Kourovka Notebook Problem 11.121

Definitions and mathematical statement

All groups below are finite. A formation is a nonempty class of groups closed under isomorphisms, homomorphic images, and finite subdirect products. For a formation $\mathcal H$, define the $\mathcal H$-residual of $G$ by \[ G^{\mathcal H}=\bigcap\{N\trianglelefteq G:G/N\in\mathcal H\}. \] Use the standard residual convention for the product of formations: \[ \mathcal A\mathcal B=\{G:G^{\mathcal B}\in\mathcal A\}. \] A chief factor of $G$ is a quotient $H/K$, where $K<H$ are normal subgroups of $G$ and no normal subgroup of $G$ lies strictly between them. Put \[ C_G(H/K)=\{g\in G:ghg^{-1}K=hK\text{ for every }h\in H\}. \] A formation $\mathcal F$ is local if there is an assignment $f$ taking each prime $p$ to a formation or the empty class such that \[ G\in\mathcal F \quad\Longleftrightarrow\quad G/C_G(H/K)\in f(p) \quad\text{for every chief factor }H/K\text{ of }G \text{ and every prime }p\mid |H/K|. \] A formation is non-local if it is not local.

For a formation $\mathcal F$, write $D_{\mathrm{nt}}(\mathcal F)$ for the ordered pairs $(\mathcal A,\mathcal B)$ of formations satisfying $\mathcal F=\mathcal A\mathcal B$ for which this decomposition is non-trivial in the intended source convention. The supplied record does not define that convention; consequently this part of the formulation remains unresolved.

The question attributed to L. A. Shemetkov is whether there exists a local formation $\mathcal F$ such that \[ D_{\mathrm{nt}}(\mathcal F)\ne\varnothing \quad\text{and}\quad \forall(\mathcal A,\mathcal B)\in D_{\mathrm{nt}}(\mathcal F),\qquad \mathcal A\text{ and }\mathcal B\text{ are both non-local}. \] Here "in any such a decomposition" is read as referring to non-trivial decompositions. A requirement covering every decomposition without qualification would include $\mathcal F=\{1\}\mathcal F=\mathcal F\{1\}$, whose factors are local. The exact intended exclusion of trivial decompositions requires source clarification.

The source additionally states that local products of non-local formations exist. This is retained as a supplied background assertion, without an independent verification or proof.

Short English statement

Does there exist a local formation of finite groups that admits a non-trivial product decomposition, but for which every non-trivial product decomposition has two non-local factors? The source does not specify exactly which decompositions count as non-trivial. It also states that some local formations are products of two non-local formations.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700813$, exact source code catalogue entry, title Kourovka Notebook Problem 11.121. The registry labels the problem open; that label is supplied metadata, not an independently verified mathematical assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 11.121, attributed to L. A. Shemetkov, 11th issue (1990). The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. These bibliographic details are supplied attribution; the cited text has not been independently consulted.
• The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors of the Kourovka Notebook collection covering Issues 1--21 (1965--2026).