LAUSR

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 ) p r , 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 ) = p r − 1 . We conjecture that the inequality l ( B ) ≤ p r 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.