Odd-Degree Characters and Self-Normalizing Sylow 2-Subgroups: A Reduction to Simple Groups

Abstract

Let G be a a finite group, p a prime, and P a Sylow p-subgroup of G. A recent refinement, due to G. Navarro, of the McKay conjecture suggests that there should exist a bijection between irreducible characters of p′-degree of G and NG(P) which commutes with certain Galois automorphisms. In this article, we explore one of the consequences of this refinement, namely a way to read off from the character table of G whether a Sylow 2-subgroup of G is self-normalizing. We provide a reduction to finite simple groups and begin an investigation of the problem for simple groups.