Kourovka Notebook Problem 18.29

Definitions and mathematical statement

All groups considered are finite. A group is soluble if its derived series terminates at the trivial group. A Fitting class $\mathfrak F$ of finite soluble groups is a nonempty class, closed under isomorphisms, such that every normal subgroup of a member of $\mathfrak F$ belongs to $\mathfrak F$, and whenever $N_1,N_2\trianglelefteq G$ belong to $\mathfrak F$, with $G$ finite soluble, their product $N_1N_2$ belongs to $\mathfrak F$.

For $H\leq G$, write $H^g=g^{-1}Hg$. The subgroup $H$ is abnormal in $G$ if \[ g\in\langle H,H^g\rangle\qquad\text{for every }g\in G. \] An $\mathfrak F$-subgroup means a subgroup whose underlying group belongs to $\mathfrak F$. For subgroups $A,B\leq G$, the equation $G=AB$ means that every element of $G$ is a product $ab$, where $a\in A$ and $b\in B$.

A formation of finite soluble groups is a nonempty isomorphism-closed class closed under homomorphic images and finite subdirect products. A subdirect product of $G_1,\ldots,G_r$, where $r\geq1$, is a subgroup of $G_1\times\cdots\times G_r$ whose projection onto each factor is surjective.

Question (A. F. Vasil'ev, Kourovka Notebook Problem 18.29). Let $\mathfrak F$ be a Fitting class of finite soluble groups. Assume that for every finite soluble group $G$ and every pair of subgroups $A,B\leq G$, \[ \left(G=AB,\quad A,B\in\mathfrak F,\quad A\text{ and }B\text{ abnormal in }G\right) \ \Longrightarrow\ G\in\mathfrak F. \] Must $\mathfrak F$ be a formation?

The phrase "abnormal $\mathfrak F$-subgroups" is read here as subgroups belonging to $\mathfrak F$ that are abnormal in the ambient group $G$. The supplied record gives no separate definition of this phrase.

Short English statement

Suppose a Fitting class of finite soluble groups contains every finite soluble group that is a product of two abnormal subgroups belonging to the class. Must the class also be closed under homomorphic images and finite subdirect products?

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $701458$, source code catalogue entry, title Kourovka Notebook Problem 18.29, and upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), Problem 18.29; also identified in the record with the 21st issue edition (2026), arXiv:1401.0300v48. Proposed by A. F. Vasil'ev in 2014. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The local record labels the problem open. This is supplied registry metadata, not an independently verified assessment of the literature. The cited source has not been independently consulted for this draft.