Characters of p′-degree and Brauer's Height-Zero Conjecture

Suppose N ⊴ G, θ ∈ Irr(G), and χ(1)/θ(1) is a p′-number for all irreducible constituents χ of θG. The bulk of work in this section will be aimed at proving that G/N has an abelian Sylow p-subgroup, provided G/N is solvable. With little extra work, we see that p can be replaced by a set of primes. As a consequence of this and Fong reduction (Lemma 0.25 and Theorem 0.28), we then prove Brauer's height-zero conjecture for solvable G. Namely, if B is a p-block of a solvable group, then all the ordinary characters in B have height zero if and only if the defect group for B is abelian. The contents of this section are [Wo 3, GW 1], and while the arguments are essentially the same, some improvements and refinements should improve the reading thereof. Brauer's height-zero conjecture was extended to p-solvable G in [GW 2], with the help of the classification of simple groups.

In the key Theorem 12.9 of this section, we have N ⊴ G, θ ∈ Irr(N) and χ(1)/θ(1) a p′-number for all χ ∈ Irr(G|θ). The aim is to show that G/N has abelian Sylow p-subgroup, at least when G/N is solvable. In a minimal counterexample, there exists an abelian chief factor M/N of G such that each λ ∈ Irr(M/N) is invariant under some Sylow p-subgroup of G/M. Consequently, the results of Sections 9 and 10 play an important role.