Kourovka Notebook Problem 18.20

Definitions and mathematical statement

Let $G$ be a finite group. Under the standard interpretation of characters as ordinary complex characters, a character is the trace function of a finite-dimensional complex representation of $G$. Write $\operatorname{Irr}_{{\mathbb{C}}}(G)$ for the irreducible such characters. The degree of $\chi\in\operatorname{Irr}_{{\mathbb{C}}}(G)$ is $\chi(1_G)$.

A normal subset $M\subseteq G$ means a conjugation-invariant subset: $gMg^{-1}=M$ for every $g\in G$. It need not be a subgroup. For the formulation below, two functions on the same domain are called proportional if one is a nonzero complex scalar multiple of the other.

Characters $\varphi,\psi$ are semiproportional if they are not proportional on $G$, but there exist a normal subset $M\subseteq G$ and scalars $a,b\in{\mathbb{C}}\setminus\{0\}$ such that \[ \varphi(g)= \begin{cases} a\psi(g),&g\in M,\\ b\psi(g),&g\in G\setminus M. \end{cases} \] No assumption that $M$ is a subgroup, or that either part contains the identity, is imposed.

Conjecture (V. A. Belonogov, Kourovka Notebook Problem 18.20). For every finite group $G$ and every pair of semiproportional characters $\varphi,\psi\in\operatorname{Irr}_{{\mathbb{C}}}(G)$, \[ \varphi(1_G)=\psi(1_G). \]

Unresolved conventions. The supplied record does not explicitly specify the character field or whether proportionality permits a zero scalar. The displayed formulation uses ordinary complex characters and nonzero scalar proportionality; these conventions require confirmation against the cited source. The registry labels the conjecture open, but that status has not been independently verified.

Short English statement

Belonogov's conjecture says that two irreducible characters of a finite group have equal degrees whenever they are not globally proportional but become proportional separately on a conjugation-invariant subset and its complement. The complex-character and proportionality conventions stated above remain subject to source confirmation.

Sources

• Local UnsolvedMath registry, permanent ID $701449$, exact source problem code catalogue entry, title Kourovka Notebook Problem 18.20. The supplied record attributes the problem to V. A. Belonogov, gives the proposed year as 2014, and labels its status open. Its supplied creation and update timestamps are both 2026-10-16T14:00:00Z.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), Problem 18.20; the record also cites 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 references and attribution are supplied evidence, not independently checked sources.