Characters of p'-degree and Thompson's character degree theorem

TL;DR

This paper improves classical results on finite group character degrees by analyzing the average of p'-degrees and exploring fields of character values, leading to stronger conclusions about group structure.

Contribution

It introduces new bounds based on the average p'-degrees of irreducible characters and extends results concerning fields of character values.

Findings

Improved bounds for the average p'-degrees of irreducible characters.

Enhanced criteria for the existence of normal p-complements.

Several results on fields of character values and their implications.

Abstract

A classical theorem of John Thompson on character degrees asserts that if the degree of every ordinary irreducible character of a finite group $G$ is 1 or divisible by a prime $p$, then $G$ has a normal $p$-complement. We obtain a significant improvement of this result by considering the average of $p'$-degrees of irreducible characters. We also consider fields of character values and prove several improvements of earlier related results.