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

TL;DR

This paper establishes an upper bound on the number of simple modules in a block of a finite p-solvable group based on the sectional rank of its defect group, and explores related conjectures.

Contribution

It proves an inequality relating simple modules and sectional rank for p-solvable groups and proposes a conjecture extending this to all p-blocks.

Findings

The inequality ll(B) < p^r holds for p-solvable groups.

Counterexamples with ll(B) = p^r - 1 exist.

The inequality ll(B)  p^r is conjectured to hold generally.

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.