As an application of this result, a logarithmic bound for the derived length of the quotient group G/F(G) of a solvable group G by its Fitting subgroup F(G) is obtained in terms of the number of irreducible characters of G.
Also, a logarithmic bound for the derived length of a solvable group A acting on a solvable group G with coprime order is found in terms of the number of orbits of A acting on the set of irreducible characters of G, as well as the number of orbits of A acting on the set of irreducible Brauer characters of G.
An attempt is made to bound the derived length of the linear group G of a solvable permutation group in terms of the number of orbit sizes of G; partial results are obtained. The derived length of the linear group G of a solvable permutation group S with one nontrivial G-orbit size (i.e., S is a 3/2-transitive permutation group) is less than or equal to 6.
Access since 3/11/96: