Kourovka Notebook Problem 19.32

Definitions and mathematical statement

Let $p\geq 673$ be a prime and let $r\geq 2$ be an integer. Let $F_r$ be the free group with free generators $a_1,\ldots,a_r$, and put \[ B=B(r,p):=F_r/\langle\!\langle g^p:g\in F_r\rangle\!\rangle, \] where the denominator denotes normal closure. Write $a_1,\ldots,a_r$ also for their images in $B$. Thus $B$ is the free group of rank $r$ in the variety of groups satisfying $y^p=1$.

A one-variable equation over $B$ is an equation $w(a_1,\ldots,a_r,x)=1$, where $w$ is a group word with coefficients in $B$, represented by an element of the free product $B*\langle x\rangle$, with $\langle x\rangle\cong{\mathbb{Z}}$. For $b\in B$, evaluation fixes the coefficients and substitutes $b$ for $x$. Define \[ S_w:=\{b\in B:w(a_1,\ldots,a_r,b)=1\}. \] Interpreting the source's term "non-identical" as meaning that the equation does not hold for every substitution in $B$, the question is whether, for every such $p,r$ and every such word $w$, \[ \left(\exists b\in B\; w(a_1,\ldots,a_r,b)\neq 1\right) \quad\Longrightarrow\quad |S_w|<\infty. \] The source does not explicitly define "non-identical". The displayed formulation uses the interpretation just stated; whether the intended condition instead refers to nontriviality of a word in a relatively free Burnside group requires confirmation. Formal nontriviality in $B*\langle x\rangle$ alone is a different condition. No uniform bound on the finite number of solutions is requested.

Short English statement

For every prime $p\geq 673$ and integer $r\geq 2$, must every one-variable equation over the free Burnside group $B(r,p)$ that is not satisfied by all elements have only finitely many solutions? This formulation interprets the source's undefined term "non-identical" as failure under at least one substitution; the intended meaning requires review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701580$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 19.32"; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The supplied record attributes Problem 19.32 to A. Gaglione, proposed in 2018, and identifies its source as The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), with the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels the problem "open" and records an update dated 2026-10-16. These are supplied metadata, not an independently verified assessment of the literature or mathematical status. The cited source has not been independently consulted for this draft.