Kourovka Notebook Problem 10.78

Definitions and mathematical statement

For a prime $p$, the quasicyclic $p$-group is \[ C_{p^\infty}=\{x\in{\mathbb{Q}}/{\mathbb{Z}} : p^n x=0\text{ for some integer }n\geq 0\}. \] A group $H$ is called a Chernikov group if it has a normal subgroup $D$ of finite index such that \[ D\cong\prod_{i=1}^{r}C_{p_i^\infty} \] for some integer $r\geq 0$ and primes $p_1,\ldots,p_r$, which need not be distinct. The empty product is the trivial group, so finite groups are included.

Does there exist an abstract group $G$ and subgroups $A,B\leq G$ such that $A$ and $B$ are Chernikov groups, $G$ is not a Chernikov group, and \[ G=AB=\{ab:a\in A,\ b\in B\}? \] Here the product is a setwise product: every element of $G$ must have such a factorization. Neither normality of the factors nor uniqueness of the factorization is assumed.

Short English statement

Can a group that is not Chernikov be the setwise product of two Chernikov subgroups? This is the question attributed to V. P. Shunkov in Kourovka Notebook Problem 10.78.

Sources

• Local UnsolvedMath registry, supplied source record: permanent ID $700693$, exact source problem code catalogue entry, title Kourovka Notebook Problem 10.78; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The supplied record attributes the problem to V. P. Shunkov, proposed in 1986, and cites The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), Problem 10.78, as carried in the 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 and records an update date of 2026-10-16. This is supplied registry information, not an independently verified assessment of the literature or current mathematical status.