Kourovka Notebook Problem 19.99

Definitions and mathematical statement

A finite group $H$ is called soluble if its derived series, defined by $H^{(0)}=H$ and $H^{(i+1)}=[H^{(i)},H^{(i)}]$, reaches the trivial group. For a finite group $H$, let $F(H)$ denote its Fitting subgroup, the largest nilpotent normal subgroup of $H$. Define $F_0(H)=1$ and, recursively, let $F_{i+1}(H)$ be the inverse image of $F(H/F_i(H))$ under the quotient map. For soluble $H$, its Fitting height is \[ h(H)=\min\{r\in{\mathbb{Z}}_{\geq 0}:F_r(H)=H\}. \] In particular, $h(1)=0$.

For a finite group $G$ and a positive integer $t$, put \[ \mathcal M_t(G)=\{M<G:M\text{ is a maximal proper subgroup of }G,\ M\text{ is soluble, and }h(M)\leq t\}. \] Here maximality is among all proper subgroups of $G$. Two members $M,N$ of $\mathcal M_t(G)$ are identified when $N=gMg^{-1}$ for some $g\in G$. Write $c_t(G)$ for the number of these conjugacy classes.

The question is whether \[ \forall t\in{\mathbb{Z}}_{\geq1}\ \exists n(t)\in{\mathbb{Z}}_{\geq1}\ \forall\text{ finite groups }G,\qquad c_t(G)\geq n(t)\ \Longrightarrow\ \left(G\text{ is soluble and }h(G)\leq t\right). \] The supplied record labels this question open; that status is not independently verified here.

Short English statement

For each positive integer $t$, does a sufficiently large number of conjugacy classes of soluble maximal subgroups of Fitting height at most $t$ force a finite group itself to be soluble of Fitting height at most $t$? The threshold may depend on $t$, but must work for every finite group.

Sources

• Local UnsolvedMath registry: permanent ID $701647$, exact source code catalogue entry, title Kourovka Notebook Problem 19.99. The supplied record has status open and upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 19.99, 19th issue (2018); the background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• Proposed by A. F. Vasil'ev and T. I. Vasil'eva; the supplied proposed year is 2018. These bibliographic details and the supplied open status have not been independently verified.