A basis for augmentation quotients of finite Abelian groups.
type: Item
instance of: scholarly article
title: A basis for augmentation quotients of finite Abelian groups.
zbMATH Open document ID: 1232.20006
author: Shan Chang
author: Guoping Tang
published in: Journal of Algebra
publication date: 2011-05-16T00:00:00Z
review text: For a group \(G\), let \(\mathbb Z[G]\) denote its integral group ring and \(\Delta(G)\) the augmentation ideal of \(\mathbb Z[G]\). In case \(G\) is a finite Abelian group, the augmentation quotients \(\Delta^n(G)/\Delta^{n+1}(G)\), \(n\geq 1\), are all finite Abelian groups [\textit{I. B. S. Passi}, J. Algebra 9, 121-151 (1968; Zbl 0159.31502)], and it is of interest to determine their precise structure. It has been known that the structure becomes stationary for all large \(n\) [\textit{F. Bachmann} and \textit{L. Grünenfelder}, J. Pure Appl. Algebra 5, 253-264 (1974; Zbl 0301.16011)]. The structure of the terminal quotient has been computed by \textit{A. W. Hales} [Pac. J. Math. 118, 401-410 (1985; Zbl 0573.20008)]. In the present paper the authors determine an explicit basis for these quotients and also their structure in general, thus completing the solution of this problem [\textit{G. Karpilovsky}, Commutative group algebras. Pure Appl. Math. 78. New York-Basel: Marcel Dekker (1983; Zbl 0508.16010)].
reviewed by: Inder Bir S. Passi
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20K01
zbMATH DE Number: 5888186
zbMATH Keywords: integral group rings
zbMATH Keywords: augmentation ideals
zbMATH Keywords: augmentation quotients
zbMATH Keywords: finite Abelian groups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.08.020
OpenAlex ID: W2052589555
cites work: The periodicity in the graded ring associated with an integral group ring
cites work: Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion Abelian groups.
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cites work: Q3035674
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cites work: Structure of augmentation quotients of finite homocyclic Abelian groups.
cites work: Presenting powers of augmentation ideals of elementary \(p\)-groups
DOI: 10.1016/J.JALGEBRA.2010.08.020
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description: scientific article; zbMATH DE number 5888186