P-Blocks relative to a character of a normal subgroup

Abstract

Let G be a finite group, let N◃G, and let θ∈Irr(N) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr(G|θ) relative to p. We call each member Bθ of this partition a θ-block, and to each θ-block Bθ we naturally associate a conjugacy class of p-subgroups of G/N, which we call the θ-defect groups of Bθ. If N is trivial, then the θ-blocks are the Brauer p-blocks. Using θ-blocks, we can unify the Gluck–Wolf–Navarro–Tiep theorem and Brauer's Height Zero conjecture in a single statement, which, after work of B. Sambale, turns out to be equivalent to the Height Zero conjecture. We also prove that the k(B)-conjecture is true if and only if every θ-block Bθ has size less than or equal the size of any of its θ-defect groups, thus bringing normal subgroups to the k(B)-conjecture.