Abstract :
We continue the investigation, that began in [M. Bianchi, A. Gillio and P. P. Palfy, A note on nite groups in which the conjugacy class sizes form an arithmetic progression, Ischia group theory 2010, World Sci. Publ., Hackensack, NJ (2012) 20{25.] and [M. Bianchi, S. P. Glasby and Cheryl E. Praeger, Conjugacy class sizes in arithmetic progression, J. Group Theory, 23 no. 6 (2020) 1039{1056.], into finite groups whose set of nontrivial conjugacy class sizes form an arithmetic progression. Let G be a finite group and denote the set of conjugacy class sizes of G by cs(G). Finite groups satisfying cs(G) = f1; 2; 4; 6g and f1; 2; 4; 6; 8g are classied in [M. Bianchi, S. P. Glasby and Cheryl E. Praeger, Conjugacy class sizes in arithmetic progression, J. Group Theory, 23 no. 6 (2020) 1039{1056.] and [M. Bianchi, A. Gillio and P. P. P alfy, A note on finite groups in which the conjugacy class sizes form an arithmetic progression, Ischia group theory 2010, World Sci. Publ., Hackensack, NJ (2012) 20{25.], respectively, we demonstrate these examples are rather special by proving the following. There exists a finite group G such that cs(G) = f1; 2; 2+1; 23g if and only if = 1. Furthermore, there exists a finite group G such that cs(G) = f1; 2; 2+1; 23; 2+2g and is odd if and only if = 1.
