Kourovka Notebook Problem 17.118

Definitions and mathematical statement

For a prime $p$, a finite $p$-group is a finite group whose order is a power of $p$. A subgroup $H\leq G$ has exponent $p$ if $H\neq\{1\}$ and $h^p=1$ for every $h\in H$. Write $[G:H]$ for its index. A subgroup $C\leq G$ is characteristic if $\alpha(C)=C$ for every $\alpha\in\operatorname{Aut}(G)$.

Is there a function \[ f:\{p\in{\mathbb{N}}:p\text{ is prime}\}\longrightarrow{\mathbb{N}}_{>0} \] such that, for every prime $p$ and every finite $p$-group $G$, \[ \left(\exists H\leq G:\ \exp(H)=p\ \text{and}\ [G:H]=p\right) \ \Longrightarrow\ \left(\exists C\leq G:\ C\text{ is characteristic in }G,\ \exp(C)=p,\ [G:C]\leq f(p)\right)? \] The bound must depend only on $p$, uniformly over all such groups $G$.

Short English statement

E. I. Khukhro asks whether every finite $p$-group with an exponent-$p$ subgroup of index $p$ has a characteristic exponent-$p$ subgroup whose index is bounded solely in terms of $p$.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $701419$, source problem code catalogue entry, title Kourovka Notebook Problem 17.118. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied attribution: E. I. Khukhro, proposed in 2010; Problem 17.118 in The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), with the supplied source also identifying the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied registry labels this problem open and records an update timestamp of 2026-10-16T14:00:00Z. This is supplied registry metadata, not an independently verified mathematical or literature assessment.