On some rational extension properties for $G{L}_n(q)$ and even-degree characters fixed by order-2 Galois automorphisms

A. A. Schaeffer Fry

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, w…