On the coperiodic hull of a separable \(p\)-group
type: Item
instance of: scholarly article
title: On the coperiodic hull of a separable \(p\)-group
zbMATH Open document ID: 0701.20031
DOI: 10.1007/BF01979377
author: A. I. Moskalenko
published in: Algebra and Logic
publication date: 1989-01-01T00:00:00Z
review text: It is well-known that every reduced abelian group \(G\) embeds into the group \(G^*=Ext(Q/Z,G)\) called the cotorsion hull of \(G\). In order to reveal the structure of \(G^*\), the author investigates the structure of \(T^*\), T being a separable \(p\)-group. Clearly, \(T^*=Ext(Z(p^{\infty}),T)\) and the main result of section 1 states that \(T^*\) is isomorphic to the group of all sequences of the form \((a_ 0,a_ 1+T,...,a_ i+T,...)\), \(a_ i\) being elements of the \(p\)-adic completion of \(T\) satisfying \(p(a_{i+1}+T)=a_ i+T\). The result enables to calculate easily heights and indicators of elements of \(T^*\). Section 3 is devoted to the case when \(T\) is a direct sum of cyclic groups. Then \(T^*\) is shown to be fully transitive and there is a one-to-one correspondence between fully characteristic subgroups of \(T^*\) and dual ideals of the semilattice of certain sequences. An example of a separable \(p\)-group \(T\) such that \(T^*\) is not fully transitive is presented.
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20K35
zbMATH DE Number: 4148419
zbMATH Keywords: reduced abelian groups
zbMATH Keywords: cotorsion hulls
zbMATH Keywords: separable \(p\)-groups
zbMATH Keywords: \(p\)-adic completions
zbMATH Keywords: heights
zbMATH Keywords: indicators
zbMATH Keywords: direct sums of cyclic groups
zbMATH Keywords: fully transitive subgroups
zbMATH Keywords: fully characteristic subgroups
reviewed by: Tomáš Kepka
MaRDI profile type: Publication
cites work: Q5609457
cites work: Q5673086
cites work: Q5639910
full work available at URL: https://doi.org/10.1007/bf01979377
OpenAlex ID: W2087990631
description: scientific article; zbMATH DE number 4148419