Kourovka Notebook Problem 20.51

Definitions and mathematical statement

For a finite group $G$ and $g\in G$, define the conjugacy class of $g$ and the number of conjugacy classes of $G$ by \[ [g]_G=\{xgx^{-1}:x\in G\},\qquad k(G)=\left|\{[g]_G:g\in G\}\right|. \] The group $G$ is non-abelian if there exist $a,b\in G$ such that $ab\ne ba$, and it has odd order if $|G|$ is an odd integer.

The question proposed by D. MacHale is whether, for every odd integer $t>3$, there exists a finite non-abelian group $G$ of odd order such that \[ k(G)=t. \]

Short English statement

Does every odd integer greater than $3$ occur as the number of conjugacy classes of some finite non-abelian group of odd order?

Sources

• Local UnsolvedMath registry: permanent ID $701709$, exact source code catalogue entry, title Kourovka Notebook Problem 20.51. The supplied record labels the problem open; this status has not been independently verified.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 20.51, proposed by D. MacHale in 2022. The supplied background identifies the 20th issue (2022) and the 21st issue edition (2026), with reference arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution, not independently verified source information.