Kourovka Notebook Problem 10.57

Definitions and mathematical statement

All groups considered are finite. A formation is a nonempty class of finite groups, closed under isomorphisms, homomorphic images, and finite subdirect products. Here a subgroup $H\leq G_1\times\cdots\times G_n$ is a subdirect product if each coordinate projection maps $H$ onto $G_i$.

Define \[ \mathcal A=\{G: G\text{ is a finite group and every Sylow subgroup of }G\text{ is abelian}\}. \] The supplied source calls this the $A$-formation. To interpret the requested minimality, call a formation an $A$-formation when it is contained in $\mathcal A$, and order formations by inclusion. Under this interpretation, a minimal non-$A$-formation is a formation $\mathcal F$ such that \[ \mathcal F\nsubseteq\mathcal A, \qquad \text{for every formation }\mathcal E\subsetneq\mathcal F, \quad \mathcal E\subseteq\mathcal A. \] Equivalently, $\mathcal F$ contains a group with a nonabelian Sylow subgroup, whereas every group in every proper subformation of $\mathcal F$ has abelian Sylow subgroups.

Problem (A. N. Skiba, 1986). Classify all minimal non-$A$-formations.

Formulation qualification. The source defines the formation of all finite groups with abelian Sylow subgroups, but does not explicitly define the adjective $A$-formation for other formations or specify the order underlying minimality. The displayed classification target uses containment in $\mathcal A$ and inclusion of formations; these interpretive choices require confirmation against the original problem.

Short English statement

Under the stated inclusion interpretation, classify the formations containing a finite group with a nonabelian Sylow subgroup for which every proper subformation consists entirely of groups with abelian Sylow subgroups. The source's terminology and minimality convention require confirmation.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700672$, exact source problem code catalogue entry, title Kourovka Notebook Problem 10.57, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 10.57, 10th issue (1986); proposed by A. N. Skiba. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. These bibliographic details are supplied attribution and have not been independently verified.
• The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. The registry labels the problem open; this is a supplied status label, not an independently verified assessment of the literature.