Kourovka Notebook Problem 16.3

Definitions and mathematical statement

Let $G$ be a finite group. For $x\in G$, define its conjugacy class and centralizer by \[ x^G=\{gxg^{-1}:g\in G\},\qquad C_G(x)=\{g\in G:gx=xg\}. \] Thus $|x^G|=[G:C_G(x)]$. Say that $G$ has pairwise distinct conjugacy class sizes if \[ \forall x,y\in G,\qquad x^G\ne y^G\ \Longrightarrow\ |x^G|\ne |y^G|. \] Let $\mathbb S_3$ denote the symmetric group on three letters, and let $F(G)$ denote the Fitting subgroup of $G$, namely its largest nilpotent normal subgroup.

The question attributed to Z. Arad asks whether every finite group $G$ with pairwise distinct conjugacy class sizes satisfies \[ G\cong\mathbb S_3. \] Unresolved boundary condition. The supplied statement does not exclude the trivial group. Pairwise distinctness of conjugacy class sizes is vacuous when there is only one conjugacy class, so whether a nontriviality hypothesis was intended must be clarified. No such hypothesis has been silently inserted here.

The supplied source additionally reports that the proposed conclusion holds when $G$ is solvable, citing J. Zhang, and that $F(G)\ne\{1\}$ is known under the problem's hypotheses, citing Z. Arad, M. Muzychuk, and A. Oliver. These are supplied literature claims, not independently verified assertions; their intended scope is subject to the same unresolved nontriviality qualification.

Short English statement

The problem asks whether a finite group whose distinct conjugacy classes all have different sizes must be the symmetric group on three letters. The source reports a positive result for solvable groups and a nontriviality result for the Fitting subgroup. The omission of a possible assumption that the group is nontrivial requires clarification.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701195$, exact source code catalogue entry, title Kourovka Notebook Problem 16.3; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels the problem open; this label is supplied metadata, not an independently verified assessment.
• Z. Arad, proposer, 2006. Supplied attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 16.3, 16th issue (2006); the record also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• J. Zhang, J. Algebra 170 (1994), 608--624. Supplied citation for the solvable case; no article title or link is supplied.
• Z. Arad, M. Muzychuk, A. Oliver, J. Algebra 280 (2004), 537--576. Supplied citation for the assertion that $F(G)$ is nontrivial; no article title or link is supplied.