Character correspondeces in solvable groups with a self-normalizing Sylow subgroup

Abstract: In 1973, I. M. Isaacs described a correspondence between characters of degree not divisible by a fixed prime $p$ of a finite solvable group $G$ and those of the normalizer of Sylow $p$-subgroup of $G$, whenever the index of the normalizer in $G$ is odd. 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. Later on, for $p$-solvable groups with self-normalizing Sylow $p$-subgroup, G. Navarro showed that every irreducible character of degree not divisible by $p$ has a unique linear constituent when restricted to a Sylow $p$-subgroup. Furthermore, the process of choosing the unique linear constituent of the restriction 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.