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On a theorem of Artin and the dimension of the space spanned by the rational valued characters of a group

Mark L. Lewis

TL;DR

The paper sharpens Artin's theorem by showing that the subspace of class functions spanned by rational-valued characters has dimension equal to the number of conjugacy classes of cyclic subgroups, and by constructing a concrete basis from induced class functions tied to cyclic subgroups. It provides a detailed, two-pronged argument: a direct representation-theoretic approach that identifies a basis $\{|N_G(H_i)|\,\Phi_i\}$ corresponding to cyclic-subgroup conjugacy classes, and a Galois-theoretic viewpoint that interprets rational characters as sums over orbits of irreducible characters under $\mathrm{Gal}(F/\mathbb{Q})$, with a precise correspondence between orbits and cyclic-subgroup conjugacy classes. The results yield a refined understanding of rational-valued characters, connect classical Artin-type results to explicit bases, and reveal a tight link between subgroup conjugacy data and field automorphism actions on irreducible characters. This contributes practical tools for character-theoretic computations and enriches the conceptual bridge between representation theory and Galois theory.

Abstract

In this paper, we sharpen a theorem of Artin to show that for a finite group, the dimension of the subspace of class functions spanned by the rational valued characters equals the number of conjugacy classes of cyclic subgroups.

On a theorem of Artin and the dimension of the space spanned by the rational valued characters of a group

TL;DR

The paper sharpens Artin's theorem by showing that the subspace of class functions spanned by rational-valued characters has dimension equal to the number of conjugacy classes of cyclic subgroups, and by constructing a concrete basis from induced class functions tied to cyclic subgroups. It provides a detailed, two-pronged argument: a direct representation-theoretic approach that identifies a basis corresponding to cyclic-subgroup conjugacy classes, and a Galois-theoretic viewpoint that interprets rational characters as sums over orbits of irreducible characters under , with a precise correspondence between orbits and cyclic-subgroup conjugacy classes. The results yield a refined understanding of rational-valued characters, connect classical Artin-type results to explicit bases, and reveal a tight link between subgroup conjugacy data and field automorphism actions on irreducible characters. This contributes practical tools for character-theoretic computations and enriches the conceptual bridge between representation theory and Galois theory.

Abstract

In this paper, we sharpen a theorem of Artin to show that for a finite group, the dimension of the subspace of class functions spanned by the rational valued characters equals the number of conjugacy classes of cyclic subgroups.
Paper Structure (3 sections, 6 theorems, 2 equations)

This paper contains 3 sections, 6 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Proofs
  3. Another viewpoint

Key Result

Theorem 1

Let $\chi$ be a rational valued character of $G$. Then where $H$ runs over the cyclic subgroups of $G$ and $a_H \in Z$.

Theorems & Definitions (9)

  • Theorem 1: Isaacs' version of Artin's theorem
  • Theorem 2
  • Theorem 3
  • Theorem 4: Serre's version of Artin's theorem
  • Theorem 5: Restatement of Isaacs' version of Artin's Theorem
  • proof : Proof of Theorem \ref{['dimension']}
  • proof : Proof of Theorem \ref{['basis']}
  • proof : Proof of Theorem \ref{['restatement']}
  • Proposition 3.1