New Refinements of the McKay Conjecture for Arbitrary Finite Groups

Abstract

Let G be an arbitrary finite group and fix a prime number p.The McKay conjecture asserts that G and the normalizer in G of a Sylow p-subgroup have equal numbers of irreducible characters with degrees not divisible by p.The Alperin-McKay conjecture is version of this as applied to individual Brauer p-blocks of G.We offer evidence that perhaps much stronger forms of both of these conjectures are true.