A Reduction Theorem for the Alperin-McKay-Navarro Conjecture

Abstract

We reduce the Alperin-McKay-Navarro conjecture to an inductive condition on the universal covering groups of non-abelian finite simple groups. The reduction yields height-zero character bijections compatible with Brauer correspondence and block relations between H-triples for arbitrary finite ambient groups. We record the resulting arithmetic and structural consequences, including preservation of $p$-rationality levels and character-theoretic criteria for Sylow subgroups. The proof combines semilinear centralization and transfer through quasisimple components with induction on the central index. It uses the Clifford and gluing theorems for block relations and a central-defect Dade-Glauberman-Nagao correspondence established in two related papers. We also verify the inductive condition for several sporadic groups at primes for which the Sylow subgroups have prime order, and in defining characteristic for the Suzuki groups ${}^{2}B_{2}(2^{2m+1})$ with $m \geq 2$ and the small Ree groups ${}^{2}G_{2}(3^{2m+1})$ with $m \geq 1$.