ON THE DEGREES OF IRREDUCIBLE CHARACTERS FIXED BY SOME FIELD AUTOMORPHISM IN p-SOLVABLE GROUPS

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. This result is generalized 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.