A $\sigma$-McKay theorem for $\pi$-separable groups

Abstract

We prove a Hall $\sigma$-subgroup analogue of the McKay conjecture for $\pi$-separable groups proposed by G. Navarro. This result simultaneously generalizes the classical McKay conjecture and its $\pi$-separable version. More precisely, if $G$ is $\pi$-separable, $p$ is a prime and $\sigma=\pi\cup\{p\}$, then it is known that there exists $H$ a Hall $\sigma$-subgroup of $G$. In this case, we prove that $|{\rm Irr}_{\sigma'}(G)|=|{\rm Irr}_{\sigma'}(\mathbf{N}_{G}(H))|$.