Kourovka Notebook Problem 14.28

Definitions and mathematical statement

All groups considered are finite. A group is soluble if its derived series terminates at the trivial group. A formation is a nonempty class $\mathfrak F$, closed under isomorphisms and homomorphic images, such that, whenever $N_1,N_2\unlhd G$ and $G/N_1,G/N_2\in\mathfrak F$, one has $G/(N_1\cap N_2)\in\mathfrak F$. A Fitting formation is a formation that is also closed under taking normal subgroups and under normal products: if $N_1,N_2\unlhd G$ and $N_1,N_2\in\mathfrak F$, then $N_1N_2\in\mathfrak F$. It is soluble if every member is soluble.

For subgroups $A,B\leq G$, write $AB=\{ab:a\in A,\ b\in B\}$. Say that $\mathfrak F$ has Kegel's property if, for every finite group $G$ and all subgroups $A,B,C\leq G$, \[ G=AB=BC=CA\quad\text{and}\quad A,B,C\in\mathfrak F \quad\Longrightarrow\quad G\in\mathfrak F. \] No normality of $A,B,C$ is assumed. Let $\Phi(G)$ be the Frattini subgroup, the intersection of all maximal proper subgroups of $G$, with $\Phi(1)=1$. A formation is saturated if, for every finite group $G$, membership of $G/\Phi(G)$ in the formation implies membership of $G$.

Question (A. F. Vasiliev, 1999). Is every soluble Fitting formation with Kegel's property saturated? Explicitly, must every such $\mathfrak F$ satisfy \[ \forall\,G\text{ finite},\qquad G/\Phi(G)\in\mathfrak F\ \Longrightarrow\ G\in\mathfrak F? \] The supplied record labels this problem open. Its 2013 comment reports some progress in the cited paper of Ballester-Bolinches and Ezquerro, but supplies no precise result or additional hypotheses from that paper; no such result is incorporated into this statement.

Short English statement

A. F. Vasiliev asks whether a soluble Fitting formation must be saturated if it contains every finite group admitting three pairwise factorizations by subgroups belonging to the formation. Saturation means that membership can be inferred from the quotient by the Frattini subgroup. The supplied open label and the 2013 progress comment are source reports, not independently verified assessments.

Sources

• Local UnsolvedMath registry, permanent ID $701013$, exact source problem code catalogue entry, title Kourovka Notebook Problem 14.28. The supplied record attributes the problem to A. F. Vasiliev, proposed in 1999, and labels it open; its update date is 2026-10-16T14:00:00Z.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 14.28, 14th issue (1999). The supplied background identifies the reference edition as the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• Supplied comment dated 2013: some progress was made in A. Ballester-Bolinches and L. M. Ezquerro, J. Group Theory 8, no. 5 (2005), 605--611. The supplied record gives no article title or link and does not specify the progress obtained. This dated comment is supplied source information and has not been independently verified.