Kourovka Notebook Problem 6.55

Definitions and mathematical statement

Let $p$ be a prime. A $p$-group means a group in which every element has finite order equal to a power of $p$; the group itself is not assumed finite. For a subgroup $H\leq G$, write $H^g=g^{-1}Hg$.

An internal semidirect product $G=F\rtimes H$ means that $F\trianglelefteq G$, $H\leq G$, $G=FH$, and $F\cap H=\{1\}$. Following the supplied definition, such a group is called a Frobenius group with kernel $F$ and complement $H$ if \[ H\cap H^g=\{1\}\qquad\text{for every }g\in G\setminus H \] and \[ F\setminus\{1\}=G\setminus\bigcup_{g\in G}H^g. \]

The existence question is whether there are a prime $p$, a $p$-group $G$, and subgroups $F,H\leq G$ satisfying these conditions.

Unresolved qualification. The supplied definition does not explicitly require $F\neq\{1\}$ or $H\neq\{1\}$. These nontriviality requirements affect the intended existence question and must be confirmed. If both are intended, the question is the same existence statement with the additional conditions $F\neq\{1\}$ and $H\neq\{1\}$. They have not been silently added to the supplied definition. The source also gives no explicit finiteness restriction; the formulation above permits infinite $p$-groups.

Short English statement

V. P. Shunkov asks whether Frobenius $p$-groups exist. Such a group must split into a kernel and complement satisfying the displayed conjugate-intersection and complement-union conditions. The intended nontriviality requirements for the kernel and complement need clarification.

Sources

• Local UnsolvedMath registry, permanent ID $700397$, exact source problem code catalogue entry, title Kourovka Notebook Problem 6.55. Supplied status: open; this is a registry label, not an independently verified mathematical assessment. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 6.55, sixth issue (1978); proposed by V. P. Shunkov. The supplied background identifies the twenty-first issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. This attribution and edition information are supplied source data and have not been independently checked.