Kourovka Notebook Problem 10.17

Definitions and mathematical statement

A group $G$ is an $FC$-group if every conjugacy class $\{g^{-1}xg:g\in G\}$, for $x\in G$, is finite. Write \[ Z(G)=\{z\in G:zg=gz\text{ for every }g\in G\},\qquad G'=[G,G]=\langle x^{-1}y^{-1}xy:x,y\in G\rangle. \] An embedding means an injective group homomorphism. A direct product of finite groups means $\prod_{i\in I}F_i$, where $I$ is an arbitrary indexing set and each $F_i$ is a finite group; the factors need not be abelian.

The question is whether, for every $FC$-group $G$, the following implication holds: \[ \begin{gathered} \text{there exist a set }I,\text{ finite groups }(F_i)_{i\in I},\text{ and an embedding} \\ G'\hookrightarrow\prod_{i\in I}F_i \end{gathered} \quad\Longrightarrow\quad \begin{gathered} \text{there exist a set }J,\text{ finite groups }(H_j)_{j\in J},\text{ and an embedding} \\ G/Z(G)\hookrightarrow\prod_{j\in J}H_j. \end{gathered} \] No finiteness, finite-generation, or countability assumption on $G$ is supplied. The mathematical question is stated as supplied; its attribution is unresolved because the statement names M. J. Tomkinson whereas the proposer metadata and background name L. A. Kurdachenko.

Short English statement

If every conjugacy class of a group $G$ is finite and its derived subgroup embeds in a direct product of finite groups, must the quotient $G/Z(G)$ also admit such an embedding? The supplied registry labels this question open; that label is not independently verified.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $700634$, exact source code catalogue entry, title Kourovka Notebook Problem 10.17, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record was created and updated on 2026-10-16T14:00:00Z and supplies the status open.
• Supplied bibliographic source: The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 10.17, 10th issue (1986); cited edition: 21st issue (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. This bibliographic information is supplied source data and has not been independently checked.
• Attribution as supplied: the statement begins with the parenthetical name M. J. Tomkinson; the background and proposer field instead name L. A. Kurdachenko, with proposed year 1986. The record does not explain the relationship between these attributions.