On the degrees of irreducible characters fixed by some field automorphism in finite groups

TL;DR

This paper investigates the properties of finite groups where certain Galois automorphisms of prime order do not divide the degrees of any irreducible characters fixed by them, extending the classical Ito-Michler theorem.

Contribution

It proves a variant of the Ito-Michler theorem focusing on the behavior of irreducible characters under specific Galois automorphisms in finite groups.

Findings

Identifies conditions under which prime p does not divide degrees of fixed irreducible characters.

Extends classical results to automorphisms of order p.

Provides new insights into the structure of finite groups with automorphism constraints.

Abstract

We prove a variant of the Theorem of Ito-Michler, investigating the properties of finite groups where a prime number $p$ does not divide the degree of any irreducible character left invariant by some Galois automorphism $\sigma$ of order $p$.