A result on \(B_ 1\)-groups
type: Item
instance of: scholarly article
title: A result on \(B_ 1\)-groups
zbMATH Open document ID: 0857.20036
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1997-03-05T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108382
review text: This paper repairs a gap in the proof of a theorem of \textit{D. Arnold} [in Boll. Unione Mat. Ital., VI. Ser., A 5, 175-184 (1986; Zbl 0601.20050)] as well as the proof of Bican-Salce that a countable \(B_1\)-group is finitely Butler. Later results relying on these two are thereby validated as well. Recall that a torsion-free Abelian group \(G\) is a \(B_1\)-group if \(\text{Bext}(G,T)=0\) for all torsion-groups \(T\); and \(G\) is finitely Butler if any pure finite rank subgroup is a Butler group. A subgroup \(K\) of a torsion-free Abelian group \(G\) is called generalized regular if \(G/K\) is torsion and for any rank-one pure subgroup \(X\) of \(G\), the \(p\)-component of \(X/(X\cap K)\) is zero for almost all primes \(p\). Theorem. Let \(H\) be an arbitrary \(B_1\)-group and \(B\) a generalized regular subgroup of \(H\). Then for any finite rank pure subgroup \(L\) of \(H\), the torsion group \((L+B)/B\) has at most finitely many nonzero \(p\)-components.
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 20K27
zbMATH DE Number: 889832
zbMATH Keywords: countable \(B_ 1\)-groups
zbMATH Keywords: torsion-free Abelian groups
zbMATH Keywords: pure finite rank subgroups
zbMATH Keywords: Butler groups
zbMATH Keywords: generalized regular subgroups
zbMATH Keywords: torsion groups
author: Ladislav Bican
author: Kulumani M. Rangaswamy
reviewed by: Charles Vinsonhaler
MaRDI profile type: Publication
cites work: Q3735945
cites work: Q3661801
cites work: A characterization of countable Butler groups
cites work: On some subgroups of infinite rank Butler groups
cites work: Infinite Rank Butler Groups
cites work: Q5609457
cites work: Q4025946
cites work: Q4025957
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description: scientific article; zbMATH DE number 889832