Weak Maximality Condition and Polycyclic Groups

Abstract

A group G is called strongly restrained if there exists an integer n such that $\langle {x^{(y)}}\rangle$ can be generated by n elements for all x, y in G. We show that a group G is polycyclic-by-finite if and only if G is a finitely generated strongly restrained group in which every nontrivial finitely generated subgroup has a nontrivial finite quotient. This provides a general setting for various results in soluble and residually finite groups that have appeared recently.