Kourovka Notebook Problem 10.44

Definitions and mathematical statement

Let $R$ and $S$ be associative rings with identity, and assume that $2\cdot 1_S$ is invertible in $S$. For positive matrix sizes $n,m$, put $V_R=R^n_R$ and $W_S=S^m_S$. Write $GL_n(R)=\operatorname{Aut}_R(V)$, and use the usual elementary subgroup \[ E_n(R)=\left\langle I_n+r e_{ij}:r\in R,\ 1\leq i,j\leq n,\ i\neq j\right\rangle, \] where $e_{ij}$ is the matrix unit; define $E_m(S)$ similarly. The source gives no further restrictions on $n$ or $m$.

The source asks to prove that every standard homomorphism \[ \Lambda:E_n(R)\longrightarrow E_m(S) \] originates from a collineation and a correlation: there should exist maps described in the source as semilinear isomorphisms \[ g:V_R\longrightarrow W_S, \qquad h:V_R\longrightarrow {}_S\!W'_{S^0}, \] and an object $f$ such that, for every $x\in E_n(R)$, \[ \Lambda(x)=(g^{-1}xg)f+(h^{-1}xh)(1-f). \] Here $g$ is called a collineation and $h$ a correlation.

Unresolved qualifications. This is a transcription of the proposed assertion, not a fully specified proposition. The supplied record does not define "standard homomorphism," specify the scalar maps underlying semilinearity, define $W'$ or $S^0$, or state the ambient algebra and required properties of $f$. In particular, no idempotence or centrality condition on $f$ is supplied. The bimodule notation in the target of $h$, the induced action on that target, and the composition convention needed to interpret the displayed conjugations are also unspecified. Reading $S^0$ as an opposite ring and $W'$ as a dual module would require confirmation. These omissions prevent a precise, unqualified interpretation of the asserted formula.

Short English statement

Petechuk's problem asks whether every standard homomorphism between the stated elementary matrix groups over unital associative rings, with $2$ invertible in the target ring, has the displayed decomposition using a collineation and a correlation. Essential terminology and the parameter $f$ are undefined in the supplied record, so the intended assertion requires clarification. The registry labels the problem open and separately reports an unverified AI counterexample claim.

Sources

• Local UnsolvedMath registry, permanent ID $700659$, source code catalogue entry, title "Kourovka Notebook Problem 10.44." The supplied record attributes the problem to V. M. Petechuk, proposed in 1986.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), Problem 10.44; 21st issue edition (2026), arXiv:1401.0300v48. The registry's collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied background and status-review notes report: an unverified AI claim that a counterexample was constructed, attributed to Achyuth Jayadevan, Preprint, 2026, https://zenodo.org/records/22847412. The status-review notes say that the problem is marked with $\#$ for this unverified claim in the October 2026 21st issue. This is supplied attribution, not verification of a counterexample.
• The supplied status review is dated 2026-10-16, records canonical status open, and classifies its evidence as unrefereed_claim. This assessment is supplied dated curation, not independently verified. No cited source was independently consulted for this draft.