Groups whose proper quotients have finite derived subgroups
Let \({\mathfrak X}\) be a class of groups. A group G is said to be just-non- \({\mathfrak X}\) if it is not an \({\mathfrak X}\)-group but every of its proper quotients belongs to \({\mathfrak X}\). Just-non-abelian groups were considered by \textit{M. F. Newman} in two articles of 1960 [Proc. Lond. Math. Soc., III. Ser. 10, 354-364, 365-375 (1960; Zbl 0099.252)]. In the paper under review the authors consider just-non-(finite-by-abelian) groups (or JNFA- groups) with nontrivial Fitting subgroup (i.e. soluble-by-finite JNFA- groups). In the first part it is proved that JNFA-groups with nontrivial centre are precisely the torsion-free nilpotent groups of class 2 for which the commutator subgroup is infinite cyclic and the centre is isomorphic with a subgroup of the additive group of rational numbers. The main result about JNFA-groups with trivial centre and non-trivial Fitting subgroup is that the Fitting subgroup A of such a group G is abelian, and it is a (faithful) just-infinite module over the finite-by-abelian group G/A. In particular, A is either torsion-free or of prime exponent. Conversely, if A is a faithful just-infinite module over a finite-by- abelian group Q, every extension of A by Q is a JNFA-group with trivial centre and Fitting subgroup A. Therefore just-infinite modules over finite-by-abelian groups are naturally involved in the study of JNFA- groups, and Section 4 of the paper is devoted to the description of groups admitting such type of modules. Finally, just-non-(central-by- finite) groups are characterized in the last Section of the paper.
scientific article; zbMATH DE number 4075385
scientific article