Conjugacy class sizes in arithmetic progression

Abstract Let cs⁢(G){\mathrm{cs}(G)} denote the set of conjugacy class sizes of a group G, and let cs*⁢(G)=cs⁢(G)∖{1}\mathrm{cs}^{*}(G)=\mathrm{cs}(G)\setminus\{1\} be the sizes of non-central classes. We prove three results. We classify all finite groups for which (1) cs⁢(G)={a,a+d,…,a+r⁢d}{\mathrm{cs}(G)=\{a,a+d,\dots,a+rd\}} is an arithmetic progression with r⩾2{r\geqslant 2}; (2) cs*⁢(G)={2,4,6}{\mathrm{cs}^{*}(G)=\{2,4,6\}} is the smallest case where cs*⁢(G){\mathrm{cs}^{*}(G)} is an arithmetic progression of length more than 2 (our most substantial result); (3) the largest two members of cs*⁢(G){\mathrm{cs}^{*}(G)} are coprime. For (3), it is not obvious, but it is true that cs*⁢(G){\mathrm{cs}^{*}(G)} has two elements, and so is an arithmetic progression.