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

Abstract 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 $$3\not \mid |G|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>∤</mml:mo> <mml:mo>|</mml:mo> <mml:mi>G</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> or where the semidirect product GV has abelian Sylow 2-subgroups, respectively. We then use these results to study a conjecture of Navarro.