A McKay bijection for projectors

Abstract

If \mathfrak F is a saturated formation of groups, we define a canonical subset \text{Irr}_{\mathfrak F'}(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'}(G)|=|\text{Irr}(\mathbf{N}_G(H)/H')| , where H'=[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.