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.
Corollary 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.
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- 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/v1/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/v1/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.