: 10.1006/JABR.1998.7539
description: scientific article; zbMATH DE number 1230356