Some augmentation quotients of integral group rings.
Let \(\mathbb{Z} G\) be the integral group ring of a group \(G\) and \(\Delta(G)\) be its augmentation ideal. Denote the \(n\)-th associative power of \(\Delta(G)\) by \(\Delta^n(G)\) and let \(\Delta^0(G)=\mathbb{Z} G\). In Section 2 the authors give a complete description of the quotient \(\Delta(G)\Delta^n(H)/\Delta^{n+1}(H)\) for all natural \(n\), where \(H\) is a subgroup of \(G\), and the structures of \(\Delta(G)/\Delta(H)\) and \(\Delta^2(G)/\Delta^2(H)\) are consequently obtained. Besides, if \(K\) is a subgroup of \(H\) a complete description of the quotient \(\Delta(G)\Delta^{n-1}(H)/\Delta(G)\Delta^n(K)\) is given, too. The previous result generalizes a result of \textit{R. Karan} and \textit{L. R. Vermani} [Bull. Lond. Math. Soc. 18, 5-6 (1986; Zbl 0557.20003)]. When \(H\) is a normal subgroup of \(G\) and \(H/[H,G]\) is a direct sum of cyclic groups, or divisible or torsion free and completely decomposable, then in Section 3 the structure of the quotient \[ (\Delta(G)\Delta(H)+\Delta(H)\Delta(G))/(\Delta(G)\Delta^2(H)+\Delta(H)\Delta(G)\Delta(H)+\Delta^2(H)\Delta(G)) \] is determined. In Section 4 a complete description of the quotient \(\Delta^n(F)/\Delta^n(R)\) for all natural \(n\) is given in the case when \(F\) is a free group and \(R\) is any subgroup of \(F\).
scientific article; zbMATH DE number 5615370
scientific article