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Sibirskii Matematicheskii Zhurnal, 2013, Volume 54, Number 1, Pages 225–239
(Mi smj2415)
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Counterexamples to a rank analog of the Shepherd–Leedham-Green–Mckay theorem on finite $p$-groups of maximal nilpotency class
E. I. Khukhro Sobolev Institute of Mathematics, Novosibirsk, Russia
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.
Keywords:
finite $p$-group, nilpotency class, derived length, lower central series, rank.
Received: 08.10.2012
Citation:
E. I. Khukhro, “Counterexamples to a rank analog of the Shepherd–Leedham-Green–Mckay theorem on finite $p$-groups of maximal nilpotency class”, Sibirsk. Mat. Zh., 54:1 (2013), 225–239; Siberian Math. J., 54:1 (2013), 173–183
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https://www.mathnet.ru/eng/smj2415 https://www.mathnet.ru/eng/smj/v54/i1/p225
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Abstract page: | 196 | Full-text PDF : | 66 | References: | 31 | First page: | 2 |
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