The McKay conjecture and Brauer's induction theorem. (Q2840178)

Let \(p\) be a prime, let \(P\) be a Sylow \(p\)-subgroup of a finite group \(G\), and let \(H:=N_G(P)\). The McKay conjecture predicts the existence of a bijection \(F\) between \(\text{Irr}_{p'}(G):=\{\chi\in\text{Irr}(G):p\nmid\chi(1)\}\) and \(\text{Irr}_{p'}(H)\). Various reformulations, refinements and reductions of this conjecture are known. Also, the conjecture has been proved in many cases.NEWLINENEWLINE In the paper under review, the author presents new refinements of the McKay conjecture, in several variants. The new feature of his conjectures is that it should be possible to choose \(F\) in such a way that \(F(\chi)\equiv\pm\text{Res}^G_H(\chi)\pmod J\) for \(\chi\in\text{Irr}_{p'}(G)\); here \(J\) is a certain group of virtual characters. The author shows that his versions of the McKay conjecture imply the refinement by Isaacs and Navarro.NEWLINENEWLINE The author also proposes a refinement of Broué's Abelian defect group conjecture and verifies his conjectures in several special cases.