Derived Length and the Number of Character Degrees

We let cd(G) = {χ(1)|χ ∈ Irr(G)}. I. M. Isaacs proved that if |cd(G)|≤3, then G is solvable and dl(G) ≤ |cd(G)| (see [Is, 12.6 and 12.15]). Since |cd(A5)| = 4, we cannot improve the first conclusion, but it has been conjectured by G. Seitz that dl(G) ≤ |cd(G)| for all solvable groups G. Isaacs gave the first general bound, namely dl(G) ≤ 3· |cd(G)| (or 2·|cd(G)| if |G| is odd). These are proved in Theorem 16.5 below. Lemma 16.4 is important here and further analysis allows us to present Gluck's improvement to dl(G) ≤ 2 · |cd(G)| in Theorem 16.8. Using Theorem 8.4, we give Berger's proof of Seitz's conjecture for groups of odd order. The key result here is Theorem 16.6, which does not hold for arbitrary solvable groups.

The first proposition if quite important to this section. For χ ∈ Irr(G), we let D(χ) = ∩{ker(ψ) | ψ > ∈ Irr(G) and ψ(1) < χ(1)}. Should χ be linear, then D(χ) = G.

Proposition. Let X ∈ Irr (G) and write X = θGfor some H ≤ G and θ ∈ Irr(H). Then D(χ) ≤ |D(θ) ≤ H.

Proof. Note that when χ is linear, then χ = θ and H = G = D(χ). If ψ ∈ Irr (H) and ψ(1) < θ(1), then ψG(1) < θG(1) = χ(1) and every irreducible constituent of ψG has degree less than χ(1). Thus D(χ) ≤ ker(ψG) ≤ ker(ψ) ≤ H. Hence D(χ) ≤ D(θ), except possibly when θ is linear and H < G. But in this case, observe that 1HG(1) = θG(1) = χ(1) and 1HG reduces.