Kourovka Notebook Problem 18.2

Definitions and mathematical statement

A group $H$ is called a Chernikov group if it has a normal subgroup $D$ of finite index such that $D$ is a finite direct product of quasicyclic groups. For a prime $p$, the quasicyclic group is \[ C_{p^\infty}=\{z\in{\mathbb{C}}^\times : z^{p^n}=1\text{ for some integer }n\geq 0\}, \] with multiplication. The empty direct product is permitted, so finite groups are included among Chernikov groups.

For a group $G$, define $G^{(0)}=G$ and $G^{(n+1)}=[G^{(n)},G^{(n)}]$ for every integer $n\geq 0$, where $[H,H]$ is the subgroup generated by all commutators $x^{-1}y^{-1}xy$ with $x,y\in H$. The group $G$ is soluble if $G^{(m)}=\{1\}$ for some integer $m\geq 0$.

Question (B. Amberg). Let $G$ be a group with Chernikov subgroups $A,B\leq G$ satisfying \[ G=AB=\{ab:a\in A,\ b\in B\}. \] Suppose there exist abelian subgroups $A_0\leq A$ and $B_0\leq B$ such that \[ [A:A_0]\leq 2,\qquad [B:B_0]\leq 2. \] Must $G$ be soluble? The factorization is a setwise product; no normality of $A$ or $B$ in $G$, and no finiteness of $G$, is assumed. The supplied registry labels the question open; this status has not been independently verified.

Short English statement

Must every group that is the product of two Chernikov subgroups, each containing an abelian subgroup of index at most $2$, be soluble? This is the question attributed to B. Amberg in Kourovka Notebook Problem 18.2.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701431$, source code catalogue entry, titled Kourovka Notebook Problem 18.2; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record supplies the status open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 18.2, proposed by B. Amberg in 2014. The supplied background identifies the 18th issue (2014) and the 21st issue edition (2026), with identifier arXiv:1401.0300v48. The supplied collection description attributes editorship to E. I. Khukhro and V. D. Mazurov.