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 p ep(G) divides χ(1) for some χ ∈ Irr(G). We show that |G : F(G)|p ≤ p kep(G) for a constant k. This settles a conjecture of A. Moretó [17, Conjecture 4].We also study the related problems of the p-parts of conjugacy class sizes of finite groups.