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 2B2(22m+1) with m ≥ 2 and the small Ree groups 2G2(32m+1) with m ≥ 1.
Related
- The inductive Navarro-Alperin weight condition for the Suzuki and Ree groups
- A reduction theorem for Lê's conjecture
- The Brauer A(∞) condition and Navarro's conjecture on Brauer characters under coprime actions
- Elements of the Green ring related to Alperin's weight conjecture
- A 3× 3 counterexample to Lin and Wimmer's rank-minimization conjecture associated with Roth's similarity theorem
BibTeX
@misc{chen2026reduction,
title = {{A Reduction Theorem for the Alperin-McKay-Navarro Conjecture}},
author = {Shi Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33507},
eprint = {2609.33507},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33507},
}
RIS
TY - UNPB
TI - A Reduction Theorem for the Alperin-McKay-Navarro Conjecture
AU - Shi Chen
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33507
UR - https://arxiv.org/abs/2609.33507
AB - 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 ^2B_2(2^2m+1) with m ≥ 2 and the small Ree groups ^2G_2(3^2m+1) with m ≥ 1.
ER -