Kourovka Notebook Problem 15.31

Definitions and mathematical statement

Let $p$ be a prime and let $G$ be a finite $p$-group. For $g\in G$, define its conjugacy class and centralizer by \[ g^G=\{x^{-1}gx:x\in G\},\qquad C_G(g)=\{x\in G:xg=gx\}. \] The breadth of $g$ is the nonnegative integer \[ b_G(g)=\log_p|g^G|=\log_p[G:C_G(g)]. \] Define $\gamma_1(G)=G$ and $\gamma_{i+1}(G)=[\gamma_i(G),G]$, where $[H,G]$ is generated by the commutators $h^{-1}x^{-1}hx$ with $h\in H$ and $x\in G$. Write $c(G)$ for the nilpotency class, with $c(\{1\})=0$; for nontrivial $G$, this is the least positive integer $c$ such that $\gamma_{c+1}(G)=\{1\}$.

For a nonnegative integer $n$, assume that there exists a generating set $S\subseteq G$ such that \[ G=\langle S\rangle,\qquad b_G(s)\le n\quad\text{for every }s\in S. \] Thus the breadth restriction applies to the chosen generators, not necessarily to every element of $G$.

Question. Do there exist real constants $A,B\ge0$, independent of $p$, $n$, and $G$, such that every group satisfying these hypotheses obeys \[ c(G)\le An+B? \] Here a linear bound is expressed as an affine upper bound. The supplied wording does not explicitly state whether the constants must be independent of the prime $p$. The displayed question uses the uniform interpretation; under a prime-dependent interpretation, the quantifiers would instead be: for every prime $p$, there exist $A_p,B_p\ge0$ such that $c(G)\le A_pn+B_p$ for every admissible $n$ and finite $p$-group $G$.

The supplied record attributes the bound $c(G)\le n^2+1$ to M. R. Vaughan-Lee and J. Wiegold, and the improvement $c(G)\le n^2-n+1$ to A. Mann. These are supplied background claims, not independently verified here.

Short English statement

If a finite $p$-group has a generating set whose individual generators each have at most $p^n$ conjugates, must its nilpotency class admit a linear upper bound in $n$? The source does not explicitly specify whether the bound must be uniform over all primes. The registry labels this problem open; that label is supplied source data, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, supplied source record with permanent ID $701119$, exact source problem code catalogue entry, titled Kourovka Notebook Problem 15.31. The supplied record labels the problem open.
• Ya. G. Berkovich, proposer, 2002. The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), Problem 15.31; the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• M. R. Vaughan-Lee and J. Wiegold, Proc. R. Soc. Edinburgh Sect. A 95 (1983), 215--221. Cited by the supplied statement for the bound $c(G)\le n^2+1$; no article title or link is supplied.
• A. Mann, J. Group Theory 4, no. 3 (2001), 241--246. Cited by the supplied statement for the bound $c(G)\le n^2-n+1$; no article title or link is supplied.