Determinability of periodic Abelian groups by their endomorphism groups
type: Item
instance of: scholarly article
title: Determinability of periodic Abelian groups by their endomorphism groups
zbMATH Open document ID: 0845.20043
DOI: 10.1007/BF02304413
author: S. Ya. Grinshpon
author: A. M. Sebel'din
published in: Mathematical Notes
publication date: 1996-05-20T00:00:00Z
review text: Let \(\mathfrak X\) be some class of abelian groups. It is said that an abelian group \(G\) is determined by its endomorphism group in the class \(\mathfrak X\) if there is no group \(H\) in \(\mathfrak X\) such that \(E^+(G)\cong E^+(H)\) and \(H\not\cong G\) (\(E^+(G)\) and \(E^+(H)\) are the endomorphism groups of the groups \(G\) and \(H\)). Denote by \(G_p\) the \(p\)-component of the group \(G\). The main result of the paper is as follows. Theorem (ZFC + G.C.H.). A torsion abelian group \(A\) is determined by its endomorphism group in the class of all abelian groups if and only if 1) \(A\) is a bounded group; 2) if \(f_{A_p}(k)\geq\aleph_0\), then for \(\ell<k\) \(f_{A_p}(\ell)=0\) or \(f_{A_p}(\ell)>f_{A_p}(k)\) (\(k,\ell\in\mathbb{N}\)), \(f_{A_p}(\ell)\) the \(\ell\)-th Ulm-Kaplansky invariant.
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K10
zbMATH DE Number: 857372
zbMATH Keywords: endomorphism groups
zbMATH Keywords: torsion Abelian groups
zbMATH Keywords: Ulm-Kaplansky invariants
reviewed by: I. Kh. Bekker
MaRDI profile type: Publication
cites work: Endomorphisms of abelian groups and the theorem of Baer and Kaplansky
description: scientific article; zbMATH DE number 857372