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

TL;DR

This work extends Ito–Michler–type connections between a finite group’s normal structure and the degrees of its irreducible characters to the setting of $p$-solvable groups under a field automorphism of order $p$. The authors develop technical tools for $p$-group actions on irreducible modules and leverage Deligne–Lusztig theory to analyze simple groups of Lie type, establishing the existence of $\sigma$-invariant irreducible characters with degrees divisible by $p$ in many cases. They then prove that for a finite $p$-solvable group $G$ and a $\sigma$ of order $p$, if $p mid heta(1)$ for all $\sigma$-invariant $ heta eq 1$ in $ ext{Irr}(G)$, then $G$ has a normal Sylow $p$-subgroup; this generalizes the known primes_dividing result to the $p$-solvable setting. The paper also discusses weaker outcomes in the non-$p$-solvable context and lays out a framework for extending these results to broader classes via a detailed simple-group analysis, thereby advancing the understanding of how field automorphisms constrain character degrees and group structure.

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 $p$.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4