The relative second Fox and third dimension subgroup of arbitrary groups. (Q999249)

Let \(I(G)\) be the augmentation ideal of the group ring \(KG\) of a group \(G\) over a commutative ring \(R\) with identity. In this paper the author gives a complete description of the third relative dimension subgroup \(G\cap(1+I(K)I(G)+I^3(G))\) and the second relative Fox subgroup \(G\cap(1+I(K)I(H)+I^2(G)I(H))\) for any subgroups \(K\) and \(H\) of \(G\) and coefficient ring \(R\). If \(R\) is the ring of integers \(\mathbb{Z}\), \(H\) is a normal subgroup of \(G\) and \(K\) is a certain subgroup of \(G\) containing the subgroup \([H,G]\), then these subgroups are studied by different methods by \textit{K.-I. Tahara}, \textit{L. R. Vermani} and \textit{A. Razdan} [Can. Math. Bull. 41, No. 1, 109-117 (1998; Zbl 0910.20004)].