Kourovka Notebook Problem 8.23

Definitions and mathematical statement

A group $G$ has a section isomorphic to a group $K$ if there exist a subgroup $H\leq G$ and a normal subgroup $N\trianglelefteq H$ such that $H/N\cong K$. Let \[ D=\langle r,s\mid r^9=s^2=1,\ srs^{-1}=r^{-1}\rangle \] be the dihedral group of order $18$.

Is the following implication true for every pair of groups $A$ and $B$? \[ \begin{aligned} &\exists H\leq A\times B\;\exists N\trianglelefteq H:\ H/N\cong D\\ &\quad\Longrightarrow \bigl(\exists H_A\leq A\;\exists N_A\trianglelefteq H_A:\ H_A/N_A\cong D\bigr)\\ &\qquad\qquad\lor \bigl(\exists H_B\leq B\;\exists N_B\trianglelefteq H_B:\ H_B/N_B\cong D\bigr)? \end{aligned} \] No finiteness assumption on $A$ or $B$ is stated in the supplied source record.

Short English statement

If the dihedral group of order $18$ is a quotient of a subgroup of $A\times B$, must it also be a quotient of a subgroup of at least one factor?

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700475$, exact source problem code catalogue entry, titled Kourovka Notebook Problem 8.23.
• Attributed in the supplied record to L. G. Kovács, proposed in $1982$.
• Supplied reference: The Kourovka Notebook (Unsolved Problems in Group Theory), problem 8.23, 8th issue (1982); 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels the problem open. This status and the bibliographic details are supplied source information and have not been independently verified.