Abstract
In this paper we complete the proof of Brauer’s height zero conjecture for two primes proposed by G. Malle and G. Navarro.
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References
Alperin, J.L.: Isomorphic blocks. J. Algebra 43, 694–698 (1976)
Aschbacher, M., Kessar, R., Oliver, B.: Fusion Systems in Algebra and Topology. Cambridge University Press, Cambridge (2011)
Bender, H.: On the normal \(p\)-structure of a finite group and related topics I. Hokkaido Math. J. 7, 271–288 (1978)
Brauer, R.: Number theoretical investigations on groups of finite order. In: Proceedings of the International Symposium on Algebraic Number Theory, Tokyo and Nikko (1955), Science Council of Japan, Tokyo, pp. 55–62 (1956)
Buturlakin, A.A., Grechkoseeva, M.A.: The cyclic structure of maximal tori in finite classical groups. Algebra Logic 46, 73–89 (2007)
Carter, R.W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley, Chichester (1985)
Feit, W., Seitz, G.M.: On finite rational groups and related topics. Ill. J. Math. 33, 103–131 (1989)
Gorenstein, D., Lyons, R.: The local structure of finite groups of characteristic 2 type. Mem. Am. Math. Soc. 42, 276 (1983)
Gorenstein, D., Lyons, R., Solomon, R.: The Classification of the Finite Simple Groups. Number 3. Part I. Chapter A. Almost Simple K-Groups. Mathematical Surveys and Monographs, vol. 40. no. 3. American Mathematical Society, Providence (1998)
Huppert, B., Blackburn, N.: Finite Groups III. Springer, Berlin (1982)
Isaacs, I.M.: Character Theory of Finite Groups. Academic Press, New York (1994)
Kessar, R., Malle, G.: Quasi-isolated blocks and Brauer’s height zero conjecture. Ann. Math. (2) 178, 321–384 (2013)
Kessar, R., Malle, G.: Brauer’s height zero conjecture for quasi-simple groups. J. Algebra 475, 43–60 (2017)
Malle, G.: Height 0 characters of finite groups of Lie type. Represent. Theory 11, 192–220 (2007)
Malle, G.: On the inductive Alperin–McKay and Alperin weight conjecture for groups with abelian Sylow subgroups. J. Algebra 397, 190–208 (2014)
Malle, G., Moretó, A., Navarro, G.: Element orders and Sylow structure of finite groups. Math. Z. 252, 223–230 (2006)
Malle, G., Navarro, G.: Brauer’s height zero conjecture for two primes. Math. Z. 295, 1723–1732 (2020)
Malle, G., Navarro, G.: Brauer’s height zero conjecture for principal blocks. J. Reine Angew. Math. 778, 119–125 (2021)
Malle, G., Navarro, G., Schaeffer Fry A.A., Tiep, P.H.: Brauer’s height zero conjecture. arXiv:2209.04736
Malle, G., Testerman, D.: Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge Studies in Advanced Mathematics, vol. 133. Cambridge University Press, Cambridge (2011)
Michler, G.O.: Brauer’s conjectures and the classification of finite simple groups. In: Representation Theory II, Groups and Orders. Lecture Notes in Mathematics, pp. 129–142. Springer, Heidelberg (1986)
Navarro, G.: Characters and Blocks of Finite Groups. London Mathematical Society Lecture Note Series, vol. 250. Cambridge University Press, Cambridge (1998)
Navarro, G., Späth, B.: On Brauer’s height zero conjecture. J. Eur. Math. Soc. (JEMS) 16, 695–747 (2014)
Navarro, G., Tiep, P.H.: Brauer’s height zero conjecture for the 2-blocks of maximal defect. J. Reine Angew. Math. 669, 225–247 (2012)
Navarro, G., Tiep, P.H.: Characters of relative \(p^{\prime }\)-degree over normal subgroups. Ann. Math. (2) 178, 1135–1171 (2013)
Revin, D.O., Vdovin, E.P.: An existence criterion for Hall subgroups of finite groups. J. Group Theory 14, 93–101 (2011)
Ruhstorfer, L.: The Alperin–McKay conjecture for the prime \(p = 2\). arXiv:2204.06373
Sambale, B.: Brauer’s height zero conjecture for metacyclic defect groups. Pac. J. Math. 262, 481–507 (2013)
Steinberg, R.: Endomorphisms of linear algebraic groups. In: Mem. Amer. Math. Soc., vol. 80. American Mathematical Society, Providence (1968)
Taylor, J.: Action of automorphisms on irreducible characters of symplectic groups. J. Algebra 505, 211–246 (2018)
Wielandt, H.: Zum Satz von Sylow. Math. Z. 60, 407–408 (1954)
Willems, W.: \(p^*\)-theory and modular representation theory. J. Algebra 104, 135–140 (1986)
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We are deeply grateful to Gunter Malle and Gabriel Navarro for helpful communications during writing this paper.
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Y. Liu and W. Willems were supported by NSFC (12171211) and the Natural Science Foundation of Jiangxi Province (20192ACB21008), and L. Wang and J. Zhang by NSFC (11631001 & 11871083). Also, Y. Liu gratefully acknowledges the support by an Alexander von Humboldt Fellowship for Experienced Researchers.
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Liu, Y., Wang, L., Willems, W. et al. Brauer’s height zero conjecture for two primes holds true. Math. Ann. 388, 1677–1690 (2024). https://doi.org/10.1007/s00208-022-02543-0
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DOI: https://doi.org/10.1007/s00208-022-02543-0