Kourovka Notebook Problem 16.28

Definitions and mathematical statement

Let $F$ be a field of characteristic $p>0$, and let $G$ be a connected linear reductive algebraic group over $F$. The source does not specify whether its notation $G$ denotes $F$-rational points or geometric points over an algebraic closure. In the formulas below, $G$ denotes the group of points intended by the source; this choice remains unresolved. Closed and open subsets are understood in the Zariski topology on that point set.

For a subset $X\subseteq G$ and an integer $k\geq 1$, define \[ X^k=\{x_1\cdots x_k\mid x_1,\ldots,x_k\in X\}. \] The source asks the following two questions, with the field and algebraic group universally quantified as above.

1. (a) For every closed subset $X\subseteq G$, does there exist an integer $c=c(X)>1$ such that $X^c$ is closed?
2. (b) For every conjugacy class \[ X=\{gxg^{-1}\mid g\in G\},\qquad x\in G, \] if $X^2$ contains an open subset of $G$, must $X^2=G$?

The source does not say that the open subset in (b) is nonempty. Literally, the hypothesis allows the empty open subset and is therefore automatic. Requiring a nonempty open subset would give a different, potentially intended formulation; that requirement has not been silently inserted here. The point domain also affects the meaning of a conjugacy class and must be clarified before a definitive formulation is accepted.

Short English statement

For a connected reductive linear algebraic group in positive characteristic, (a) asks whether every closed subset has a closed product power with exponent greater than one. Part (b) asks whether the square of a conjugacy class containing an open subset equals the whole group. The intended point domain and the possible requirement that the open subset be nonempty remain unresolved.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701220$, exact source code catalogue entry, title Kourovka Notebook Problem 16.28. Proposed by E. P. Vdovin in 2006.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), Problem 16.28; also cited in the supplied record through the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• Achyuth Jayadevan, Preprint of 12 September 2026, https://zenodo.org/records/22729229. The supplied discussion reports a negative answer and describes this as an AI solution confirmed by E. P. Vdovin. This attribution and confirmation are supplied claims, not independently verified here.
• M. Kinyon and J. Urban, Preprint of 17 September 2026, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf. The supplied discussion explicitly describes this as an unverified AI solution.
• The registry labels the record partially_solved. Its status review, dated exactly 2026-10-16, reports that a subproblem is confirmed solved in the maintained 21st-issue source, while the background uses the label "[Solved] No, not always." This assessment is supplied dated curation, not independently verified. The supplied excerpts do not identify which subproblem the negative answer resolves.