# The average character degree and an improvement of the Ito-Michler theorem

## TL;DR

This paper extends the classical Ito-Michler theorem by using average character degree to provide a comprehensive improvement applicable to all primes, enhancing understanding of the relationship between character degrees and Sylow p-subgroups.

## Contribution

The paper generalizes previous results by applying average character degree to improve the Ito-Michler theorem for all primes, not just p=2.

## Key findings

- Full improvement of Ito-Michler theorem for all primes
- Character degree conditions imply Sylow subgroup properties
- Enhanced criteria for group structure analysis

## Abstract

The classical It\^{o}-Michler theorem states that the degree of every ordinary irreducible character of a finite group $G$ is coprime to a prime $p$ if and only if the Sylow $p$-subgroups of $G$ are abelian and normal. In an earlier paper, we used the notion of average character degree to prove an improvement of this theorem for the prime $p=2$. In this follow-up paper, we obtain a full improvement for all primes.