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

TL;DR

This paper refines Artin's theorem to precisely determine the dimension of the space spanned by rational valued characters of a finite group, linking it to the count of conjugacy classes of cyclic subgroups.

Contribution

The paper provides a sharper version of Artin's theorem, establishing an exact relationship between rational characters and cyclic subgroup conjugacy classes.

Findings

Dimension equals the number of conjugacy classes of cyclic subgroups.

Provides a precise characterization of rational characters in finite groups.

Enhances understanding of the structure of class functions.

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.