Character degrees in blocks and defect groups

Abstract

A recent question of Gabriel Navarro asks whether it is true that the derived length of a defect group is less than or equal to the number of degrees of irreducible characters in a block. In this article, we bring new evidence towards the validity of this statement. Reduction to simple groups2.1. Auxiliary lemmas.Lemma 2.1. Let G be a finite group, N Ÿ G, and θ P IrrpB 0 pNqq. Assume θ extends to χ P IrrpB 0 pGqq. Then tµχ | µ P IrrpB 0 pG{Nqqu Ď IrrpG|θq X IrrpB 0 pGqq.Proof. By Gallagher's correspondence, IrrpG|θq " tµχ | µ P IrrpG{Nqu. We have χ " 1 G{N χ and then by [Riz18, Lemma 2.4], if µ P IrrpB 0 pG{Nqq then µχ P IrrpB 0 pGqq. Lemma 2.2. Let G be a finite group and let N " S 1 ˆ¨¨¨ˆS t Ÿ G, where the S i 's are subgroups permuted by G. Let S " S 1 . Assume there exists α P IrrpN G pSq{C G pSqq in the principal block and such that S is not contained in kerpαq. Then α G P IrrpB 0 pGqq.