On the Nilpotent Length of Polycyclic Groups

AbstractLetGbe a polycyclic group. We prove that if the nilpotent length of each finite quotient ofGis bounded by a fixed integern, then the nilpotent length ofGis at mostn. The casen=1 is a well-known result of Hirsch. As a consequence, we obtain that if the nilpotent length of each 2-generator subgroup is at mostn, then the nilpotent length ofGis at mostn. A more precise result in the casen=2 permits us to prove that if each 3-generator subgroup is abelian-by-nilpotent, thenGis abelian-by-nilpotent. Furthermore, we show that the nilpotent length ofGequals the nilpotent length of the quotient ofGby its Frattini subgroup