Kourovka Notebook Problem 20.100

Definitions and mathematical statement

For a group $G$ with identity $e$, the order of an element $g\in G$ is the least positive integer $m$ such that $g^m=e$, if such an integer exists, and is infinite otherwise. For a positive integer $i$, write $g^i$ for the product of $i$ copies of $g$.

Conjecture (Z.-W. Sun, Kourovka Notebook Problem 20.100). For every positive integer $n$ and every group $G$ containing no element of order in $\{2,\ldots,n+1\}$, every subset $A\subseteq G$ of cardinality $n$ admits a bijective enumeration $A=\{a_1,\ldots,a_n\}$ such that \[ a_i^{\,i}\ne a_j^{\,j}\qquad\text{for all }1\le i<j\le n. \] The group is not assumed to be finite or abelian, and the identity is allowed to belong to $A$.

The supplied record states that the conjecture holds for $n\le 3$. This small-case assertion and the record's open status are supplied source assessments, not independently verified here.

Short English statement

Sun conjectures that any $n$-element subset of a group with no elements of orders $2,\ldots,n+1$ can be ordered so that raising its $i$-th element to the $i$-th power gives pairwise distinct elements. The supplied record reports the cases $n\le 3$ as known.

Sources

• Local UnsolvedMath registry, permanent ID $701758$, exact source code catalogue entry, title Kourovka Notebook Problem 20.100; supplied status: open. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), original problem number 20.100; proposed by Z.-W. Sun in 2022. The supplied background 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 the Kourovka Notebook collection covering Issues 1--21 (1965--2026). These bibliographic details and the reported small-case result are supplied attribution and have not been independently checked.