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. © 2020 Walter de Gruyter GmbH, Berlin/Boston 2020 Australian Research Council DP190100450 Engineering and Physical Sciences Research Council EP/R014604/1 SG and CP gratefully acknowledge support from the Australian Research Council Discovery Project DP190100450. MB acknowledges support from G.N.S.A.G.A. (Indam) and thanks the Centre for the Mathematics of Symmetry and Computation (CMSC) for its hospitality. This work began in the CMSC Research Retreat of 2019. CP also thanks the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Groups, Representations and applications: New Perspectives". This program was supported by EPSRC grant number EP/R014604/1.