# Primitive characters of odd order groups

## TL;DR

This paper investigates the properties of primitive characters in finite groups of odd order, establishing divisibility relations between character degrees and conjugacy class sizes, with specific results for certain group classes.

## Contribution

It proves divisibility properties of primitive character degrees by conjugacy class sizes in odd order groups, extending understanding of character theory in finite groups.

## Key findings

- For all primes dividing the group order, a conjugacy class exists where the prime part of the degree divides its size.
- In some group classes, the entire degree of primitive characters divides a conjugacy class size.
- The results deepen the connection between character degrees and conjugacy class structures in odd order groups.

## Abstract

Let $G$ be a finite group of odd order. We show that if $\chi$ is an irreducible primitive character of $G$ then for all primes $p$ dividing the order of $G$ there is a conjugacy class such that the $p-$part of $\chi(1)$ divides the size of that conjugacy class. We also show that for some classes of groups the entire degree of an irreducible primitive character $\chi$ divides the size of a conjugacy class.