Kourovka Notebook Problem 10.18

Definitions and mathematical statement

For a group $G$, write $Z(G)=\{z\in G:zg=gz\text{ for all }g\in G\}$ for its centre. An $FC$-group is a group in which every conjugacy class $\{xgx^{-1}:x\in G\}$, for $g\in G$, is finite. For a group $H$, its derived subgroup is $H'=\langle a^{-1}b^{-1}ab:a,b\in H\rangle$. Say that $G$ is residually in the class of groups with finite derived subgroups if, for every $g\in G\setminus\{1\}$, there exists a normal subgroup $N\trianglelefteq G$ such that $g\notin N$ and $(G/N)'$ is finite. The quotient $G/N$ itself need not be finite.

Question. For every $FC$-group $G$ that is residually in the class of groups with finite derived subgroups, do there exist an index set $I$, finite groups $F_i$ for $i\in I$, and an injective group homomorphism \[ G/Z(G)\longrightarrow\prod_{i\in I}F_i? \] Here the product is the full direct product; no finiteness or countability restriction on $I$, and no finite generation or countability assumption on $G$, is stated in the supplied record. The record contains conflicting proposer attributions, detailed below; this leaves the mathematical question unchanged.

Short English statement

Must the quotient by the centre of every $FC$-group that is residually a group with finite derived subgroup embed in a direct product of finite groups? Each target factor must be finite, although the groups witnessing the residual hypothesis may be infinite.

Sources

• Local UnsolvedMath registry, permanent ID $700635$, title Kourovka Notebook Problem 10.18, exact source code catalogue entry; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied registry labels the problem open; this label is not an independently verified mathematical or literature assessment.
• Supplied bibliographic source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), problem 10.18; the record also identifies the 21st issue edition (2026), arXiv:1401.0300v48. Its collection description names E. I. Khukhro and V. D. Mazurov as editors. These source details are supplied attribution and have not been independently checked.
• Attribution conflict: the statement begins with M. J. Tomkinson's name, whereas the background and proposer field name L. A. Kurdachenko and give the proposed year as 1986. The supplied evidence does not resolve their respective roles.