On p-parts of character degrees and conjugacy class sizes of finite groups

Let G be a finite group and Irr(G) the set of irreducible complex characters of G. Let ep(G) be the largest integer such that pep(G) divides χ(1) for some χ∈Irr(G). We show that |G:F(G)|p≤pkep(G) for a constant k. This settles a conjecture of A. Moretó [13, Conjecture 4]. We also study the related problems of the p-parts of conjugacy class sizes of finite groups.