Kourovka Notebook Problem 18.114

Definitions and mathematical statement

Let ${\mathbb{F}}_5$ be the field with five elements. For a nonzero finite-dimensional ${\mathbb{F}}_5$-vector space $V$, let $GL(V)$ denote its group of invertible linear maps. A subgroup $G\leq GL(V)$ is a $5'$-subgroup if $5\nmid |G|$, and it acts irreducibly if its only invariant linear subspaces are $0$ and $V$. For a finite group $H$, let $k(H)$ denote the number of its conjugacy classes. Write $V\rtimes G$ for the semidirect product of the additive group of $V$ by the given action of $G$, with multiplication \[ (v,g)(w,h)=(v+g(w),gh). \] Does there exist an integer $n\geq 1$, an $n$-dimensional ${\mathbb{F}}_5$-vector space $V$, and a subgroup $G\leq GL(V)$ such that \[ 5\nmid |G|,\qquad G\text{ acts irreducibly on }V,\qquad k(V\rtimes G)=|V|=5^n, \] while $G$ is not cyclic?

The supplied source further reports that the analogous existence question, with ${\mathbb{F}}_p$ and a $p'$-subgroup in place of ${\mathbb{F}}_5$ and a $5'$-subgroup, has an affirmative answer for $p=2,3$ and a negative answer for primes $p>5$. These are supplied background assertions attributed to the journal reference below; they have not been independently verified here. The local registry labels the question for $p=5$ as open.

Short English statement

Is there a noncyclic group of order coprime to $5$ acting faithfully and irreducibly on a nonzero finite vector space over ${\mathbb{F}}_5$, such that the resulting semidirect product has exactly as many conjugacy classes as the vector space has elements?

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701542$, exact source code catalogue entry, title Kourovka Notebook Problem 18.114. The supplied status is open; this is a registry label, not an independently verified literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 18.114, proposed by P. Schmid in 2014. The supplied background identifies the 18th issue (2014) and the 21st issue edition (2026), with identifier arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• J. Group Theory, volume $14$ (2011), pp. 175--199: the supplied reference for the comparison with primes $p=2,3$ and $p>5$. No authors, article title, or URL are supplied for this reference.