Kourovka Notebook Problem 18.77

Definitions and mathematical statement

Let $p$ be a prime and let $G$ be a finite group of order $p^n$ for some integer $n\geq 0$. An irreducible complex representation of $G$ is a homomorphism $\Theta:G\to\operatorname{GL}(V_\Theta)$, where $V_\Theta$ is a nonzero finite-dimensional complex vector space with no proper nonzero $G$-invariant subspace. Its degree is $\dim_{{\mathbb{C}}}V_\Theta$, and its kernel is $\ker\Theta=\{g\in G:\Theta(g)=\operatorname{id}_{V_\Theta}\}$.

Let $e\geq 0$ be the integer such that $p^e$ is the largest degree of an irreducible complex representation of $G$, and let $\mathcal M(G)$ contain one representative from each equivalence class of irreducible complex representations of degree $p^e$. Is it true, for every such prime $p$ and finite $p$-group $G$, that \[ p>e\quad\Longrightarrow\quad \bigcap_{\Theta\in\mathcal M(G)}\ker\Theta=\{1_G\}? \] The supplied registry labels this question open; that status is not independently verified here.

Short English statement

If the largest irreducible complex representation degree of a finite $p$-group is $p^e$ and $p>e$, must every nonidentity element act nontrivially in at least one irreducible representation of that largest degree?

Sources

• Local UnsolvedMath registry, supplied source record: permanent ID $701506$, exact source code catalogue entry, title Kourovka Notebook Problem 18.77. The supplied status is open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 18.77, proposed by D. S. Passman in 2014. The supplied background identifies the 18th issue (2014) and the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. This bibliographic information is supplied attribution and has not been independently verified.