Kourovka Notebook Problem 12.35

Definitions and mathematical statement

A formation of finite groups is a nonempty class $\mathfrak F$ of finite groups, closed under isomorphisms, homomorphic images, and finite subdirect products. Here a subdirect product is a subgroup of a direct product whose projection onto each factor is surjective. For a finite group $X$, define its $\mathfrak F$-residual by \[ X^{\mathfrak F}=\bigcap\{N\trianglelefteq X:X/N\in\mathfrak F\}. \] A subgroup $H\leq G$ is subnormal in $G$ if there is a finite chain \[ H=H_0\trianglelefteq H_1\trianglelefteq\cdots\trianglelefteq H_r=G, \qquad r\geq 0. \] For subgroups $H,K\leq G$, write $\langle H,K\rangle$ for the subgroup of $G$ generated by $H\cup K$.

Source qualification. The source requires $\mathfrak F$ to be a radical composition formation of finite groups. The supplied record does not define the additional technical qualifications "radical" and "composition". They are retained as hypotheses in the source's intended sense; their precise definitions require confirmation before this becomes a fully self-contained statement.

Requested assertion (S. F. Kamornikov, Kourovka Notebook Problem 12.35). For every radical composition formation $\mathfrak F$ of finite groups, every finite group $G$, and every pair of subnormal subgroups $H,K$ of $G$, prove that \[ \langle H,K\rangle^{\mathfrak F} = \langle H^{\mathfrak F},K^{\mathfrak F}\rangle. \] Each residual is computed in the group appearing as its base: respectively $\langle H,K\rangle$, $H$, and $K$. The equality is an equality of subgroups of $G$.

Short English statement

For any radical composition formation $\mathfrak F$, the $\mathfrak F$-residual of the subgroup generated by two subnormal subgroups of a finite group should equal the subgroup generated by their individual $\mathfrak F$-residuals. The source does not define the technical qualifications on the formation, so their intended meaning requires review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700854$, exact source code catalogue entry, title Kourovka Notebook Problem 12.35; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 12.35, 12th issue (1992); proposed by S. F. Kamornikov. The supplied record also identifies the 21st issue edition (2026), arXiv identifier 1401.0300v48. Its collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The local record labels the problem open and records an update dated 2026-10-16. These are supplied registry metadata, not an independently verified mathematical or literature assessment. No separate literature-verification date or reference defining "radical composition formation" is supplied.