## Abstract

By the Shepherd-Leedham-Green-McKay theorem on finite *p*-groups of maximal nilpotency class, if a finite *p*-group of order *p*
^{n} has nilpotency class *n*−1, then *f* has a subgroup of nilpotency class at most 2 with index bounded in terms of *p*. Some counterexamples to a rank analog of this theorem are constructed that give a negative solution to Problem 16.103 in *The Kourovka Notebook*. Moreover, it is shown that there are no functions *r(p)* and *l(p)* such that any finite 2-generator *p*-group whose all 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 torsion-free divisible nilpotent groups that are in the Mal’cev correspondence with “truncated” Witt algebras.

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*To Victor Danilovich Mazurov on the occasion of his 70th birthday.*

Original Russian Text Copyright © 2013 Khukhro E.I.

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Translated from *Sibirskiĭ Matematicheskiĭ Zhurnal*, Vol. 54, No. 1, pp. 225–239, January–February, 2013.

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Khukhro, E.I. Counterexamples to a rank analog of the Shepherd-Leedham-Green-Mckay theorem on finite *p*-groups of maximal nilpotency class.
*Sib Math J* **54, **173–183 (2013). https://doi.org/10.1134/S0037446613010217

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### Keywords

- finite
*p*-group - nilpotency class
- derived length
- lower central series
- rank