Kourovka Notebook Problem 18.3

Definitions and mathematical statement

For a group $H$, let $Z(H)=\{z\in H:zh=hz\text{ for every }h\in H\}$ denote its centre. The group $H$ is central-by-finite if $[H:Z(H)]<\infty$. Write $H'=[H,H]$ and $H''=[H',H']$; the group $H$ is metabelian if $H''=\{1\}$. A group is metabelian-by-finite if it has a normal metabelian subgroup of finite index. A group is soluble-by-finite if it has a normal subgroup of finite index whose derived series reaches the trivial subgroup after finitely many steps.

Question (B. Amberg, 2014). For every group $G$ and subgroups $A,B\leq G$ satisfying \[ G=AB=\{ab:a\in A,\ b\in B\},\qquad [A:Z(A)]<\infty,\qquad [B:Z(B)]<\infty, \] does there exist a subgroup $M\trianglelefteq G$ such that \[ [G:M]<\infty\qquad\text{and}\qquad M''=\{1\}? \] No finiteness or finite generation assumption on $G$, and no normality assumption on $A$ or $B$, is imposed.

The supplied source attributes to N. Chernikov a theorem that every such $G$ is soluble-by-finite. This is supplied background, not an independently verified assertion here; the record does not give a separate citation for that theorem.

Short English statement

Must every group that is a product of two subgroups, each having centre of finite index, contain a normal metabelian subgroup of finite index? The supplied record labels this question open; that status has not been independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $701432$, source problem code catalogue entry, title Kourovka Notebook Problem 18.3.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 18.3, proposed by B. Amberg in 2014. The supplied record identifies the 18th issue (2014) and the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied statement attributes the soluble-by-finite conclusion to N. Chernikov, without a separate bibliographic reference. The references and the supplied open status have not been independently checked.