Kourovka Notebook Problem 9.83

Definitions and mathematical statement

Let $p$ be a prime. A $p$-subgroup of a group $G$ is a subgroup whose elements all have finite order equal to a power of $p$. A Sylow $p$-subgroup means a $p$-subgroup maximal under inclusion. Write $\operatorname{Syl}_p(G)$ for the set of these subgroups. A group is periodic if every element has finite order.

The source asks: suppose that $G$ is a "(periodic) $p$-conjugacy biprimitively finite group" in the sense of Kourovka Notebook Problem 6.59, and suppose that \[ \exists S\in\operatorname{Syl}_p(G)\quad |S|<\infty. \] Must it follow that \[ \forall P,Q\in\operatorname{Syl}_p(G)\quad \exists g\in G\quad g^{-1}Pg=Q? \] The defining condition for "$p$-conjugacy biprimitively finite" is not included in the supplied record, so that hypothesis remains an unresolved reference to Problem 6.59. The parentheses around "periodic" are also retained from the source: the supplied record does not determine whether periodicity is required or whether periodic and unrestricted versions are both intended. Thus this is a conditional formulation pending clarification of the source hypothesis, rather than a complete standalone statement.

Short English statement

V. P. Shunkov asks whether a group satisfying the source's "(periodic) $p$-conjugacy biprimitively finite" hypothesis and having one finite Sylow $p$-subgroup must have all its Sylow $p$-subgroups conjugate. The technical hypothesis and the role of periodicity require clarification from the referenced Problem 6.59.

Sources

• UnsolvedMath local registry, permanent ID $700616$, exact source problem code catalogue entry, titled "Kourovka Notebook Problem 9.83"; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Attributed in the supplied record to V. P. Shunkov, proposed in 1984. The stated source is The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), Problem 9.83, as included in the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The statement refers to Kourovka Notebook Problem 6.59 for its technical hypothesis; that problem's text was not supplied.
• The registry labels this problem "open". This is supplied source metadata, not an independently verified mathematical or literature assessment. The cited source has not been independently consulted for this draft.