Derived length and conjugacy class sizes

Let G be a finite solvable group, and let F(G) be its Fitting subgroup. We prove that there is a universal bound for the derived length of G/F(G) in terms of the number of distinct conjugacy class sizes of G. This result is asymptotically best possible. It is based on the following result on orbit sizes in finite linear group actions: If G is a finite solvable group and V a finite faithful irreducible G-module of characteristic r, then there is a universal logarithmic bound for the derived length of G in terms of the number of distinct r′-parts of the orbit sizes of G on V. This is a refinement of the author's previous work on orbit sizes.