A family of finite balanced $p$-groups with unbounded nilpotency class
For every prime $p$ we construct a family of balanced finite two-generator $p$-groups whose nilpotency class is arbitrarily large. This gives an affirmative answer to Problem 17.113 in the Kourovka Notebook.
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Substantive mathematical contributions
Show proposalsConcrete proposals are tied to an immutable base version. When the paper owner confirms a contribution, it can be linked to the later manuscript version that incorporated it, was revised because of it, or resolved the issue independently. If the owner has not decided within 90 days, an administrator or a Managing Editor of the paper's subject area can take an editorial decision. Proposals by co-authors are recorded but give no contribution credit.
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- External referenceD. L. Johnson., Presentations of groups, Cambridge University Press, 2012.LaTeX source\bibitem{presentations} D. L. Johnson., \emph{Presentations of groups}, Cambridge University Press, 2012.
- External referenceE. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, No. 21, Novosibirsk, 2026; arXiv:1401.0300.LaTeX source\bibitem{kourovka} E. I. Khukhro and V. D. Mazurov (eds.), \emph{Unsolved Problems in Group Theory: The Kourovka Notebook}, No. 21, Novosibirsk, 2026; arXiv:1401.0300.
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- PF:2026.000002/v3/THM-2.1
For $m,r,s,t\in \N$ such that $r^t\equiv 1\pmod{m}$ and $s=m/\gcd(m,r-1)$, consider the group \[ G = \langle a,b \mid a^m=1,\; a^b = a^r,\; b^t =a^s \rangle. \] Then $G$ is a finite metacyclic group of order $mt$, and there exists an integer $u$ such that $G$ is presented by \[ \langle a,b \mid [b,a^{-u}] = a^{\gcd(m,r-1)},\; b^t =a^s \rangle. \]
- PF:2026.000002/v3/THM-3.1
For every $n\geq1$ and every prime $p$, the group $G_{p,n}$ is a finite balanced $p$-group with nilpotency class $n+1$.