Kourovka Notebook Problem 10.43

Definitions and mathematical statement

Let $R$ and $S$ be associative rings with identity, with $2\cdot 1_S$ invertible in $S$, and let $n\geq 3$ and $m\geq 2$ be integers. No commutativity assumption is imposed. Write $GL_n(R)$ for the group of invertible $n\times n$ matrices over $R$. For $i\neq j$ and $r\in R$, put \[ t_{ij}(r)=I_n+rE_{ij},\qquad E_n(R)=\langle t_{ij}(r):i\neq j,\ r\in R\rangle, \] where $E_{ij}$ is the matrix unit, and put \[ a_{ij}=t_{ij}(1)t_{ji}(-1)t_{ij}(1). \] For a two-sided ideal $I$ of $R$, the source also introduces the natural reduction homomorphism $\Lambda_I:GL_n(R)\to GL_n(R/I)$; no particular ideal or condition involving this homomorphism is imposed in the requested problem.

Let \[ q:GL_n(R)\longrightarrow GL_n(R)/Z(GL_n(R)) \] be the quotient map, write $\overline{x}=q(x)$, and, for $G\leq GL_n(R)$, write $PG=q(G)$. In particular, $PE_n(R)$ denotes the image of $E_n(R)$ in this quotient by the centre of $GL_n(R)$.

The source uses $GL(W)=GL_m(S)$, with $W$ a free $S$-module of rank $m$. It calls a homomorphism $\Lambda:E_n(R)\to GL(W)$ standard when there are $S$-modules $P,Q$, a decomposition \[ S^m\cong P^{\oplus n}\oplus Q, \] a unital ring homomorphism $\delta:R\to\operatorname{End}P$, its entrywise induced homomorphism \[ \delta^*:GL_n(R)\longrightarrow GL_n(\operatorname{End}P), \] a central idempotent $f$ of $\delta(R)$, an antiisomorphism $\nu$ of $\delta(R)$, an isomorphism $g:W\to S^m$, and an embedding \[ \tau:GL_n(\operatorname{End}P)\longrightarrow GL(gW) \] such that, for every $x\in E_n(R)$, \[ \Lambda(x)=g^{-1}\tau\!\left( \delta^*(x)f+ \left({}^{t}\delta^*(x)^{\nu}\right)^{-1}(1-f) \right)g. \] Here ${}^{t}$ denotes transposition, and multiplication by $f$ and $1-f$ in the matrix expression uses the corresponding scalar diagonal matrices. The formula is reproduced with the source's notation: the codomain and precise matrix action of $\nu$, the handedness of the modules, and the compatibility required of $\tau$ with the displayed module decomposition are not fully specified in the supplied record. These qualifications remain unresolved.

For $i\neq j$, let \[ a^*_{ij}=t_{ij}(1_{\operatorname{End}P}) t_{ji}(-1_{\operatorname{End}P}) t_{ij}(1_{\operatorname{End}P}) \in GL_n(\operatorname{End}P). \] This gives the signed matrix meant by the source's replacement of the scalar entries of $a_{ij}$ by the corresponding entries over $\operatorname{End}P$.

The source states the following sufficient criterion: a homomorphism \[ \Lambda_0:PE_n(R)\longrightarrow GL(W)=GL_m(S) \] is induced by a standard homomorphism if there are common choices of $g$ and $\tau$, with module data as in the preceding setup, such that \[ \Lambda_0(\overline{a}_{ij}) =g^{-1}\tau(a^*_{ij})g \qquad\text{for every }i\neq j. \tag{C} \] Here being induced means that $\Lambda_0\circ(q|_{E_n(R)})$ is standard. This criterion is supplied background, not an independently verified assertion in this draft.

Problem (V. M. Petechuk, 1986). Describe the form, at least in particular cases, of homomorphisms $\Lambda_0:PE_n(R)\to GL_m(S)$ for which condition $(C)$ fails. Thus no single admissible choice of the module data, $g$, and $\tau$ makes the displayed equalities hold simultaneously for all $i\neq j$. The source does not specify which particular cases should be treated or require existence for every allowed $R,S,n,m$. Moreover, failure of this sufficient criterion is the requested condition; the supplied statement does not assert that it is equivalent to failure of standardness. A fully intrinsic formulation of the admissible choices in $(C)$ requires clarification of the source's module and embedding conventions.

Short English statement

For associative unital rings $R,S$, with $2$ invertible in $S$, and integers $n\geq3$, $m\geq2$, describe, at least in particular cases, homomorphisms $PE_n(R)\to GL_m(S)$ whose images of all $\overline{a}_{ij}$ cannot simultaneously have the prescribed form under one common choice of $g$ and $\tau$. The source gives that form as a sufficient criterion for being induced by a standard homomorphism. Some conventions in its definition of standardness remain unspecified. The registry labels the problem open and records an unverified AI solution claim; neither assessment is independently verified here.

Sources

• Local UnsolvedMath registry, permanent ID $700658$, exact source problem code catalogue entry, title Kourovka Notebook Problem 10.43. The supplied record attributes the problem to V. M. Petechuk, proposed in 1986.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 10.43, 10th issue (1986); supplied reference edition: 21st issue (2026), arXiv:1401.0300v48. The registry's collection description identifies E. I. Khukhro and V. D. Mazurov as editors. This is the supplied authoritative source attribution; the publication has not been independently consulted for this draft.
• Supplied discussion and status-review note: [Unverified AI claim] such homomorphisms are found, attributed to Achyuth Jayadevan, Preprint, 2026, https://zenodo.org/records/22852216. The record describes this as an unverified AI solution claim and says the problem is marked with $\#$ in the October 2026 version of the 21st issue, arXiv:1401.0300v48. No validity or resolution is asserted here.
• The supplied status review is dated 2026-10-16, assigns canonical status open, and records evidence unrefereed_claim. This assessment is supplied dated curation, not independently verified literature or source verification.