Character degrees and local subgroups revisited

TL;DR

This paper characterizes the relationship between certain irreducible characters and local subgroups in finite groups, removing previous solvability restrictions and providing new insights into block theory and subgroup normalizations.

Contribution

It generalizes existing theorems by removing the p-solvability condition, establishing new criteria for character and subgroup relationships in finite groups.

Findings

Characterization of $ ext{Irr}_{p'}(G)$ and $ ext{Irr}_{q'}(G)$ inclusion

Conditions for Sylow subgroup normalization by defect groups

Removal of p-solvability restriction in key theorems

Abstract

Let $p$ and $q$ be different primes and let $G$ be a finite $q$-solvable group. We prove that $\mathrm{Irr}_{p'}(G)\subseteq \mathrm{Irr}_{q'}(G)$ if and only if $\mathbf{N}_G(P)\subseteq \mathbf{N}_G(Q)$ and $\mathbf{C}_{Q'}(P)=1$ for some $P\in\mathrm{Syl}_p(G)$ and $Q\in\mathrm{Syl}_q(G)$. Further, if $B$ is a $q$-block of $G$ and $p$ does not divide the degree of any character in $\mathrm{Irr}(B)$ then we prove that a Sylow $p$-subgroup of $G$ is normalized by a defect group of $B$. This removes the $p$-solvability condition of two theorems of G. Navarro and T. R. Wolf.