On the nilpotent length of some residually finite groups (Q2711630)

A well-known theorem of K. A. Hirsch asserts that a polycyclic group all of whose finite images are nilpotent is itself nilpotent. The author extends this result by allowing the finite quotients to have bounded nilpotent length. Let \(\underline N_c\) denote the class of nilpotent groups with class at most \(c\) where \(c\in\mathbb{N}\cup\infty\): here \(\underline N_\infty\) is the class of all nilpotent groups. Let \(G\) be a group all of whose finite images belong to the class \(\underline N_{c_1}\underline N_{c_2}\cdots\underline N_{c_\ell}\) where \(c_i\in\mathbb{N}\cup\infty\). Then \(G\) belongs to \(\underline N_{c_1}\underline N_{c_2}\cdots\underline N_{c_\ell}\) provided \(G\) is either a finitely generated linear group or a residually finite soluble minimax group. In particular \(G\) has nilpotent length at most \(l\) if all its finite quotients do. This extends a previous result of Endimioni for polycyclic groups.