Structure of p-complements in groups with three p-regular class sizes

Let G be a finite p-solvable group for some prime p and suppose that the set of p-regular conjugacy class sizes is {1,m,mn} with (m,n)=1 and m coprime to p. We show that m=qb for some prime q and we describe the structure of the p-complements of G.This research is supported by the Spanish Government, Proyecto MTM2010-19938-C03-02 and by the Valencian Government, Proyecto PROMETEO/2011/30. The first author is also supported by grant Fundacio Caixa-Castello P11B2010-47.Beltrán, A.; Felipe Román, MJ. (2013). Structure of p-complements in groups with three p-regular class sizes. Monatshefte für Mathematik. 171(2):157-167. https://doi.org/10.1007/s00605-012-0466-x1571671712Akhlaghi, Z., Beltrán, A., Felipe, M.J., Khatami, M.: Finite $$p$$ -solvable groups with three $$p$$ -regular class sizes. Proc. Edinb. Math. Soc. (2012). doi: 10.1017/s0013091511000654Alemany, E., Beltrán, A., Felipe, M.J.: Finite groups with two $$p$$ -regular conjugacy class lengths II. Bull. Aust. Math. Soc. 79, 419–425 (2009)Alemany, E., Beltrán, A.: Felipe, M.J: Itô’s theorem on groups with two class sizes revisited. Bull. Aust. Math. Soc. 85, 476–481 (2012)Beltrán, A., Felipe, M.J.: The structure of finite groups with three class sizes. J. Group Theory 12, 539–553 (2009)Beltrán, A., Felipe, M.J.: Certain relations between $$p$$ -regular class sizes and the $$p$$ -structure of $$p$$ -solvable groups. J. Aust. Math. Soc. 77, 387–400 (2004)Beltrán, A., Felipe, M.J.: Prime powers as conjugacy class lengths of $$\pi $$ -elements. Bull. Austral. Math. Soc. 69, 317–325 (2004)Camina, A.R., Camina, R.D.: Implications of conjugacy class size. J. Group Theory 1, 257–269 (1998)Dolfi, S., Jabara, E.: The structure of finite groups of conjugate rank 2. Bull. Lond. Math. Soc. 41, 916926 (2009)Huppert, B.: Character theory of finite groups. De Gruyter Expositions in Mathematics, vol. 25. Walter de Gruyter & Co, Berlin (1998)Itô, N.: On finite groups with given conjugate types. II. Osaka J. Math. 7, 231–251 (1970)Kurzweil, K., Stellmacher, B.: The theory of finite groups. Springer, New York (2004)Schmidt, R.: Subgroup lattices of groups. Walter de Gruyter, Berlin (1994