Isomorphism of modular group algebras of direct sums of torsion-complete Abelian \(p\)-groups
type: Item
instance of: scholarly article
title: Isomorphism of modular group algebras of direct sums of torsion-complete Abelian \(p\)-groups
zbMATH Open document ID: 0959.20003
author: Peter Danchev
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2001-02-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108491
review text: The author proposes the following result (Theorem 3.1). Let \(F\) be a prime field of positive characteristic \(p\) and let \(G\) be a direct sum of torsion-complete Abelian \(p\)-groups such that each direct factor has cardinality not exceeding the first uncountable cardinal. Then the isomorphism of the group algebras \(FG\) and \(FH\) as \(F\)-algebras for some group \(H\) implies an isomorphism of \(H\) and \(G\). The necessity of Proposition 2.4 is proved falsely (p. 54, lines 8 and 9). This inaccuracy cannot be corrected if the same idea of proof is utilized. The group \(G\) of Proposition 2.5 must be torsion because otherwise its proof in the cases 2) and 3) is not true. The author groundlessly assumes in the proof of Proposition 2.6 that \(G^{(\alpha)}=G_\alpha\) (p. 56, line 1). Finally, the necessity of Proposition 2.4 is used in the proof of Proposition 2.6 for many factors and the last proposition is exploited in the proof of Theorem 3.1. Therefore, Theorem 3.1 is not proved.
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K21
Mathematics Subject Classification ID: 20K25
zbMATH DE Number: 1319592
zbMATH Keywords: modular group algebras
zbMATH Keywords: isomorphisms
zbMATH Keywords: torsion-complete Abelian \(p\)-groups
zbMATH Keywords: direct sums
reviewed by: Todor Mollov
MaRDI profile type: Publication
cites work: Group algebras of Abelian groups
cites work: Commutative group algebras of $\sigma $-summable abelian groups
cites work: Q4891273
cites work: A classification of direct sums of closed groups
cites work: The Isomorphic Refinement Theorem for Direct Sums of Closed Groups
cites work: A Note on a Theorem of May Concerning Commutative Group Algebras
cites work: Countable Direct Sums of Torsion Complete Groups
cites work: Q3995261
cites work: A Class of Primary Abelian Groups Characterized by Its Socles
cites work: Semi-Complete Primary Abelian Groups
cites work: Modular Group Algebras of Totally Projective p-Primary Groups
cites work: Modular Group Algebras of Simply Presented Abelian Groups
cites work: Q3831145
description: scientific article; zbMATH DE number 1319592