Kourovka Notebook Problem 8.85

Definitions and mathematical statement

Let $p$ be a prime. A finite $p$-group is a finite group $G$ with $|G|=p^n$ for some integer $n\geq 0$. For $x\in G$, let $|x|$ denote its order. Define the Hughes subgroup by \[ H_p(G)=\langle x\in G : |x|\neq p\rangle, \] where $\langle S\rangle$ is the subgroup generated by a subset $S\subseteq G$. The requested construction is a finite $p$-group $G$ such that \[ H_p(G)\neq\{1_G\} \qquad\text{and}\qquad [G:H_p(G)]=p^3, \] where $[G:H_p(G)]$ is the number of left cosets of $H_p(G)$ in $G$. The source does not specify whether such a construction is requested for at least one prime $p$ or for every prescribed prime $p$; the outer quantifier on $p$ remains unresolved.

Short English statement

Construct a finite group of prime power order whose Hughes subgroup, generated by all elements whose order differs from $p$, is nontrivial and has index exactly $p^3$. The intended range of primes is unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $700533$, exact source code catalogue entry, titled Kourovka Notebook Problem 8.85; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record was created and updated on 2026-10-16.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 8.85, 8th issue (1982); proposed by E. I. Khukhro. The supplied background identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied registry labels the problem open. This is supplied source metadata, not an independently verified assessment of its mathematical or literature status. The cited source has not been independently consulted for this draft.