On modules over integer-valued group rings of locally soluble groups with rank restrictions imposed on subgroups.
Let \(\mathbb ZG\) be the integral group ring of a group \(G\) and \(A\) be a \(\mathbb ZG\)-module. The author finds sufficient conditions for \(A\) and \(G\) such that a locally soluble group \(G\) is soluble [see also \textit{O. Yu. Dashkova}, J. Math. Sci., New York 169, No. 5, 636-643 (2010); translation from Fundam. Prikl. Mat. 15, No. 7, 113-125 (2009; Zbl 1238.20051)]. For example, let \(G\) be a locally soluble group and let for some prime number \(p\) or \(p=0\) the rank \(r_p(G)\) be infinite. If for any subgroup \(M\) with infinite rank \(r_p(M)\) the cocentralizer of \(M\) in \(A\) is a minimax \(\mathbb Z\)-module, then \(G\) is soluble. Some properties of soluble groups of this kind are also discussed. It is shown that under some conditions for \(A\) and \(G\) the group \(G\) contains a normal nilpotent subgroup \(H\) such that the rank \(r_p(H)\) is infinite and \(G/H\cong C_{q^\infty}\) for some prime number \(q\).
scientific article; zbMATH DE number 6110035