Fully inert subgroups of free Abelian groups.
type: Item
instance of: scholarly article
title: Fully inert subgroups of free Abelian groups.
zbMATH Open document ID: 1322.20046
published in: Periodica Mathematica Hungarica
publication date: 2015-01-07T00:00:00Z
review text: A subgroup \(H\) of an Abelian group \(G\) is fully inert if for every \(\varphi\in\text{End}(G)\), \((\varphi H+H)/H\) is finite. The notion is used to study the dynamical properties of endomorphisms. In this paper, the authors continue their previous research [J. Group Theory 16, No. 6, 915-939 (2013; Zbl 1292.20062)] on the topic by characterising the fully inert subgroups of a free group, \(G\), in the non-trivial case that \(G\) has infinite rank. The main result states that \(H\) is fully inert in \(G\) if and only if (1) \(G/H\) is bounded; (2) for each prime \(p\), at most one Ulm-Kaplansky invariant of the \(p\)-component \(M_p\) of \(G/H\) is infinite, and equal to the rank of \(G\); (3) The rank of the elementary \(p\)-group \((H+pG)/pG\) is finite for each prime \(p\) such that \(M_p\) is infinite. The proof consists of an ingenious use of a sequence of elementary lemmas.
reviewed by: Phillip Schultz
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K20
zbMATH DE Number: 6384177
zbMATH Keywords: free Abelian groups
zbMATH Keywords: fully inert subgroups
zbMATH Keywords: endomorphisms
zbMATH Keywords: Ulm-Kaplansky invariants
zbMATH Keywords: Hill-Megibben invariants
author: Dikran Dikranjan
author: Luigi Salce
author: P. Zanardo
MaRDI profile type: Publication
full work available at URL: s10998-014-0041-4
OpenAlex ID: W1982437195
cites work: Generalizations of the Stacked Bases Theorem
cites work: Stacked bases for modules over principal ideal domains
cites work: Intrinsic algebraic entropy
cites work: Fully inert subgroups of divisible Abelian groups
cites work: Q5609457
DOI: 10.1007/S10998-014-0041-4
description: scientific article; zbMATH DE number 6384177