Conjugacy class sizes in arithmetic progression

Abstract

Let $\mathrm{cs}(G)$ denote the set of conjugacy class sizes of a group $G$, and let ${\mathrm{cs}}^{\ast }(G)=\mathrm{cs}(G)\setminus \{1\}$ be the sizes of non-central classes. We prove three results. We classify all finite groups $G$ with $\mathrm{cs}(G)=\{a,a+d,\dots ,a+rd\}$ an arithmetic progression with $r⩾2$. (We show that $\mathrm{cs}(G)=\{1,2,3\}$.) Our most substantial result classifies all $G$ with ${\mathrm{cs}}^{\ast }(G)=\{2,4,6\}$. 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 ${\mathrm{cs}}^{\ast }(G)$ has two elements, and so is an arithmetic progression.)

13 pages; v4 correct typo: C_B(A) changed to C_A(B) on p3. Also last paragraph of Introduction modified slightly