Kourovka Notebook Problem 20.94

Definitions and mathematical statement

A group $G$ is periodic if every element has finite order, that is, \[ \forall g\in G\;\exists n\in{\mathbb{Z}}_{\geq 1}\quad g^n=1_G. \] It is simple if it is nontrivial and its only normal subgroups are $\{1_G\}$ and $G$. A finite Frobenius group is a finite group $F$ having a subgroup $H$ such that $\{1_F\}<H<F$ and \[ H\cap xHx^{-1}=\{1_F\}\qquad\text{for every }x\in F\setminus H. \] Let $\mathcal F$ denote the class of finite Frobenius groups.

The source asks whether there exists an infinite periodic simple group saturated with groups from $\mathcal F$, with saturation understood exactly as in Kourovka Notebook Problem 14.101. To display the requested quantifiers, write $\operatorname{Sat}_{14.101}(G,\mathcal F)$ as a placeholder for that source-defined condition. The question is whether \[ \exists\text{ a group }G\quad \left[ \begin{gathered} |G|\text{ is infinite},\\ \forall g\in G\;\exists n\in{\mathbb{Z}}_{\geq 1}\quad g^n=1_G,\\ \forall N\trianglelefteq G\quad (N=\{1_G\}\ \lor\ N=G),\\ \operatorname{Sat}_{14.101}(G,\mathcal F) \end{gathered} \right]. \] The supplied record does not include the definition in Problem 14.101. Consequently, the saturation condition remains unresolved here, and this display is not a complete self-contained formulation. In particular, no quantification over finite subgroups or embeddings is substituted for the missing definition.

Short English statement

A. I. Sozutov asks whether an infinite simple group in which every element has finite order can be saturated with finite Frobenius groups. The intended meaning of saturation must be recovered from Kourovka Notebook Problem 14.101 before this draft can be made self-contained.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701752$, exact source code catalogue entry, title Kourovka Notebook Problem 20.94; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied attribution: A. I. Sozutov, proposed in 2022. The record identifies The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), Problem 20.94, and the 21st issue edition (2026), arXiv:1401.0300v48. The collection metadata names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied statement refers to the definition of saturation in Problem 14.101; that definition is not included in the supplied evidence.
• The registry labels the problem open. This is a supplied status label, not an independently verified assessment of the literature. The cited editions and cross-reference have not been independently consulted for this draft.