Kourovka Notebook Problem 21.76

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\geq3$ and a field $K$, an elementary net consists of additive subgroups $\sigma_{ij}\leq(K,+)$ for $i\ne j$, with $\sigma_{ir}\sigma_{rj}\subseteq\sigma_{ij}$ whenever the three off-diagonal entries are defined. It is irreducible if each $\sigma_{ij}\ne\{0\}$. Write $t_{ij}(a)=I_n+a e_{ij}$, where $e_{ij}$ has its only nonzero entry $1$ in position $(i,j)$, and let \[ E(\sigma)=\langle t_{ij}(a):i\ne j,\ a\in\sigma_{ij}\rangle\leq\operatorname{GL}_n(K). \] The net is closed when $t_{ij}(a)\in E(\sigma)$ implies $a\in\sigma_{ij}$. It is completable if additive diagonal subgroups $\sigma_{ii}\leq K$ can be chosen so that $\sigma_{ir}\sigma_{rj}\subseteq\sigma_{ij}$ for every $i,r,j$, giving a complete net. No assumption that diagonal groups contain $1$ is added.

Statement recorded in the supplied source.

Do there exist a field $K$ of odd characteristic, an integer $n\geq3$, and an irreducible closed elementary net $\sigma$ over $K$ that is not completable? [S2]

Short English statement

Can an irreducible elementary net over an odd-characteristic field be closed under generated transvections while admitting no compatible diagonal completion?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2585 (KOU-21.76). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 V. A. Koibaev, Problem 21.76, in E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 21st edition, arXiv:1401.0300v48 (6 October 2026). See also source definitions in Problems 19.48. This version records an affirmative answer; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.