Degrees and p ‐rational characters

Suppose that G is a finite group and that p is a prime number. We prove that if every p ‐rational irreducible character of G is non‐zero on every p ‐element of G , then G has a normal Sylow p ‐subgroup. This yields a p ‐rational refinement of the Itô–Michler theorem: if p does not divide the degree of any irreducible p ‐rational character of G , then G has a normal Sylow p ‐subgroup