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The Fields of Values of the Isaacs' Head Characters

Gabriel Navarro

TL;DR

This work determines the fields of values of Isaacs' head characters for finite solvable groups by proving a local-to-global bijection that preserves fields of values along extension correspondences. Central to the approach is Theorem A, which gives a bijection between linear characters of a Carter subgroup $C$ and head characters $H(G)$ with ${\mathbb Q}(\chi)={\mathbb Q}(\chi^*)$, implying ${\rm Q}(\chi)$ equals ${\mathbb Q}_n$ for $n$ the order of the corresponding linear character of $C$. The key technical result (theorem in Section 2) establishes a field-preserving bijection between extension sets ${\rm Ext}(G|\theta)$ and ${\rm Ext}(H|\varphi)$ under a structured normal series, enabling the global statement. Consequences include a lower bound on rational-valued irreducibles, connections to rational groups, and a Galois-McKay analogue for nilpotent formations, with discussion on extensions to broader group classes and the role of Carter subgroups in controlling character values.

Abstract

We determine the fields of values of the Isaacs' head characters of a finite solvable group.

The Fields of Values of the Isaacs' Head Characters

TL;DR

This work determines the fields of values of Isaacs' head characters for finite solvable groups by proving a local-to-global bijection that preserves fields of values along extension correspondences. Central to the approach is Theorem A, which gives a bijection between linear characters of a Carter subgroup and head characters with , implying equals for the order of the corresponding linear character of . The key technical result (theorem in Section 2) establishes a field-preserving bijection between extension sets and under a structured normal series, enabling the global statement. Consequences include a lower bound on rational-valued irreducibles, connections to rational groups, and a Galois-McKay analogue for nilpotent formations, with discussion on extensions to broader group classes and the role of Carter subgroups in controlling character values.

Abstract

We determine the fields of values of the Isaacs' head characters of a finite solvable group.

Paper Structure

This paper contains 3 sections, 6 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. The key result
  3. Head Characters

Key Result

Lemma 2.1

Let $G$ be a finite solvable group. Let $K$ be the smallest normal subgroup of $G$ such that $G/K$ is nilpotent. If $K$ is abelian, then $K$ is complemented in $G$, and its complements are the Carter subgroups of $G$.

Theorems & Definitions (12)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 2 more