The McKay conjecture and Brauer's induction theorem

Abstract

Let $G$ be an arbitrary finite group. The McKay conjecture asserts that $G$ and the normaliser $N_G(P)$ of a Sylow $p$-subgroup $P$ in $G$ have the same number of characters of degree not divisible by $p$ (that is, of $p^{\prime }$-degree). We propose a new refinement of the McKay conjecture, which suggests that one may choose a correspondence between the characters of $p^{\prime }$-degree of $G$ and $N_G(P)$ to be compatible with induction and restriction in a certain sense. This refinement implies, in particular, a conjecture of Isaacs and Navarro. We also state a corresponding refinement of the Broué abelian defect group conjecture. We verify the proposed conjectures in several special cases.

Minor changes made throughout the paper

References in corpus (2)

• Counting p'-characters in finite reductive groups
• On equivariant bijections relative to the defining characteristic