Sylow subgroups and the number of irreducible characters of degrees divisible by a prime $p$

TL;DR

This paper links the structure of Sylow p-subgroups and defect groups in finite groups to the number of irreducible characters with degrees divisible by p, providing bounds on their derived lengths.

Contribution

It establishes new bounds on the derived length of Sylow p-subgroups and defect groups based on irreducible character degrees and heights in finite groups.

Findings

Upper bound for derived length of Sylow p-subgroups in terms of irreducible characters.

Bound on derived length of defect groups related to characters of positive height.

Connections between group structure and character theory are clarified.

Abstract

Let $G$ be a finite group and $p$ a prime. We establish an upper bound for the derived length of a Sylow $p$-subgroup of $G$ in terms of the number of irreducible characters of $G$ whose degrees are divisible by $p$. We also prove that if $B$ is a $p$-block of a finite $p$-solvable group $G$ with defect group $D$, then the derived length of $D$ is at most one more than the number of ordinary irreducible characters of positive height in $B$.