On the Reidemeister spectrum of an abelian group
type: Item
instance of: scholarly article
title: On the Reidemeister spectrum of an abelian group
zbMATH Open document ID: 1459.20042
author: B. Goldsmith
author: Fatemeh Karimi
author: Noel White
published in: Forum Mathematicum
publication date: 2019-03-20T00:00:00Z
full work available at URL: https://arrow.tudublin.ie/cgi/viewcontent.cgi
review text: \par Let \(G\) be an Abelian group throughout. For an automorphism \(\phi\) of \(G\), the Reidemeister number \(R(\phi)\) of \(\phi\) is the cardinality of the quotient group \(G/(\phi -1_G)(G)\) and the Reidemeister spectrum of \(G\) is the set of all Reidemeister numbers for all automorphisms of \(G\). A group \(G\) is said to have the \(R_{\infty}\) property if \(R(\phi)\) is infinite for every automorphism \(\phi\) of \(G\). Reidemeister numbers for infinite abelian groups were first studied in [\textit{K. Dekimpe} and \textit{D. Gonçalves}, Topol. Methods Nonlinear Anal. 46, No. 2, 773--784 (2015; Zbl 1372.20051)] where the \(R_{\infty}\) property played a key role. Their work was a motivating factor for the current paper. In the present paper, the authors generalize the \(R_{\infty}\) property in order to use the potentially different cardinalities of the infinite Reidemeister numbers. Section 3 works on finding the (quite complicated) Reidemeister spectrum of torsion-free abelian groups of rank 1 and offers several illustrative examples. Section 4 explores the Reidemeister spectrum of direct sums and direct products, broken into homogeneous and non-homogeneous sums and products. A complete description is very difficult owing in part to set theoretic issues that arise, but some results are obtained by assuming the generalized continuum hypothesis. Section 5 utilizes the area of abelian group theory that identifies certain rings as being endomorphism rings of abelian groups. Here, the authors extend results of [loc. cit.] by providing reduced groups of arbitrarily large cardinality whose spectrum consists of a single cardinal-the cardinality of \(G\). The authors also provide an answer to a comment in [loc. cit.] by exhibiting an abelian 2-group which has the \(R_{\infty}\) property.
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20K30
zbMATH DE Number: 7039549
zbMATH Keywords: Reidemeister number
zbMATH Keywords: Reidemeister spectrum
zbMATH Keywords: \(R_{\infty}\)-property
zbMATH Keywords: rank one group
zbMATH Keywords: type
reviewed by: Ross P. Abraham
MaRDI profile type: Publication
OpenAlex ID: W2891684369
cites work: Finite rank torsion free Abelian groups and rings
cites work: Every Countable Reduced Torsion-Free Ring is an Endomorphism Ring
cites work: ON ENDOMORPHISM RINGS OF PRIMARY ABELIAN GROUPS
cites work: The \(R_{\infty}\) property for abelian groups
cites work: Summands of Separable Abelian Groups: Dedicated to Paul Turán on his 60th Birthday
cites work: Note on extensions of Abelian groups by primary groups
cites work: Approximations and endomorphism algebras of modules.
Wikidata QID: Q129212030
DOI: 10.1515/FORUM-2017-0184
description: scientific article; zbMATH DE number 7039549