Kourovka Notebook Problem 20.12

Definitions and mathematical statement

Let $G$ be a group, with identity element $1_G$, and let $n\geq 1$ be an integer. Say that $G$ has the unique $n$-fold product property, denoted $\mathrm{U}_n(G)$, if for every ordered tuple of finite nonempty subsets $A_1,\ldots,A_n\subseteq G$, there exists $g\in G$ such that \[ \left|\left\{(a_1,\ldots,a_n)\in A_1\times\cdots\times A_n : a_1\cdots a_n=g\right\}\right|=1. \] The multiplication is taken in the displayed order. The usual unique product property is $\mathrm{U}_2(G)$.

The source observes that $\mathrm{U}_n(G)$ implies $\mathrm{U}_m(G)$ whenever $1\leq m\leq n$, by taking the remaining subsets to be $\{1_G\}$.

Question (G. M. Bergman). Is it true that, for every group $G$ and every integer $n\geq 2$, \[ \mathrm{U}_2(G)\ \Longrightarrow\ \mathrm{U}_n(G)? \] Equivalently, are all the unique $n$-fold product conditions, for integers $n\geq 2$, equivalent to the usual unique product condition?

Short English statement

If every pair of finite nonempty subsets of a group has a product element with exactly one representation, must the same hold for every ordered finite collection of at least two finite nonempty subsets?

Sources

• Local UnsolvedMath registry: permanent ID $701670$, exact source problem code catalogue entry, title Kourovka Notebook Problem 20.12. The supplied registry labels the problem open; that status is not independently verified here.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), Problem 20.12; proposed by G. M. Bergman in 2022. The record also identifies the 21st issue edition (2026), arXiv:1401.0300v48.
• The supplied collection description credits E. I. Khukhro and V. D. Mazurov as editors of Issues 1--21 (1965--2026).