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 $\ell(B) < p^r$, where $r$ is the sectional rank of $D$. We remark that there are infinitely many $p$-blocks $B$ with non-Abelian defect groups and $\ell(B) = p^r - 1$. We conjecture that the inequality $\ell(B) \leq p^r$ 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.