On the degrees of irreducible characters fixed by some field automorphism, p-solvable groups

TL;DR

This paper generalizes a known result about real-valued irreducible characters with odd degree to p-solvable groups, showing conditions under which Sylow p-subgroups are normal based on character degrees fixed by field automorphisms.

Contribution

It extends the classical result to p-solvable groups for any prime p, linking character degrees fixed by automorphisms to the normality of Sylow p-subgroups.

Findings

A p-solvable group has a normal Sylow p-subgroup if p does not divide the degree of any irreducible character fixed by a field automorphism of order p.

The result generalizes the case for real-valued characters with odd degree to a broader class of characters and groups.

The paper establishes a new criterion for the normality of Sylow p-subgroups based on automorphism-invariant character degrees.

Abstract

It is known that, if all the real-valued irreducible characters of a finite group have odd degree, then the group has normal Sylow $2$-subgroup. We generalize this result for Sylow $p$-subgroups, for any prime number $p$, while assuming the group to be $p$-solvable. In particular, it is proved that a $p$-solvable group has a normal Sylow $p$-subgroup if $p$ does not divide the degree of any irreducible character of the group fixed by a field automorphism of order $p$.