Character kernels and degree ratios in finite groups

Let K=ker(χ) be the kernel of an irreducible character χ of a finite group G, and let S be the largest solvable normal subgroup of K. We show that if the degree of χ is large compared to the degrees of other irreducible characters of G, or if the kernel K is small compared to the kernels of other irreducible characters of G, then the Fitting height of S is small. Also, we show that the derived length of a nonabelian solvable group is bounded by a logarithmic function of the ratio b/c, where b is the largest irreducible character degree of G and c is the smallest nonlinear irreducible character degree. Finally, using the classification of simple groups, we show that certain kernels of irreducible characters of large degree must be solvable.