_p-rational characters and self-normalizing sylow_p-subgroups

Let G be a finite group, p a prime, and P a Sylow p-subgroup of G. Several recent refinements of the McKay conjecture suggest that there should exist a bijection between the irreducible characters of pʹ-degree of G and the irreducible characters of pʹ-degree of N_G(P), which preserves field of values of correspondent characters (over the p-adics). This strengthening of the McKay conjecture has several consequences. In this paper we prove one of these consequences: If p 2, then G has no non-trivial pʹ-degree p-rational irreducible characters if and only if N_G(P) = P.