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

Khukhro, Evgeny (2013) Counterexamples to a rank analog of the Shepherd-Leedham-Green-Mckay theorem on finite p-groups of maximal nilpotency class. Siberian Mathematical Journal, 54 (1). pp. 173-183. ISSN 0037-4466

Full content URL: http://link.springer.com/article/10.1134%2FS003744...

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

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Abstract

By the Shepherd-Leedham-Green-McKay theorem on finite p-groups of maximal nilpotency class, if a finite p-group of order pn 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. © 2013 Pleiades Publishing, Ltd.

Additional Information:• Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 54, No. 1, pp. 225–239, January–February, 2013.
Keywords:Finite p-group, Nilpotency class, Derived length, Lower central series, Rank
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
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ID Code:15578
Deposited On:28 Oct 2014 11:03

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