Kourovka Notebook Problem 18.8

Definitions and mathematical statement

All groups under consideration are finite. A formation $\mathfrak F$ is a nonempty class of finite groups closed under homomorphic images and finite subdirect products. A subdirect product of groups $G_1,\ldots,G_n$ is a subgroup of $\prod_{i=1}^n G_i$ whose projection onto each $G_i$ is surjective. For a finite group $H$, let $\Phi(H)$ denote its Frattini subgroup, the intersection of its maximal proper subgroups, with $\Phi(1)=1$.

Let $\mathcal X$ be a class of finite simple groups. Write $O_{\mathcal X}(G)$ for the largest normal subgroup of $G$ all of whose composition factors belong to $\mathcal X$. Here composition factors are the simple factors in a composition series, considered up to isomorphism. The trivial subgroup satisfies this condition vacuously.

The source imposes the hypothesis $\pi(\mathcal X)=\operatorname{char}(\mathcal X)$ without defining either symbol. Under their conventional interpretation in this setting, these are \[ \pi(\mathcal X)=\{p\text{ prime}:p\mid |S|\text{ for some }S\in\mathcal X\},\qquad \operatorname{char}(\mathcal X)=\{p\text{ prime}:C_p\in\mathcal X\}, \] where $C_p$ is the cyclic group of order $p$. This interpretation requires confirmation against the cited source.

For a class $\mathcal X$ satisfying the stated equality, a formation $\mathfrak F$ is $\mathcal X$-saturated if \[ \text{for every finite group }G,\qquad G/\Phi\left(O_{\mathcal X}(G)\right)\in\mathfrak F \ \Longrightarrow\ G\in\mathfrak F. \] The question is whether, for every such class $\mathcal X$ and every formation $\mathfrak F$, \[ \mathfrak F\text{ is }\mathcal X\text{-saturated} \quad\Longrightarrow\quad \mathfrak F\text{ is }\mathcal X\text{-local in the sense of F\"orster}. \] The supplied record refers to Förster's 1985 publication for the definition of $\mathcal X$-locality but does not reproduce it. Consequently, the conclusion is retained as a source-dependent predicate; a self-contained definition of that predicate remains unresolved.

Short English statement

For every class $\mathcal X$ of finite simple groups satisfying the source's condition $\pi(\mathcal X)=\operatorname{char}(\mathcal X)$, must every $\mathcal X$-saturated formation be $\mathcal X$-local in Förster's sense? The locality definition is absent from the supplied record, and the interpretation of the prime-set notation requires confirmation.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $701437$, exact source code catalogue entry, title Kourovka Notebook Problem 18.8. The supplied status is open; this label is not an independently verified mathematical assessment.
• A. Ballester-Bolinches, proposer, 2014. The supplied background identifies Problem 18.8 in The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), and cites the 21st issue edition (2026), arXiv identifier 1401.0300v48. The registry describes the collection as edited by E. I. Khukhro and V. D. Mazurov.
• P. Förster, Publ. Sec. Mat. Univ. Autònoma Barcelona, 29, no. 2--3 (1985), 39--76. The supplied statement directs the reader to this reference for the definition of $\mathcal X$-locality. No article title or URL is supplied, and the reference has not been independently consulted.