Kourovka Notebook Problem 11.120

Definitions and mathematical statement

A finite group $G$ is soluble if its derived series terminates at the trivial group. A formation is a nonempty class of finite groups closed under homomorphic images and finite subdirect products. It is soluble if all its members are soluble, saturated if $G/\Phi(G)\in\mathfrak F$ implies $G\in\mathfrak F$, where $\Phi(G)$ is the Frattini subgroup, and Fitting if it is closed under normal subgroups and under products of normal $\mathfrak F$-subgroups of a finite group.

For a finite group $H$, write $\ell_{\mathrm{comp}}(H)=r$ when \[ 1=H_0\triangleleft H_1\triangleleft\cdots\triangleleft H_r=H \] is a composition series: each inclusion is proper and each factor $H_i/H_{i-1}$ is simple. The number $r$ is independent of the composition series; in particular, $\ell_{\mathrm{comp}}(1)=0$.

Let $\mathfrak F$ be a soluble saturated Fitting formation. The supplied question asks whether, for finite soluble groups $G$, there is a bound \[ l_{\mathfrak F}(G)\leq f\left(c_{\mathfrak F}(G)\right), \qquad c_{\mathfrak F}(G)=\ell_{\mathrm{comp}}(H), \] where $H$ is some $\mathfrak F$-covering subgroup of $G$, and $l_{\mathfrak F}(G)$ denotes the $\mathfrak F$-length defined in Shemetkov's cited book.

Unresolved qualifications. The supplied record does not reproduce the definitions of $\mathfrak F$-covering subgroup or $\mathfrak F$-length. These terms are retained as reference-dependent terms; no replacement definition is imposed here. The record also does not specify the domain and codomain of $f$, whether $f$ may depend on $\mathfrak F$, or whether the bound must hold for every covering subgroup or for a suitable choice. Consequently, the displayed question preserves the source formulation but is not a complete standalone quantified conjecture.

Short English statement

Shemetkov asks whether the $\mathfrak F$-length of a finite soluble group can be bounded in terms of the composition length of some $\mathfrak F$-covering subgroup, for a soluble saturated Fitting formation $\mathfrak F$. The definitions referred to the book and the precise uniformity and subgroup-choice requirements remain unresolved.

Sources

• Local UnsolvedMath registry: permanent ID $700812$, exact source code catalogue entry, title Kourovka Notebook Problem 11.120. The supplied registry labels the problem open; this is supplied status information, not an independently verified mathematical assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 11.120, proposed by L. A. Shemetkov in the 11th issue (1990). The supplied background identifies the consulted edition as the 21st issue edition (2026), arXiv:1401.0300v48. The collection description credits E. I. Khukhro and V. D. Mazurov as editors.
• L. A. Shemetkov, Formations of Finite Groups, Moscow, Nauka, 1978 (Russian), Problem 17. The supplied statement identifies this as the earlier occurrence of the problem and refers to the same book for the definition of $\mathfrak F$-length. No page number or reproduction of that definition is supplied.