Orders generated by character values

TL;DR

This paper studies the algebraic structure of orders generated by character values of finite groups within their associated number fields, revealing divisibility properties and proposing a conjecture for all finite groups.

Contribution

It establishes divisibility results for primes dividing the quotient of the ring of integers and the suborder generated by character values, and proves a property for nilpotent groups with a conjecture for all groups.

Findings

Prime divisors of the order of the quotient divide |G|.

For nilpotent G, the exponent of the quotient divides |G|.

Conjecture: the exponent divides |G| for all finite groups.

Abstract

Let $K:=\mathbb{Q}(G)$ be the number field generated by the complex character values of a finite group $G$. Let $\mathbb{Z}_K$ be the ring of integers of $K$. In this paper we investigate the suborder $\mathbb{Z}[G]$ of $\mathbb{Z}_K$ generated by the character values of $G$. We prove that every prime divisor of the order of the finite abelian group $\mathbb{Z}_K/\mathbb{Z}[G]$ divides $|G|$. Moreover, if $G$ is nilpotent, we show that the exponent of $\mathbb{Z}_K/\mathbb{Z}[G]$ is a proper divisor of $|G|$ unless $G=1$. We conjecture that this holds for arbitrary finite groups $G$.