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

TL;DR

This paper demonstrates that two character correspondences in solvable groups with self-normalizing Sylow p-subgroups, originally developed separately, are actually the same under certain conditions, unifying their theoretical frameworks.

Contribution

It proves the equivalence of Isaacs' and Navarro's character correspondences in the context of solvable groups with self-normalizing Sylow p-subgroups.

Findings

The two correspondences coincide under the given hypotheses.

The unification simplifies understanding of character theory in these groups.

Provides a clearer framework for character correspondences in solvable groups.

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.