Character Values of P-Solvable Groups on Picky Elements

Very recently, the first and third authors proposed a new conjecture on characters of finite groups, related to the McKay conjecture. Let p be a prime number and G a finite group. We say that a p-element x ∈ G is picky in G if it is contained in a unique Sylow p-subgroup of G. The simplest formulation of this conjecture predicts the existence of a bijection between the set of irreducible characters of G that do not vanish on a picky p-element x, and the corresponding set of irreducible characters of the normalizer of the unique Sylow p-subgroup of G containing x. Moreover, this bijection is expected to satisfy several natural conditions. For example, the p-parts of the degrees of corresponding characters should coincide, and their values at x should also be suitably related. In this paper, we prove this conjecture for p-solvable groups when p is odd.