Character correspondences in solvable groups with a self-normalizing sylow subgroup

Let G be a finite solvable group and let P Sylp(G) for some prime p. Whenever |G: NG(P)| is odd, Isaacs described a correspondence between irreducible characters of degree not divisible by p of G and NG(P). This correspondence is natural in the sense that an algorithm is provided to compute it, and the result of the application of the algorithm does not depend on choices made. In the case where NG(P) = P, G. Navarro showed that every irreducible character χ of degree not divisible by p has a unique linear constituent χ when restricted to P, and that the map χchi; defines a bijection. Navarro's bijection is obviously natural in the sense described above. We show that these two correspondences are the same under the intersection of the hypotheses.