A McKay bijection for projectors

Abstract

If $\mathfrak{F}$ is a saturated formation of groups, we define a canonical subset ${\text{Irr}}_{{\mathfrak{F}}^{\prime }}(G)$ of the irreducible complex characters of a finite solvable group $G$. If $H$ is an $\mathfrak{F}$-projector of $G$, we show that $|{\text{Irr}}_{{\mathfrak{F}}^{\prime }}(G)|=|\text{Irr}({\mathbf{N}}_G(H)/H^{\prime })|$, where $H^{\prime }=[H,H]$ is the derived subgroup of $H$. In particular, if $\mathfrak{F}$ is the class of $p$-groups, this reproves the solvable case of the celebrated McKay conjecture.