Balanced two-generator $p$-groups of deficiency zero and unbounded nilpotency class
For every prime $p>3$ we construct an explicit family of finite two-generator $p$-groups of deficiency zero whose nilpotency classes tend to infinity. The construction gives, for every $n\geq1$, a group of order $p^{2n+1}$ and class $n+1$. This gives an affirmative answer to Problem 17.113 of 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.
- External referenceM. Hall, Jr., The theory of groups, Macmillan, New York, 1959.LaTeX source\bibitem{hall} M. Hall, Jr., The theory of groups, Macmillan, New York, 1959.
- External referenceD. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Springer, 1996.LaTeX source\bibitem{robinson} D. J. S. Robinson, \emph{A Course in the Theory of Groups}, 2nd ed., Springer, 1996.
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- PF:2026.000002/v2/THM-2.1
For every $n\geq1$, the group $G_n$ is a finite two-generator $p$-group of order $p^{2n+1}$, nilpotency class $n+1$, and deficiency zero.
- PF:2026.000002/v2/COR-2.2
By Theorem 2.1, for every prime $p>3$ there exist two-generator finite $p$-groups of deficiency zero and arbitrarily large nilpotency class.