Special quasi-triads and integral group rings of finite representation type. II
type: Item
instance of: scholarly article
title: Special quasi-triads and integral group rings of finite representation type. II
zbMATH Open document ID: 0790.20014
author: Jeremy Haefner
author: Lee Klingler
published in: Journal of Algebra
publication date: 1994-06-27T00:00:00Z
review text: Let \(G\) be a finite group with cyclic Sylow \(p\)-subgroups for all primes \(p\) dividing \(| G|\). In the paper under review, first the rational group algebra \(\mathbb{Q} G\) is described as a direct sum of cyclic crossed product algebras. As a consequence, it is shown that, for such \(G\), the isomorphism type of \(\mathbb{Q} G\) determines that of \(G\). Then, for every prime \(p\), the localization \(\mathbb{Z}_{(p)}G\) and the completion \(\mathbb{Z}_ pG\) are described as direct sums of iterated pullbacks. When the Sylow \(p\)-subgroups of \(G\) have order dividing \(p^ 2\), the indecomposable \(\mathbb{Z}_ pG\)-lattices are determined using the results of part I [see the preceding review Zbl 0790.20013)] of this paper. This leads to a method of constructing the genera of all indecomposable \(\mathbb{Z} G\)-lattices. As an application, the authors are able to classify all finite groups \(G\) having the property that every indecomposable \(\mathbb{Z} G\)-lattice is isomorphic to a right ideal. Moreover, they determine those finite groups \(G\) with cyclic Sylow subgroups having the property that every \(\mathbb{Z} G\)-lattice has a unique number of indecomposable direct summands.
reviewed by: Burkhard Külshammer
Mathematics Subject Classification ID: 20C11
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 480476
zbMATH Keywords: finite lattice type
zbMATH Keywords: isomorphism problem
zbMATH Keywords: cyclic Sylow \(p\)-subgroups
zbMATH Keywords: rational group algebras
zbMATH Keywords: cyclic crossed product
zbMATH Keywords: finite groups
MaRDI profile type: Publication
full work available at URL: jabr.1993.1136
OpenAlex ID: W2047294650
DOI: 10.1006/JABR.1993.1136
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description: scientific article; zbMATH DE number 480476