Kourovka Notebook Problem 20.34

Definitions and mathematical statement

All groups below are finite. A formation $\mathfrak F$ is a nonempty class of groups, closed under isomorphisms, homomorphic images, and finite subdirect products. Here a subdirect product of groups $G_1,\ldots,G_n$ is a subgroup of $\prod_{i=1}^nG_i$ whose coordinate projections are surjective. The formation is saturated if, for every finite group $X$, \[ X/\Phi(X)\in\mathfrak F\ \Longrightarrow\ X\in\mathfrak F, \] where $\Phi(X)$ is the intersection of the maximal subgroups of $X$, with $\Phi(1)=1$. A nonempty isomorphism-closed class $\mathfrak F$ is a Fitting class if it is closed under taking normal subgroups and if, whenever $H,K\trianglelefteq X$ belong to $\mathfrak F$, their product $HK$ also belongs to $\mathfrak F$.

Let $f$ be an inner screen of a saturated formation $\mathfrak F$, as in the source. An action of $A$ on $G$ is a homomorphism $\rho:A\to\operatorname{Aut}(G)$, and it is faithful if $\ker\rho=1$. The semidirect product $G\rtimes_\rho A$ has multiplication \[ (g,a)(h,b)=(g\rho(a)(h),ab). \] For purposes of recording the question, write $\operatorname{Hyp}_f(G,A,\rho)$ for the source condition that this action is $f$-hypercentral. The supplied record gives no technical definition of either an inner screen or an $f$-hypercentral action; these terms remain unresolved source qualifications, and this predicate is not an independent definition of hypercentrality.

Under the interpretation that the acting group and its action are universally quantified, the proposed statement is \[ \begin{aligned} \mathfrak F\text{ is a Fitting class} \quad\Longleftrightarrow\quad &\text{for every finite }G\in\mathfrak F,\text{ every finite }A,\\ &\text{and every injective }\rho:A\to\operatorname{Aut}(G),\\ &\operatorname{Hyp}_f(G,A,\rho) \ \Longrightarrow\ G\rtimes_\rho A\in\mathfrak F. \end{aligned} \] The source explicitly says "for all $G\in\mathfrak F$" but does not explicitly give the scope of the quantifier on $A$ and its action. Thus the displayed universal interpretation requires confirmation, together with the technical meanings of the two screen-dependent terms.

Short English statement

For a saturated formation $\mathfrak F$ with an inner screen $f$, the question asks whether closure under semidirect products arising from faithful, $f$-hypercentral finite group actions characterizes $\mathfrak F$ being a Fitting class. The definitions of the screen-dependent terms and the quantifier scope for the acting group require source confirmation.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701692$, exact source code catalogue entry, title Kourovka Notebook Problem 20.34, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record labels the problem open; this is supplied registry status, not an independently verified mathematical assessment.
• Attributed to W. Guo, proposed in 2022. The supplied background cites The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), original problem 20.34, and the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The record supplies a bibliographic source attribution but no source excerpt defining inner screens or $f$-hypercentral actions, no literature-source links, and no literature-verification date. The cited edition and its terminology have not been independently consulted for this draft.