Kourovka Notebook Problem 9.78

Definitions and mathematical statement

A group $G$ is periodic if every element has finite order. It is residually finite if, for every $g\in G\setminus\{1\}$, there are a finite group $Q$ and a homomorphism $\varphi:G\to Q$ with $\varphi(g)\ne 1$. For a prime $p$, a $p$-subgroup is a subgroup all of whose elements have order a power of $p$; a Sylow $p$-subgroup means a subgroup maximal under inclusion among the $p$-subgroups of $G$.

The supplied question asks whether there exists a group $G$ satisfying all of the following conditions:

1. $G$ is periodic and residually finite.
2. $G$ is an $F^{*}$-group, in the sense referenced by the source as Archive, Problem 7.42.
3. For every prime $p$, every Sylow $p$-subgroup of $G$ is finite.
4. $G$ is not binary finite.

The supplied record does not include the referenced definition of an $F^{*}$-group or a definition of binary finite. These two conditions retain the source terminology and remain unresolved predicates in this draft; a complete formal statement requires their intended definitions. No uniform bound on the orders of the finite Sylow subgroups is stated.

Short English statement

Does there exist a periodic, residually finite $F^{*}$-group with every Sylow subgroup finite, but which is not binary finite? The meanings of $F^{*}$-group and binary finite require source clarification. The supplied registry labels the problem open; that label is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700611$, title Kourovka Notebook Problem 9.78, exact source code catalogue entry, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied status is open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 9.78, proposed by V. P. Shunkov, ninth issue (1984); the record identifies the twenty-first issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The statement refers to Archive, Problem 7.42, for the meaning of $F^{*}$-group. The contents of that archive entry were not supplied. The cited sources have not been independently consulted for this draft.