Kourovka Notebook Problem 17.66

Definitions and mathematical statement

Let $G$ be a finite group and $k$ a field. A $kG$-module $V$ is faithful if its action homomorphism $G\to\operatorname{GL}_k(V)$ is injective, and absolutely irreducible if $\overline{k}\otimes_k V$ is irreducible over $\overline{k}G$, where $\overline{k}$ is an algebraic closure of $k$. Define \[ Z^1(G,V)=\{f:G\to V: f(gh)=f(g)+g\cdot f(h)\text{ for all }g,h\in G\}, \] \[ B^1(G,V)=\{f_v:G\to V: f_v(g)=g\cdot v-v\text{ for some }v\in V\}, \qquad H^1(G,V)=Z^1(G,V)/B^1(G,V). \] Dimensions are taken over the coefficient field $k$.

The source does not specify the coefficient field, its characteristic, or whether it is fixed or varies. To expose this unresolved qualification, let $\mathcal K$ denote the intended, unspecified class of coefficient fields. For a specified choice of $\mathcal K$, the question is whether \[ \exists d\in{\mathbb{Z}}_{>0}\ \forall k\in\mathcal K\ \forall\text{ finite groups }G\ \forall\text{ faithful absolutely irreducible }kG\text{-modules }V, \qquad \dim_k H^1(G,V)\le d. \] Thus the bound must be independent of $G$ and $V$, and also independent of $k$ if coefficient fields vary. The choice of $\mathcal K$ remains unresolved; the supplied record does not determine a fully specified coefficient-field quantifier. The source also directs the reader to Archive Problem 16.55, whose statement is not supplied.

Short English statement

The question asks whether a single positive integer bounds the dimension of the first cohomology of every finite group with coefficients in any faithful absolutely irreducible module. The coefficient-field scope is unspecified and requires clarification. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, permanent ID $701368$, exact source code catalogue entry, title Kourovka Notebook Problem 17.66; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied record labels the problem open and records the proposal year as 2010.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original Problem 17.66, 17th issue (2010); the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. These bibliographic details are supplied source data and have not been independently verified.
• Attribution: the statement begins with an attribution to R. Guralnick, whereas the background and proposer field identify V. D. Mazurov. The precise relationship between these attributions is not supplied.
• Supplied cross-reference: Archive Problem 16.55. Its contents and bibliographic details are not included in the record.
• The supplied collection description identifies E. I. Khukhro and V. D. Mazurov as editors of the Kourovka Notebook collection covering Issues 1--21 (1965--2026).