A basis for augmentation quotients of finite Abelian groups.
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)].
scientific article; zbMATH DE number 5888186
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