Some remarks on subgroups determined by certain ideals in integral group rings
This paper is a contribution to the problem of identification of (normal) subgroups of a group \(G\) determined by the two-sided ideals contained in the augmentation ideal \(I(G)\) of its integral group ring \(\mathbb{Z} G\). Let \(H\) be a normal subgroup of \(G\). The authors study the ideals \(I^ n(G) + I(G)I(H)\) and \(I(H_ 1) I(H_ 2) \dots I(H_ m)\) (and sums of such ideals), where \(H_ i \in \{H,G\}\) with exactly one \(H_ i = H\). A typical result from Sections 2 and 3 is that \(G \cap (1 + I(G) I(H) I(G)) = \gamma_ 3(H)\), the third term in the lower central series of \(G\), provided \(G = HK\) with \(H \cap K\) central in \(H\). The main result of Section 4 (Theorem 4.4) states that \(G \cap (1 + I^ n(G) + I(G) I(H)) = \gamma_ n(G) \gamma_ 2(H)\), provided \(G/H\) is free Abelian [see also \textit{N. D. Gupta, A. W. Hales} and the reviewer, J. Reine Angew. Math. 346, 194-198 (1984; Zbl 0513.20003)].
scientific article; zbMATH DE number 598344
scientific article