Kourovka Notebook Problem 5.1

Definitions and mathematical statement

A group $G$ is locally finite if every finitely generated subgroup of $G$ is finite. For $g\in G$, write $g^G=\{x^{-1}gx:x\in G\}$. An $FC$-group is a group for which $g^G$ is finite for every $g\in G$. A $BFC$-group is a group for which there exists an integer $M\geq 1$ such that $|g^G|\leq M$ for every $g\in G$.

Using the standard subgroup-minimal interpretation, a minimal non-$FC$-group is a group that is not an $FC$-group but whose every proper subgroup is an $FC$-group. Minimal non-$BFC$-groups are defined analogously. These conventions are not explicitly defined in the supplied record. Let \[ G'=[G,G]=\langle x^{-1}y^{-1}xy:x,y\in G\rangle \] be the derived subgroup of $G$.

Part (b), the remaining question in the supplied record. Is it true that every locally finite minimal non-$FC$-group $G$ satisfies \[ G'\neq G? \] Equivalently, must every such group have nontrivial abelianization $G/G'$?

The source additionally reports that the question has an affirmative answer for minimal non-$BFC$-groups. This is retained as a supplied qualification: the sentence does not explicitly repeat the local finiteness hypothesis or provide a separate citation for this assertion, so its precise scope requires source review.

Part (a), archived as solved in the supplied record. Is every locally finite minimal non-$FC$-group non-simple? Here a simple group means a nontrivial group whose only normal subgroups are the identity subgroup and the whole group. The supplied archive reports an affirmative answer to part (a), attributed to M. Kuzucuoḡlu and R. E. Phillips (1989); that reported resolution is not independently verified here.

Short English statement

The remaining question asks whether every locally finite group that fails the $FC$ property, while all its proper subgroups satisfy it, has nontrivial abelianization. The source reports an affirmative answer for minimal non-$BFC$-groups and archives the related non-simplicity question as solved. These are supplied source assessments requiring human review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700285$, source problem code catalogue entry, title Kourovka Notebook Problem 5.1, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 5.1, proposed by V. V. Belyaev and N. F. Sesekin in the 5th issue (1976). The supplied record describes an update in the 21st issue (2026), identified as arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. No source was independently visited or checked.
• For the archived affirmative answer to part (a), the supplied record cites M. Kuzucuoḡlu and R. E. Phillips, Proc. Cambridge Philos. Soc., 105, no. 3 (1989), 417--420. No article title or URL is supplied.
• The supplied status review is dated 2026-10-16 and labels the record partially_solved, stating that solved parts occur in the archive and remaining parts in the main section. This is supplied dated curation, not an independently verified mathematical or literature assessment.