Kourovka Notebook Problem 19.47

Definitions and mathematical statement

Let $k$ be a field of characteristic $p>2$, and let $K/k$ be a finite field extension of degree $n=[K:k]\geq 1$. Regard $K$ as an $n$-dimensional vector space over $k$, and fix an ordered $k$-basis $\mathcal B$. Write $\operatorname{GL}_k(K)$ for the group of invertible $k$-linear maps on $K$; here the source notation $\operatorname{Aut}_k(K)$ denotes linear automorphisms, not necessarily field automorphisms. For $t\in K^{\times}=K\setminus\{0\}$, define $m_t(x)=tx$, and put \[ T_{\mathcal B}(K/k)=\{[m_t]_{\mathcal B}:t\in K^{\times}\}\leq\operatorname{GL}(n,k). \] This is the subgroup denoted by $T(K/k)$ and called a "nonsplit maximal torus" in the supplied source. No separability hypothesis is supplied; the explicit multiplication subgroup is the definition used here, without asserting an algebraic-group torus interpretation in every case.

For distinct indices $i,j\in\{1,\ldots,n\}$, let $E_{ij}$ be the matrix with entry $1$ in position $(i,j)$ and zero elsewhere. The elementary transvections relative to $\mathcal B$ are \[ u_{ij}(a)=I_n+aE_{ij},\qquad a\in k\setminus\{0\},\quad i\ne j. \] A subgroup $H\leq\operatorname{GL}(n,k)$ is rich in transvections if $u_{ij}(a)\in H$ for every such $i,j,a$. When $n=1$, the set of distinct index pairs is empty; the source supplies no restriction $n\geq2$.

The source asks whether every subgroup $H\leq\operatorname{GL}(n,k)$ containing $T_{\mathcal B}(K/k)$ and a "one-dimensional transformation" is rich in transvections. The supplied record does not define "one-dimensional transformation," so this hypothesis remains unresolved. If the intended meaning is an invertible transformation $g$ with $\operatorname{rank}_k(g-I_n)=1$, then the corresponding provisional formulation is: for every $k,K,\mathcal B$ as above and every $H\leq\operatorname{GL}(n,k)$, is the implication \[ \left(T_{\mathcal B}(K/k)\leq H\ \text{and}\ \exists g\in H:\operatorname{rank}_k(g-I_n)=1\right) \Longrightarrow \left(\forall i\ne j\ \forall a\in k\setminus\{0\},\ I_n+aE_{ij}\in H\right) \] valid? This rank interpretation is conditional and requires source confirmation; it is not a supplied definition.

Short English statement

For a finite extension of a field of odd characteristic, consider the subgroup of invertible linear maps given by multiplication by nonzero extension elements. Must every linear group containing this subgroup and a "one-dimensional transformation" contain every elementary transvection? The meaning of "one-dimensional transformation" is missing; the displayed rank formulation is provisional.

Sources

• Local UnsolvedMath registry, permanent ID $701595$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 19.47." The supplied registry status is open; this is a source label, not an independently verified literature assessment.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), problem 19.47; the background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. These references have not been independently consulted. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied proposer field is malformed and reads V. A. Ko\textbackslash u{\textbackslash i. The background repeats this fragment and places baev} after the heading "Discussion and literature." The intended proposer spelling cannot be recovered reliably from this record. The supplied proposed year is 2018.
• The record was created and updated on 2026-10-16T14:00:00Z, with upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. These are registry timestamps, not a supplied literature-verification date. No literature_checked_at value or literature_sources links are supplied.