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

TL;DR

The paper establishes a direct link between the structure of Sylow $p$-subgroups and the distribution of $p$-divisible irreducible character degrees. It proves the general bound $\mathrm{dl}(P)\le n_p(G)+1$ for finite groups by first handling $p$-solvable groups and then reducing to finite nonabelian simple groups, where a stronger bound $\mathrm{dl}(P)\le n_p^*(G)$ is obtained, with $n_p^*(G)$ the number of $\mathrm{Aut}(G)$-orbits on $p$-singular characters. A comprehensive case analysis is carried out for sporadic, alternating, exceptional, and classical groups (in both defining and non-defining characteristics) using tools such as Jordan decomposition, Lusztig series, and block theory. The block version shows that for a $p$-block $B$ with defect group $D$ in a $p$-solvable group, $\mathrm{dl}(D)\le n_p(B)+1$, connecting block heights to defect-group structure. These results advance conjectures linking character degrees to subgroup structure and provide broad, type-specific bounds across the landscape of finite groups.

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 $B$.

Theorems & Definitions (44)

• Theorem 1
• Theorem 2
• Conjecture 3
• Theorem 4
• Example
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3