Counterexamples to a rank analogue of the Shepherd–Leedham-Green–McKay theorem on finite p-groups of maximal class

By the Shepherd–Leedham-Green–McKay theorem on finite p-groups of maximal class, if a finite p-group of order pn has nilpotency class n−1, then it has a subgroup of nilpotency class at most 2 with index bounded in terms of p. Counterexamples to a rank analogue of this theorem are constructed, which give a negative solution to Problem 16.103 in Kourovka Notebook. Moreover, it is shown that there are no functions r(p) and l(p) such that any 2-generator finite p-group all of whose factors of the lower central series, starting from the second, are cyclic would necessarily have a normal subgroup of derived length at most l(p) with quotient of rank at most r(p). The required examples of finite p-groups are constructed as quotients of torsion-free nilpotent groups, which are abstract 2-generator subgroups of nilpotent divisible torsion-free groups that are in the Mal’cev correspondence with “truncated” Witt algebras.