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

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 2 -subgroup. This result is generalized for Sylow p p -subgroups, for any prime number p p , while assuming the group to be p p -solvable. In particular, it is proved that a p p -solvable group has a normal Sylow p p -subgroup if p p does not divide the degree of any irreducible character of the group fixed by a field automorphism of order p p .