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

Abstract

Let [Formula: see text] be a finite solvable group and let [Formula: see text] for some prime [Formula: see text]. Whenever [Formula: see text] is odd, Isaacs described a correspondence between irreducible characters of degree not divisible by [Formula: see text] of [Formula: see text] and [Formula: see text]. 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 [Formula: see text], G. Navarro showed that every irreducible character [Formula: see text] of degree not divisible by [Formula: see text] has a unique linear constituent [Formula: see text] when restricted to [Formula: see text], and that the map [Formula: see text] 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.