Conjugacy class sizes in arithmetic progression

Let cs(G) denote the set of conjugacy class sizes of a group G, and let cs ∗ (G) = cs(G)\{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} is an arithmetic progression with r ≥ 2 (2) cs ∗ (G) = { 2, 4, 6 } (G)= {2,4,6} is the smallest case where cs ∗ (G) is an arithmetic progression of length more than 2 (our most substantial result); (3) the largest two members of cs ∗ (G) are coprime. For (3), it is not obvious, but it is true that cs ∗(G) has two elements, and so is an arithmetic progression.