Kourovka Notebook Problem 13.13

Definitions and mathematical statement

For a finite group $K$, its Burnside ring $B(K)$ is the Grothendieck ring of isomorphism classes of finite $K$-sets, with addition induced by disjoint union and multiplication induced by Cartesian product with diagonal action. Interpret an isomorphism of Burnside rings as an isomorphism of rings; preservation of the distinguished classes of transitive sets is not stipulated.

Define the lower central series by $\gamma_1(K)=K$ and $\gamma_{i+1}(K)=[\gamma_i(K),K]$, where $[A,K]$ is generated by commutators $a^{-1}k^{-1}ak$ with $a\in A$ and $k\in K$. For a finite nilpotent group, let $c(K)$ be the least integer $c\geq 0$ such that $\gamma_{c+1}(K)=1$; thus $c(1)=0$.

The proposed question is whether there exists a function \[ f:\mathbb Z_{\geq 0}\longrightarrow\mathbb Z_{\geq 0} \] such that, for every prime $p$ and all finite $p$-groups $G,H$, \[ B(G)\cong B(H)\quad\Longrightarrow\quad c(H)\leq f(c(G)). \] This expresses a bound depending only on the nilpotency class of $G$. The supplied wording does not explicitly clarify uniformity in $p$. If a prime is intended to be fixed before choosing the bound, the alternative quantifier order is: for every prime $p$, there exists a function $f_p$ satisfying the same implication for all finite $p$-groups $G,H$.

The source additionally reports an example of finite $p$-groups with isomorphic Burnside rings and $c(G)=2$, $c(H)=3$. Neither the prime nor the groups are specified in the supplied record.

Short English statement

Does an isomorphism between the Burnside rings of two finite $p$-groups force the nilpotency class of one group to be bounded in terms of the class of the other? The source reports a class $2$/class $3$ example. Whether the requested bound must be uniform over primes is not explicit.

Sources

• Local UnsolvedMath registry, permanent ID $700931$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 13.13." The supplied record labels the problem open; this is supplied registry status, not an independently verified assessment.
• R. Brandl, proposer, 1995. The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), Problem 13.13; cited in the supplied record through the 21st issue edition (2026), arXiv identifier 1401.0300v48.
• The supplied collection description attributes the editing of The Kourovka Notebook, Issues 1--21 (1965--2026), to E. I. Khukhro and V. D. Mazurov. The cited source and the reported example have not been independently inspected for this draft.