Character correspondences, degrees and derived length of certain solvable linear groups

Suppose that G is a finite solvable group, V is a finite faithful completely reducible G -module over a field of characteristic p . In this paper, we first give explicit bounds for the degrees of the irreducible characters of G in terms of | V | in the two cases where 3 |G| 3 ∤ | G | or where the semidirect product GV has abelian Sylow 2-subgroups, respectively. We then use these results to study a conjecture of Navarro.