Bounding the Number of Character Degrees of a Solvable Group

A difficult problem in the character theory of solvable groups is to show that the number c.d. (G) of irreducible character degrees of a solvable group G is equal to or greater than d.l. (G), the derived length of G. Isaacs [4] has shown that d.l. (G) ^ 3 c.d. (G)-2 for every solvable group. Berger [1] subsequently proved that d.l. (G) < c.d. (G) when | G | is odd. This problem belongs to the class of length-type problems for solvable groups; see [2]. It has connections with M-groups and other major topics in the character theory of solvable groups. While there is no obvious way to attack it, neither is there any reason to suppose that its solution is beyond the range of present day techniques. Here we show that d.l. (G) ^ 2 c.d. (G) for every solvable group G. Following [4], we let 1 —fx < / 2 < . . .< /„ be the distinct character degrees of G. For 1 ̂ r ^ n, let a(r) denote