Khukhro, E. I. (2013) Counterexamples to a rank analogue of the Shepherd--Leedham-Green--McKay theorem on finite $p$-groups of maximal class. [MIMS Preprint]
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Abstract
By the Shepherd--Leedham-Green--McKay theorem on finite $p$-groups of maximal class, if a finite $p$-group of order $p^n$ 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.
| Item Type: | MIMS Preprint |
|---|---|
| Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 20 Group theory and generalizations |
| Depositing User: | Professor Evgeny Khukhro |
| Date Deposited: | 24 Jan 2013 |
| Last Modified: | 20 Oct 2017 14:13 |
| URI: | https://eprints.maths.manchester.ac.uk/id/eprint/1937 |
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