Conjugacy class sizes in arithmetic progression

Abstract

Let ${\rm cs}(G)$ denote the set of conjugacy class sizes of a group $G$, and let ${\rm cs}^*(G)={\rm cs}(G)\setminus\{1\}$ be the sizes of non-central classes. We prove three results. We classify all finite groups $G$ with ${\rm cs}(G)=\{a, a+d, \dots ,a+rd\}$ an arithmetic progression with $r\geqslant 2$. (We show that ${\rm cs}(G)=\{1,2,3\}$.) Our most substantial result classifies all $G$ with ${\rm cs}^*(G)=\{2,4,6\}$. Finally, we classify all groups $G$ whose largest two non-central conjugacy class sizes are coprime. (Here it is not obvious but it is true that ${\rm cs}^*(G)$ has two elements, and so is an arithmetic progression.)