Abstract
The main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group $G$ can be computed locally. The simplest of these conjectures is the “McKay conjecture” which asserts that the number of irreducible complex characters of $G$ of degree not divisible by $p$ is the same if computed in a $p$-Sylow normalizer of $G$. In this paper, we propose a much stronger version of this conjecture which deals with Galois automorphisms. In fact, the same idea can be applied to the celebrated Alperin and Dade conjectures.
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