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Hall $π$-subgroups and characters of $π'$-degree

Eugenio Giannelli, Nguyen N. Hung

TL;DR

The paper addresses the link between Hall $\pi$-subgroups and irreducible characters of $\pi'$-degree with restricted fields of values in finite groups, extending Navarro–Tiep's prime-by-prime result to arbitrary sets of odd primes. The authors reduce the problem to the classification of finite simple groups and perform a detailed analysis for alternating, sporadic, and Lie-type simple groups, using semisimple characters from Lusztig series to control fields of values and degrees. The main contribution is a general theorem: if a finite group $G$ has a nontrivial Hall $\pi$-subgroup for a set $\pi$ of odd primes, then $G$ possesses a nontrivial $\pi'$-degree irreducible character with values in $\mathbb{Q}(\zeta_p)$ for some $p\in\pi$. This result enhances the understanding of how subgroup structure influences character theory and provides a robust reduction framework for $E_\pi$-groups in the context of odd primes.

Abstract

We study the relationship between the existence of Hall $π$-subgroups and that of irreducible characters of $π'$-degree with prescribed fields of values in finite groups. This work extends a result of Navarro and Tiep from a single odd prime to multiple odd primes.

Hall $π$-subgroups and characters of $π'$-degree

TL;DR

The paper addresses the link between Hall -subgroups and irreducible characters of -degree with restricted fields of values in finite groups, extending Navarro–Tiep's prime-by-prime result to arbitrary sets of odd primes. The authors reduce the problem to the classification of finite simple groups and perform a detailed analysis for alternating, sporadic, and Lie-type simple groups, using semisimple characters from Lusztig series to control fields of values and degrees. The main contribution is a general theorem: if a finite group has a nontrivial Hall -subgroup for a set of odd primes, then possesses a nontrivial -degree irreducible character with values in for some . This result enhances the understanding of how subgroup structure influences character theory and provides a robust reduction framework for -groups in the context of odd primes.

Abstract

We study the relationship between the existence of Hall -subgroups and that of irreducible characters of -degree with prescribed fields of values in finite groups. This work extends a result of Navarro and Tiep from a single odd prime to multiple odd primes.
Paper Structure (3 sections, 13 theorems, 17 equations)

This paper contains 3 sections, 13 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Non-Abelian Simple Groups
  3. A Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $G$ be a finite group of order divisible by a prime $p$. Then $G$ has a nontrivial irreducible character of degree not divisible by $p$ whose values lie in ${\mathbb Q}(e^{2\pi i/p})$.

Theorems & Definitions (25)

  • Theorem 1.1: Navarro-Tiep1, Theorem A
  • Theorem 1
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5
  • proof
  • ...and 15 more