Kourovka Notebook Problem 14.29

Definitions and mathematical statement

A soluble Fitting class is a nonempty class $\mathfrak F$ of finite soluble groups, closed under isomorphism, taking normal subgroups, and taking normal products: if $N_1,N_2\trianglelefteq H$ belong to $\mathfrak F$, then $N_1N_2\in\mathfrak F$. For a finite soluble group $H$, its $\mathfrak F$-radical is \[ H_{\mathfrak F}=\prod\{N\trianglelefteq H:N\in\mathfrak F\}. \] This is the largest normal subgroup of $H$ belonging to $\mathfrak F$.

A formation is a nonempty isomorphism-closed class of finite groups closed under homomorphic images and finite subdirect products. Here a subdirect product of $H_1,\ldots,H_r$ is a subgroup of $H_1\times\cdots\times H_r$ whose projection onto each factor is surjective.

Existence question (A. F. Vasiliev). Does there exist a soluble Fitting class $\mathfrak F$ which is not a formation and satisfies \[ \forall\,G\text{ finite soluble}\quad \forall\,A,B\leq G,\qquad G=AB\ \Longrightarrow\ A_{\mathfrak F}\cap B_{\mathfrak F}\leq G_{\mathfrak F}? \] Here $AB=\{ab:a\in A,\ b\in B\}$, and the radicals of $A$ and $B$ are regarded as subgroups of $G$. No normality assumption on $A$ or $B$ is imposed.

The supplied record does not define the radical notation or the class terminology; the definitions above use their conventional finite-group meanings and require confirmation against the original source. The registry labels the problem open; that label is supplied source metadata, not an independently verified assessment.

Short English statement

Does there exist a class $\mathfrak F$ of finite soluble groups that is a Fitting class but not a formation, such that every factorization $G=AB$ of a finite soluble group satisfies $A_{\mathfrak F}\cap B_{\mathfrak F}\leq G_{\mathfrak F}$? The subscripts are interpreted as Fitting-class radicals.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $701014$, exact source code catalogue entry, title Kourovka Notebook Problem 14.29. The supplied status is open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), Problem 14.29; proposed by A. F. Vasiliev. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection metadata attributes editorship to E. I. Khukhro and V. D. Mazurov. These bibliographic details are supplied attribution and have not been independently verified.