On modules over group rings of nilpotent groups.
Let \(R\) be a ring, \(G\) a group, and \(A\) an \(RG\)-module. If \(H\) is a subgroup of \(G\), then \(C_A(H)\) is an \(R\)-submodule of \(A\), and the \(R\)-module \(C_A(H)\) is called a cocentralizer of \(H\) in \(A\). There arises the natural question about the impact of cocentralizers of distincts important families of subgroups of \(G\) on the structure of \(A\). This approach has been successfully implemented in the theory of infinite dimensional linear groups. One of the papers dedicated to this topic [KMO = \textit{L. A. Kurdachenko, J. M. Muñoz-Escolano} and \textit{J. Otal}, Publ. Mat., Barc. 52, No. 1, 151-169 (2008; Zbl 1149.20030)] became the basis for the article under review where the author considers the family \(L_{nm}(G)\) consisting of all subgroups \(H\) such that \(A/C_A(H)\) is not minimax. More precisely, the author studies the case when \(L_{nm}(G)\) satisfies the weak minimal or maximal condition. If in this case the group \(G\) is nilpotent, then \(G\) is minimax (that is, the group \(G\) satisfies the weak minimal and maximal conditions). This is the main result of the paper under review. Note, that this result is a direct analogy of one of the results of the article [KMO, loc. cit.]. In such situations, there is always the question of how much of the technique that is used in a generalized situation is different from the technique that was developed for linear groups in [KMO, loc. cit.]. We have to admit that one cannot find a conceptual difference here. Moreover, using a particular technique, the author does not bother to specify the source from which she took it.
scientific article; zbMATH DE number 6128286
scientific article