Kourovka Notebook Problem 21.88

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finite group $G$, conjugacy is the relation $x\sim y$ if $y=g^{-1}xg$ for some $g\in G$. Let $k(G)$ be the number of its conjugacy classes, including the identity class. Odd order means that $|G|$ is odd; nonabelian means that some pair of elements fails to commute.

Statement recorded in the supplied source.

Does there exist a finite nonabelian group $G$ of odd order satisfying \[ \frac{k(G)}{|G|}=\frac1{17}? \] The ratio is the exact rational number $1/17$, rather than an asymptotic density or approximation. [S2]

Short English statement

Is there a finite nonabelian odd-order group whose conjugacy-class count is exactly one seventeenth of its order?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2597 (KOU-21.88). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 D. MacHale, Problem 21.88, in E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 21st edition, arXiv:1401.0300v48 (6 October 2026). This version records a negative answer; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.