On some rational extension properties for $GL_n(q)$ and even-degree characters fixed by order-2 Galois automorphisms

TL;DR

This paper investigates the properties of characters fixed by order-2 Galois automorphisms in finite groups, establishing conditions for the existence of normal Sylow 2-subgroups and analyzing character extensions in linear groups.

Contribution

It proves that fixed characters of odd degree imply a normal Sylow 2-subgroup and studies the extension of unipotent characters in linear groups to automorphism groups.

Findings

Characters fixed by order-2 Galois automorphisms with odd degree imply a normal Sylow 2-subgroup.

Unipotent characters of PSL_n(q) extend to rational characters of automorphism groups.

Extension properties of characters in GL_n(q) under transpose-inverse automorphisms are established.

Abstract

In this note, we prove that if every character of a finite group $G$ fixed by an order-2 Galois automorphism has odd degree, then $G$ has a normal Sylow $2$-subgroup. On the way, we study extensions of characters of $GL_n(q)$, $q$ odd, to the group extended by the transpose-inverse automorphism and prove that unipotent characters of $PSL_n(q)$ extend to rational characters of its automorphism group.