Kourovka Notebook Problem 16.95

Definitions and mathematical statement

Let $F$ be a field and let $n\geq 1$ be an integer. Write $\operatorname{GL}(n,F)$ for the group of invertible $n\times n$ matrices over $F$. A permutation matrix is a matrix $P$ for which there is a permutation $\sigma$ of $\{1,\ldots,n\}$ such that $P_{ij}=1$ when $i=\sigma(j)$, and $P_{ij}=0$ otherwise. For an $n\times n$ matrix $B$ over $F$, its characteristic polynomial is $\chi_B(t)=\det(tI_n-B)$, and its minimal polynomial $m_B(t)$ is the monic polynomial of least positive degree satisfying $m_B(B)=0$. Call $B$ cyclic if $m_B(t)=\chi_B(t)$.

Conjecture (J. G. Thompson, Kourovka Notebook 16.95). For every field $F$, every integer $n\geq 1$, and every $A\in\operatorname{GL}(n,F)$, there exists an $n\times n$ permutation matrix $P$ such that \[ m_{AP}(t)=\chi_{AP}(t)\qquad\text{in }F[t]. \] The supplied formulation places no restriction on the characteristic or cardinality of $F$.

Short English statement

The conjecture asserts that every invertible square matrix over any field can have its columns permuted so that its minimal and characteristic polynomials coincide.

Sources

• Local UnsolvedMath registry, permanent ID $701286$, source code catalogue entry, title Kourovka Notebook Problem 16.95. The supplied record labels the problem open; this status has not been independently verified.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 16.95, proposed by J. G. Thompson in the 16th issue (2006). The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48.
• The supplied collection description attributes the editing of the Kourovka Notebook, Issues 1--21 (1965--2026), to E. I. Khukhro and V. D. Mazurov.