Kourovka Notebook Problem 11.33

Definitions and mathematical statement

Let $q$ be a prime power and let $G(q)$ be a simple Chevalley group over ${\mathbb{F}}_q$. For a positive integer $m$, write $G(q^m)$ for the corresponding group of the same type over ${\mathbb{F}}_{q^m}$. The source presupposes an inclusion $\iota_m:G(q)\hookrightarrow G(q^m)$ arising from the field extension; its compatibility with the chosen simple-group quotients requires clarification. For a field $k$, a representation means a $k$-vector space with a linear group action. An irreducible representation is a nonzero representation with no nonzero proper invariant subspace. For a module of finite length, its composition factors are the simple quotients in a composition series. Write $\operatorname{Irr}_k(H)$ for the isomorphism classes of irreducible $k$-representations of $H$.

The supplied problem, part (b), asks whether there exists a positive integer $m$ such that, for every field $k$ of prime characteristic $\ell$ with $\ell\nmid q$, the restriction to $G(q)$ of every non-one-dimensional representation of $G(q^m)$ has every member of $\operatorname{Irr}_k(G(q))$ as a composition factor.

For finite-dimensional representations, the literal reading of this assertion is \[ \exists m\in{\mathbb{Z}}_{\geq1}\ \forall k\ \forall V:\quad \left[\operatorname{char}(k)=\ell>0,\ \ell\nmid q,\ V\text{ is a finite-dimensional }kG(q^m)\text{-module},\ \dim_k V\ne1\right] \Longrightarrow \left[\text{every }W\in\operatorname{Irr}_k(G(q))\text{ occurs in }\operatorname{Res}_{\iota_m}V\right]. \] Here occurrence means occurrence as a composition factor, without a prescribed multiplicity. This display is a conditional formalization under the finite-dimensional convention, which the source does not explicitly state. The source also does not explicitly require $V$ to be irreducible, to have nontrivial action, or to be nonzero. Consequently, its phrase "non-one-dimensional" leaves material formulation issues unresolved; no irreducibility hypothesis has been silently inserted. The displayed quantifier order takes $m$ to be independent of $k$ and $\ell$; this scope should be checked against the original source.

The supplied archived part (a) asks for the analogous assertion over ${\mathbb{C}}$: there exists a positive integer $m$ such that the restriction of every non-one-dimensional complex representation of $G(q^m)$ to $G(q)$ contains every irreducible complex representation of $G(q)$ as a composition factor. The same unresolved qualification concerning the representation class applies. The supplied archive reports part (a) as proved; part (b) remains in the supplied main section. These are supplied source classifications, not independently verified mathematical or literature assessments.

Short English statement

For each simple Chevalley group $G(q)$, part (b) asks for one extension degree $m$ such that every representation described by the source as non-one-dimensional, in prime characteristic not dividing $q$, contains every irreducible representation of $G(q)$ after restriction. The complex analogue, part (a), is supplied as archive-solved. The intended representation class, coefficient-field conventions, and subgroup inclusion need clarification.

Sources

• Local UnsolvedMath registry, permanent ID $700726$, exact source code catalogue entry, title "Kourovka Notebook Problem 11.33"; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), Problem 11.33; supplied background identifies an update in the 21st issue (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These references have not been independently consulted.
• Proposed in 1990 by the person recorded as A. E. Zalesski\textbackslash u{\textbackslash i; the supplied name encoding is incomplete and has not been reconstructed.
• The supplied Archive of Solved Problems entry attributes the proof associated with archived part (a) to D. Gluck, J. Algebra 155, no. 2 (1993), 221--237. No article title or external link is supplied, and the citation and its precise scope have not been independently verified.
• Supplied status review dated 2026-10-16: canonical status partially_solved, evidence label published; the review says that solved parts occur in the archive and remaining parts in the main section. This is supplied dated curation, not an independently verified assessment.