On the number of simple modules in a block of a finite group

Abstract

We prove that if B is a p-block with non-trivial defect group D of a finite p-solvable group G, then ℓ(B)<pr, where r is the sectional rank of D. We remark that there are infinitely many p-blocks B with non-Abelian defect groups and ℓ(B)=pr−1.We conjecture that the inequality ℓ(B)≤pr holds for an arbitrary p-block with defect group of sectional rank r. We show this to hold for a large class of p-blocks of various families of quasi-simple and nearly simple groups.