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Counterexamples to a rank analog of the Shepherd-Leedham-Green-Mckay theorem on finite p-groups of maximal nilpotency class

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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Correspondence to E. I. Khukhro.

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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