On subnormal subgroups of factorized groups
Let the group \(G=AB\) be the product of two subgroups \(A\) and \(B\), and let \(H\) be a subgroup of the intersection \(A\cap B\). A well-known result of Maier and Wielandt says that if \(G\) is finite and the subgroup \(H\) is subnormal in \(A\) and \(B\), then \(H\) is also subnormal in \(G\) [see the book ``Products of groups'', Oxford Univ. Press (1992; Zbl 0774.20001), by the first two authors and the reviewer]. Here the authors generalize this theorem to the case when the group \(G\) is nilpotent-by-abelian-by-finite, thus extending a result of Stonehewer for periodic groups of this type. The generalization includes in particular soluble-by-finite linear groups \(G\). The proofs in this paper involve some group ring methods of independent interest. In particular some facts about radical modules play an important role. Here a \(G\)-module \(M\) is radical if and only if the augmentation ideal \(\Delta_\mathbb{Z}(G)\) of the integral group ring \(\mathbb{Z} G\) contains a one-sided ideal \(P\) such that \(\Delta_\mathbb{Z}(G)=G-1+P\) and \(M\) is isomorphic to the \(G\)-module \(\Delta_\mathbb{Z}(G)/P\).
scientific article; zbMATH DE number 1106620
scientific article