McKay natural correspondences on characters

Abstract

Let $G$ be a finite group, let $p$ be an odd prime, and let $P\in {Syl}_p\left(G\right)$. If $N_G\left(P\right)=P{C}_G\left(P\right)$, then there is a canonical correspondence between the irreducible complex characters of $G$ of degree not divisible by $p$ belonging to the principal block of $G$ and the linear characters of $P$. As a consequence, we give a characterization of finite groups that possess a self-normalizing Sylow $p$-subgroup or a $p$-decomposable Sylow normalizer.