On iterated torsion products of Abelian \(p\)-groups
type: Item
instance of: scholarly article
title: On iterated torsion products of Abelian \(p\)-groups
zbMATH Open document ID: 0773.20021
published in: Rocky Mountain Journal of Mathematics
publication date: 1993-04-01T00:00:00Z
review text: The author considers the group \(\text{Tor}(A,B)\), where \(A\) and \(B\) are \(p\)-groups. The main results show when the group \(\text{Tor}(A,B)\) is a direct sum of countable groups (dsc). Some cases were considered by Nunke and Hill. Let \(\lambda\) be a limit ordinal (\(\lambda\leq\Omega\)), Megibben calls \(G\) a \(C_ \lambda\) group if for every \(\alpha<\lambda\) the group \(G/G(\alpha)\) is a dsc. Theorem 1. Let \(A\) and \(B\) be reduced groups with \(\lambda=\ell(A)<\ell(B)\) (\(\ell(A)\), \(\ell(B)\) are lengths of \(A\) and \(B\)). Then \(\text{Tor}(A,B)\) is a dsc if and only if \(A\) is a dsc and \(B\) is a \(C_ \lambda\) group. Theorem 2. Let \(G\) be a group and \(H\) a reduced dsc of length \(\lambda\). Then \(G\) is \(C_ \lambda\) if and only if \(\text{Tor}(G,H)\) is a dsc. The author introduces a definition which is fundamental for the results of the paper. Let \(G\) be an infinite \(C_ \lambda\) group. By a \(\lambda\)-development of \(G\) we mean an ascending chain of subgroups \(\{G_ i\}_{i\in I}\) of \(G\), where \(I=[0,\alpha]\) for some limit ordinal \(\alpha\) such that we have (a) for all \(i<\alpha\), \(G_ i\) is a \(\lambda\)-pure subgroup of \(G\); (b) \(G_ 0=0\), \(G_ \alpha=G\); (c) if \(i\leq\alpha\) is a limit ordinal, \(\bigcup_{j<i} G_ j=G_ i\). If, in addition to the above, we have (d) \(| G_ i|<| G|\) for all \(i<\alpha\), we will call the \(\lambda\)-development proper. Finally, if in addition to (a)-(d) we have (e) \((G_{i+1}/G_ i)(\lambda)=0\) if and only if \((G/G_ i)(\lambda)=0\), for all \(i<\alpha\) we will call the \(\lambda\)-development normal. Theorem 3. Suppose \(A\) and \(B\) are normal \(C_ \lambda\) groups with \(|\lambda| \leq | A| < | B|\). Then \(\text{Tor}(A,B)\) is a dsc if and only if \(B\) has a \(\lambda\)-development \(\{B_ i\}\) such that, for each \(i\), \(\text{Tor}(A,B_{i+1}/B_ i)\) is a dsc. Furthermore, if \(\text{Tor}(A,B)\) is a dsc, we may choose the \(\lambda\)-development to be normal. In the…
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20K40
zbMATH DE Number: 150393
zbMATH Keywords: \(p\)-groups
zbMATH Keywords: direct sum of countable groups
zbMATH Keywords: reduced groups
zbMATH Keywords: ascending chain of subgroups
zbMATH Keywords: \(\lambda\)-pure subgroup
zbMATH Keywords: normal \(C_ \lambda\) groups
zbMATH Keywords: iterated torsion product
zbMATH Keywords: normal \(\lambda\)-developments
author: Patrick W. Keef
reviewed by: I. Kh. Bekker
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072928
OpenAlex ID: W2035229563
cites work: On singular compactness
cites work: The Balanced-Projective Dimension of Abelian p-Groups
cites work: Isotype subgroups of direct sums of countable groups
cites work: The Recovery of Some Abelian Groups from Their Socles
cites work: When Tor(A,B) is a direct sum of cyclic groups
cites work: The balance of Tor
cites work: On the Tor functor and some classes of Abelian groups
cites work: A generalization of the classical theory of primary groups
cites work: A family of functors defined on generalized primary groups
cites work: Homology and direct sums of countable abelian groups
cites work: On the structure of Tor. II
DOI: 10.1216/RMJM/1181072928
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description: scientific article; zbMATH DE number 150393