Orbit sizes and character degrees, III

We continue our investigation of the orbit structure of finite group actions. This time we show that for any p-group G acting faithfully and irreducibly on a finite vector space V the derived length of G can be bounded logarithmically in the number of distinct orbit sizes of G on V. This improves a linear bound due to Isaacs and can be used-in combination with our former results-to prove that the derived length of any finite solvable group acting faithfully and irreducibly on a finite vector space is logarithmically bounded in terms of the number of distinct orbit sizes of G on V. As a consequence, we get a new contribution to the old question of how the derived length dl(G) of a finite solvable group is bounded in terms of the number \cd G\ of its irreducible complex character degrees, namely dl(G/F(G)) less than or equal to 24log(2)\cd G\ + 364. A consequence on modular character degrees is also discussed.