On the Reidemeister spectrum of an abelian group
\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.
scientific article; zbMATH DE number 7039549