Kourovka Notebook Problem 12.23

Definitions and mathematical statement

For a group $G$ and an integer $k\geq 2$, let $P_k(G)$ denote the property \[ \forall (x_1,\ldots,x_k)\in G^k\;\exists\sigma\in\mathbb{S}_k\setminus\{\operatorname{id}\}: \quad x_1\cdots x_k=x_{\sigma(1)}\cdots x_{\sigma(k)}, \] where $\mathbb{S}_k$ is the group of all permutations of $\{1,\ldots,k\}$. Entries of the tuple may repeat, and the permutation may depend on the tuple. The index of permutability is \[ p(G)=\min\{k\in{\mathbb{Z}}:k\geq 2\text{ and }P_k(G)\}, \] provided this set is nonempty. The source does not specify a convention when the set is empty.

For a positive integer $n$, let $\mathbb{S}_n$ be the symmetric group on $\{1,\ldots,n\}$. The problem proposed by M. Gutsan is to determine $p(\mathbb{S}_n)$ as a function of $n$. The source does not explicitly state the range of $n$; positive finite degrees are the conventional interpretation used here, and inclusion of degree zero remains unspecified. The supplied registry labels the problem open; this label is not an independently verified mathematical or literature assessment.

Short English statement

Determine the least integer $k\geq 2$ such that every ordered list of $k$ elements of $\mathbb{S}_n$, allowing repetitions, has the same product after some nonidentity permutation of its positions. The source leaves the range of $n$ implicit.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700842$, exact source problem code catalogue entry, title Kourovka Notebook Problem 12.23; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), Problem 12.23; proposed by M. Gutsan.
• The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently checked.