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Character degrees and local subgroups revisited

J. Miquel Martínez

TL;DR

This work links the global behavior of $p'$-degree and $q'$-degree irreducible characters in finite $q$-solvable groups to local subgroup structure, proving that $\mathrm{Irr}_{p'}(G)\subseteq\mathrm{Irr}_{q'}(G)$ is equivalent to a normalizer containment ${\bf N}_G(P)\subseteq{\bf N}_G(Q)$ with ${\bf C}_{Q'}(P)=1$ for some Sylow subgroups. The approach combines a refined McKay bijection for $q$-solvable groups, Glauberman correspondence, and Fong–Swan reductions, situating the result within the McKay conjecture framework. Theorem B extends block theory by showing that if $p$ does not divide any degrees in a $q$-block $B$, then the defect group normalizes a Sylow $p$-subgroup, using Nav-Wol01–type reductions and the Liebeck–Navarro–Praeger–Tiep lemma to pass to suitable quotients and then lift back to $G$. Together these results remove the $p$-solvability restriction from two Navarro–Wolf theorems, enhancing the understanding of how local subgroup data controls global character and block-theoretic properties in $q$-solvable groups.

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.

Character degrees and local subgroups revisited

TL;DR

This work links the global behavior of -degree and -degree irreducible characters in finite -solvable groups to local subgroup structure, proving that is equivalent to a normalizer containment with for some Sylow subgroups. The approach combines a refined McKay bijection for -solvable groups, Glauberman correspondence, and Fong–Swan reductions, situating the result within the McKay conjecture framework. Theorem B extends block theory by showing that if does not divide any degrees in a -block , then the defect group normalizes a Sylow -subgroup, using Nav-Wol01–type reductions and the Liebeck–Navarro–Praeger–Tiep lemma to pass to suitable quotients and then lift back to . Together these results remove the -solvability restriction from two Navarro–Wolf theorems, enhancing the understanding of how local subgroup data controls global character and block-theoretic properties in -solvable groups.

Abstract

Let and be different primes and let be a finite -solvable group. We prove that if and only if and for some and . Further, if is a -block of and does not divide the degree of any character in then we prove that a Sylow -subgroup of is normalized by a defect group of . This removes the -solvability condition of two theorems of G. Navarro and T. R. Wolf.

Paper Structure

This paper contains 3 sections, 4 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Theorem A
  3. Theorem B

Key Result

Theorem 1

Let $G$ be a $q$-solvable group and $Q\in\operatorname{Syl}_q(G)$. Then there is a bijection where $f(\chi)(1)$ divides $\chi(1)$ and $\chi(1)/f(\chi)(1)$ divides $|G:{\bf N}_{G}(Q)|$ for all $\chi\in\operatorname{Irr}_{q'}(G)$.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Theorem 3: Liebeck--Navarro--Praeger--Tiep
  • Theorem 4
  • proof