Kourovka Notebook Problem 11.15

Definitions and mathematical statement

Let $F(x,y)$ be the free group on two generators. A word $w\in F(x,y)$ defines an evaluation $w(g,h)\in G$ for every group $G$ and every $g,h\in G$. Write $w(G)=1$ when $w(g,h)=1_G$ for all $g,h\in G$. A Sylow $p$-subgroup of a finite group $G$ is a subgroup of order equal to the largest power of $p$ dividing $|G|$; this includes the trivial subgroup when $p\nmid |G|$.

The source states that for every prime $p$ there exists a sequence $(a_{p,k})_{k\geq 1}$ of words in $F(x,y)$, depending on $p$ but independent of $G$, such that every finite group $G$ satisfies \[ \text{every Sylow $p$-subgroup of $G$ is abelian} \quad\Longleftrightarrow\quad \exists N\geq 1\ \forall k\geq N\ \forall g,h\in G,\qquad a_{p,k}(g,h)=1_G. \] Here $N$ may depend on $p$ and $G$. Thus "almost all" means all but finitely many indices.

Problem (R. Brandl). For each prime $p>2$, give an explicit sequence $(a_{p,k})_{k\geq 1}$ with the displayed property. The source reports an explicit sequence for $p=2$. It does not specify the required meaning of "explicit" or whether a single algorithm taking $p$ as input is intended. Its existence assertion and the reported $p=2$ result are retained as supplied background, without independent verification.

Short English statement

For each odd prime $p$, explicitly describe a sequence of two-variable group words such that a finite group has abelian Sylow $p$-subgroups exactly when every sufficiently late word is an identity in that group. The supplied record labels the problem open and separately reports an unverified AI claim of a computable construction; that claim is not treated here as an established solution.

Sources

• Local UnsolvedMath registry: permanent ID $700708$, exact source problem code catalogue entry, title Kourovka Notebook Problem 11.15; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• R. Brandl, proposer, 1990. The supplied record attributes the problem to The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), Problem 11.15, as included in the 21st issue edition (2026), arXiv:1401.0300v48. The registry identifies the notebook editors as E. I. Khukhro and V. D. Mazurov.
• R. Brandl, J. Austral. Math. Soc. 31 (1981), 464--469. The source cites this reference for an explicit sequence when $p=2$; no article title is supplied.
• Achyuth Jayadevan, Preprint (2026), https://zenodo.org/records/22852259. The supplied background and status review explicitly label the reported computable construction an "unverified AI solution" claim. The status review says that the October 2026 notebook edition marks it with $\#$.
• Supplied status review dated 2026-10-16: canonical status open, evidence category unrefereed_claim. This is supplied dated curation, not an independently verified literature assessment.