Kourovka Notebook Problem 17.68

Definitions and mathematical statement

Let $G$ be an arbitrary group, and let $C,F\leq G$ be cyclic subgroups, with $F$ finite. No finiteness assumption is imposed on $C$ or $G$. Define \[ CFC=\{c_1fc_2:c_1,c_2\in C,\ f\in F\}. \] The index $[G:C]$ is the cardinality of the set of left cosets $\{gC:g\in G\}$.

Is the following assertion true: for every such $G,C,F$, \[ G=CFC\quad\Longrightarrow\quad [G:C]<\infty? \] The supplied source also says "Cf. 12.62"; the referenced problem is not included in the record, so its relationship to this assertion remains unspecified.

Short English statement

Must a cyclic subgroup $C$ have finite index in a group $G$ whenever every element of $G$ can be written as $c_1fc_2$, with $c_1,c_2\in C$ and $f$ in a fixed finite cyclic subgroup $F$? The supplied registry labels this problem open; that status is not independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $701370$, exact source code catalogue entry, title Kourovka Notebook Problem 17.68. The supplied record was created and updated on 2026-10-16 and labels the problem open.
• Attributed to V. D. Mazurov, proposed in 2010, in The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), Problem 17.68. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. These bibliographic details are supplied attribution and have not been independently verified.
• The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors of the Kourovka Notebook collection covering Issues 1--21 (1965--2026).
• Supplied cross-reference: "Cf. 12.62." Its contents are absent from this record.