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 $\ell (B)\le p^r$ 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.

we made some minor corrections

References in corpus (1)

• On the Decomposition Numbers of the Ree Groups 2F4(q^2) in Non-Defining Characteristic

Cited by in corpus (5)

• Weight conjectures for fusion systems
• On defects of characters and decomposition numbers
• Bounds on the number of simple modules in blocks of finite groups of Lie type
• Sectional rank and Cohomology II
• Weight conjectures for fusion systems on an extraspecial group