Kourovka Notebook Problem 18.4

Definitions and mathematical statement

Let $G$ be a group, with no finiteness assumption. A subgroup is cyclic if it is generated by one element; finite, infinite, and trivial cyclic subgroups are allowed. For subgroups $A,B\leq G$, define \[ ABA=\{a_1ba_2:a_1,a_2\in A,\ b\in B\}. \] Thus $G=ABA$ means that every element of $G$ has such an expression, without any uniqueness requirement. Define the derived series by \[ G^{(0)}=G,\qquad G^{(n+1)}=\langle [x,y]:x,y\in G^{(n)}\rangle, \qquad [x,y]=x^{-1}y^{-1}xy. \] The group $G$ is soluble if $G^{(n)}=\{1\}$ for some integer $n\geq0$.

1. (a) Is it true that, for every group $G$ and every pair of cyclic subgroups $A,B\leq G$, the equality $G=ABA$ implies that $G$ is soluble?
2. (b) Is it true that, for every group $G$ and every pair of cyclic subgroups $A,B\leq G$ such that $G=ABA$ and $B=tAt^{-1}$ for some $t\in G$, the group $G$ is soluble?

The supplied statement reports that part (a) is known to hold when $G$ is finite. This is a source-reported qualification, not an independently verified result here. The registry labels the problem open. Its metadata names B. Amberg as proposer, while the statement begins with the attribution "(A. Rhemtulla, S. Sidki)"; the precise relationship between these attribution roles is not specified.

Short English statement

Must every group that is a product $ABA$ of two cyclic subgroups be soluble? Ask the same question with the additional requirement that $A$ and $B$ are conjugate in the group. The supplied source reports an affirmative answer for finite groups.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $701433$, exact source problem code catalogue entry, title Kourovka Notebook Problem 18.4. The supplied status is open; the record was updated on 2026-10-16T14:00:00Z.
• Supplied bibliographic attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), problem 18.4; also identified in the background as the 21st issue edition (2026), arXiv:1401.0300v48. The set description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied source data and have not been independently verified here.
• Proposer metadata and background: B. Amberg, proposed in 2014. The mathematical statement separately attributes the problem to A. Rhemtulla and S. Sidki in its opening parenthesis. Both attributions are retained without assigning further roles.
• The supplied statement says that the assertion in part (a) is known for finite groups, but supplies no separate citation for that result.