Kourovka Notebook Problem 19.108

Definitions and mathematical statement

Let $p$ be an odd prime and let $P$ be a finite $p$-group, meaning that $|P|=p^n$ for some integer $n\geq 0$. Let $\chi$ be an irreducible complex character of $P$: thus $\chi(g)=\operatorname{tr}(\rho(g))$ for an irreducible representation $\rho:P\to\operatorname{GL}(V)$ on a nonzero finite-dimensional complex vector space $V$. Its degree is $\chi(1)=\dim_{{\mathbb{C}}}V$. For $x\in P$, let $\operatorname{ord}(x)$ denote the least positive integer $r$ such that $x^r=1$.

Is it true that, for every such $p,P,\chi$ and every $x\in P$, \[ \chi(x)\neq 0\quad\Longrightarrow\quad \operatorname{ord}(x)\mid\frac{|P|}{\chi(1)^2}? \] Equivalently, the proposed conclusion is that there exists an integer $k\geq 1$ satisfying \[ |P|=k\,\operatorname{ord}(x)\,\chi(1)^2 \] whenever $\chi(x)\neq 0$.

Short English statement

For every finite group of odd prime-power order and every irreducible complex character, must the order of each element on which the character is nonzero divide the group order divided by the square of the character degree? The supplied registry labels this problem open; that status is not independently verified here.

Sources

• Local UnsolvedMath registry, permanent ID $701656$, exact source problem code catalogue entry, titled Kourovka Notebook Problem 19.108. The supplied record attributes the problem to T. Wilde, proposed in 2018, and labels it open.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), Problem 19.108; also identified in the record with the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors of Issues 1--21 (1965--2026). These bibliographic details are supplied attribution and have not been independently checked.