Kourovka Notebook Problem 14.79

Definitions and mathematical statement

All groups considered are finite. A Fitting class is a nonempty class of groups, closed under isomorphisms, normal subgroups, and products of normal subgroups belonging to the class: if $A,B\trianglelefteq X$ and $A,B\in\mathfrak K$, then $AB\in\mathfrak K$. A class is soluble if every group belonging to it is soluble.

For a Fitting class $\mathfrak M$, let $X_{\mathfrak M}$ denote the largest normal subgroup of $X$ belonging to $\mathfrak M$. Under the usual Fitting-product convention, put \[ \mathfrak M\mathfrak H =\{X : X/X_{\mathfrak M}\in\mathfrak H\}. \] The supplied record does not define this product, so its convention requires confirmation against the cited source. Write $\operatorname{lFit}(G)$ for the least local Fitting class containing $G$, when such a least class exists. Thus $G\in\operatorname{lFit}(G)$, and $\operatorname{lFit}(G)\subseteq\mathfrak L$ for every local Fitting class $\mathfrak L$ containing $G$. The technical term "local Fitting class" is retained from the source: its defining local construction is not supplied and requires verification before a fully expanded formalization can be given.

The proposed assertion is the following. For every finite group $G$ and every pair of Fitting classes $\mathfrak M,\mathfrak H$, suppose that \[ \mathfrak F=\mathfrak M\mathfrak H=\operatorname{lFit}(G) \] is a soluble, one-generated local Fitting class, and that $\mathfrak M\ne\mathfrak F$. Must $\mathfrak H$ be a local Fitting class? Here "one-generated" refers to generation by the single group $G$ as a local Fitting class, not to cyclicity of $G$. The question remains subject to the unresolved source conventions described above.

Short English statement

If a soluble local Fitting class generated by one finite group is a Fitting product $\mathfrak M\mathfrak H$, with $\mathfrak M$ different from the whole class, must $\mathfrak H$ also be local? The precise local-class convention requires source verification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701064$, exact source code catalogue entry, titled "Kourovka Notebook Problem 14.79"; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record was created and updated on 2026-10-16 and supplies the status "open"; that status is not independently verified here.
• Attributed in the supplied record to A. N. Skiba, proposed in 1999, in The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), Problem 14.79. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48.
• The supplied collection description identifies E. I. Khukhro and V. D. Mazurov as editors of the Kourovka Notebook collection covering Issues 1--21 (1965--2026). These bibliographic details are supplied attribution; the cited source has not been independently consulted here.