Kourovka Notebook Problem 10.32

Definitions and mathematical statement

For a group $G$ and a group word $w$ in $k$ variables, its word map is \[ w_G:G^k\longrightarrow G,\qquad (g_1,\ldots,g_k)\longmapsto w(g_1,\ldots,g_k). \] The word is universal on $G$ if this map is surjective. In particular, for nonzero integers $r,s$, the word $x^ry^s$ is universal on $G$ precisely when \[ \forall g\in G\ \exists x,y\in G:\quad g=x^ry^s. \] For a positive integer $n$, let $\mathbb S_n$ be the symmetric group on $n$ letters and $\mathbb A_n$ its alternating subgroup.

The supplied source defines \[ m(r,s)= \begin{cases} \displaystyle\prod_{\substack{p\text{ prime}\\p\mid rs}}p, & r\notin\{-1,1\}\text{ and }s\notin\{-1,1\},\\ 1,&r\in\{-1,1\}\text{ and }s\in\{-1,1\}. \end{cases} \] As written, this definition leaves $m(r,s)$ undefined when exactly one exponent belongs to $\{-1,1\}$. No value is assigned here in those cases.

Source-reported background. The record attributes to J. L. Brenner, R. J. Evans, and D. M. Silberger the following bound: for every real $c>8/5$, there exists $N_0=N_0(c)$ such that $x^ry^s$, with $rs\ne0$, is universal on $\mathbb A_n$ whenever \[ n\ge\max\{N_0(c),\ c\log m(r,s)\}. \] The supplied examples are $N_0(5/2)=5$ and $N_0(2)=29$. These are reported claims from the source; the displayed formulation inherits its incomplete definition of $m$ and its unspecified logarithm base.

Requested statement (Yu. I. Merzlyakov, 1986). For nonzero integer exponents $r,s$ with at least one of $r,s$ odd, find an analogous bound for the degree $n$ ensuring that $x^ry^s$ is universal on $\mathbb S_n$. Thus the sought degree condition must imply \[ \forall g\in\mathbb S_n\ \exists x,y\in\mathbb S_n:\quad g=x^ry^s. \] The comparison with the alternating-group bound suggests seeking a logarithmic sufficient condition of the form \[ n\ge\max\{N,\ C\log m(r,s)\}, \] with constants independent of $r,s$; however, the source does not specify the desired admissible constants, whether the threshold $c>8/5$ is also intended for $\mathbb S_n$, or a complete definition of $m$. This displayed form records the suggested target rather than an asserted resolution of those ambiguities.

The record attributes to M. Droste the previously known sufficient condition \[ n\ge\max\{6,\ 4m(r,s)-4\} \] under the same parity hypothesis. Its phrase that only this cruder bound is known is a source-specific historical qualification, not a new assessment here. The appended curation also reports an explicitly unverified AI claim of a logarithmic bound; it does not establish a solution.

Short English statement

Merzlyakov asks for a logarithmic degree bound ensuring that every permutation in $\mathbb S_n$ can be written as $x^ry^s$, for nonzero integer exponents with at least one odd. The supplied definition of $m(r,s)$ omits mixed cases involving exactly one exponent equal to $\pm1$, and the intended constants are unspecified. The registry labels the problem open while recording an unverified AI solution claim.

Sources

• Local UnsolvedMath registry, permanent ID $700647$, exact source code catalogue entry, title Kourovka Notebook Problem 10.32. The supplied record attributes the problem to Yu. I. Merzlyakov, proposed in 1986.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), Problem 10.32; also the supplied 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• J. L. Brenner, R. J. Evans, and D. M. Silberger, Proc. Amer. Math. Soc. 96, no. 1 (1986), 23--28. Supplied attribution for the alternating-group bound; no article title is supplied.
• M. Droste, Proc. Amer. Math. Soc. 96, no. 1 (1986), 18--22. Supplied attribution for the symmetric-group bound; no article title is supplied.
• M. Kinyon and J. Urban, Preprint (2026), https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf. The supplied background and status review explicitly label the assertion that a logarithmic bound was found as an "unverified AI claim" or "unverified AI solution." The review says that the October 2026 21st issue marks it with $\#$.
• Supplied status review checked on 2026-10-16: canonical status open, evidence category unrefereed_claim. This is supplied dated curation, not an independently verified literature assessment. No cited source was visited or independently checked for this draft.