Itô's theorem on groups with two class sizes revisited

Let G be a finite p-solvable group. We prove that if G has exactly two conjugacy class sizes of p'-elements of prime power order, say 1 and m, then m = p(a)q(b), for two distinct primes p and q, and G either has an abelian p-complement or G = PQ x A, with P and Q a Sylow p-subgroup and a Sylow q-subgroup of G, respectively, and A is abelian. In particular, we provide a new extension of Ito's theorem on groups having exactly two class sizes for elements of prime power order.This research is supported by the Spanish Government, Proyecto MTM2010-19938-C03-02 and the second and third authors are supported by the Valencian Government, Proyecto PROMETEO/2011/30. The second author is also supported by grant Fundacio Caixa-Castello P11B2010-47.Alemany Martínez, E.; Beltrán, A.; Felipe Román, MJ. (2012). Itô's theorem on groups with two class sizes revisited. Bulletin of the Australian Mathematical Society. 85:476-481. https://doi.org/10.1017/S0004972711002875S47648185Huppert, B. (1998). Character Theory of Finite Groups. doi:10.1515/9783110809237Baer, R. (1953). Group elements of prime power index. Transactions of the American Mathematical Society, 75(1), 20-20. doi:10.1090/s0002-9947-1953-0055340-0Kurzweil, H., & Stellmacher, B. (2004). The Theory of Finite Groups. Universitext. doi:10.1007/b97433Beltrán, A., & Felipe, M. J. (2003). Finite groups with two p-regular conjugacy class lengths. Bulletin of the Australian Mathematical Society, 67(1), 163-169. doi:10.1017/s000497270003361xCamina, A. R. (1974). Finite groups of conjugate rank 2. Nagoya Mathematical Journal, 53, 47-57. doi:10.1017/s0027763000016019Zhao, X., & Guo, X. (2009). On Conjugacy Class Sizes of the p′-Elements with Prime Power Order. Algebra Colloquium, 16(04), 541-548. doi:10.1142/s1005386709000510Shirong, L. (1996). Finite groups with exactly two class lengths of elements of prime power order. Archiv der Mathematik, 67(2), 100-105. doi:10.1007/bf01268923Itô, N. (1953). On Finite Groups with Given Conjugate Types I. Nagoya Mathematical Journal, 6, 17-28. doi:10.1017/s0027763000016937Camina, R. D., & Camina, A. R. (1998). Implications of conjugacy class size. Journal of Group Theory, 1(3). doi:10.1515/jgth.1998.017Beltrán, A., & Felipe, M. J. (2004). Prime powers as conjugacy class lengths of π-elements. Bulletin of the Australian Mathematical Society, 69(2), 317-325. doi:10.1017/s000497270003605