Abstract
We prove that the number of conjugacy classes of a finite group G consisting of elements of odd order, is larger than or equal to that number for the normaliser of a Sylow 2-subgroup of G. This is predicted by the Alperin Weight Conjecture.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. L. Alperin, Weights for finite groups, in: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math. 47, Part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 369-379.
An, J., Dietrich, H.: The AWC-goodness and essential rank of sporadic simple groups. J. Algebra. 356, 325–354 (2012)
Broué, M., Michel, J.: Blocs et séries de Lusztig dans un groupe réductif fini. J. reine angew. Math. 395, 56–67 (1989)
Brough, J., Schaeffer Fry, A.A.: Radical subgroups and the inductive blockwise Alperin weight conditions for PSp4(q). Rocky Mountain J. Math. 50, 1181–1205 (2020)
Carter, R.: Finite Groups of Lie type: Conjugacy Classes and Complex Characters. Wiley, Chichester (1985)
F. Digne, J. Michel, Representations of Finite Groups of Lie Type, London Mathematical Society Student Texts, Vol. 21, Cambridge University Press, Cambridge, 1991.
W. Feit, Some consequences of the classification of finite simple groups, in: The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math. 37, Amer. Math. Soc., Providence, RI, 1980, pp. 175-181.
Feng, Z., Li, Z., Zhang, J.: On the inductive blockwise Alperin weight condition for classical groups. J. Algebra. 537, 381–434 (2019)
Z. Feng, G. Malle, The inductive blockwise Alperin weight condition for type C and the prime 2, J. Austral. Math. Soc., https://doi.org/10.1017/S1446788720000439.
The GAP group, GAP - Groups, Algorithms, and Programming, Version 4.4, 2004, http://www.gap-system.org.
Giannelli, E., Kleshchev, A.S., Navarro, G., Tiep, P.H.: Restriction of odd degree characters and natural correspondences. Int. Math. Res. Not. IMRN. 2017(20), 6089–6118 (2017)
D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups, Number 3. Part I., Chap. A, Mathematical Surveys and Monographs, Vol. 40, American Mathematical Society, Providence, RI, 1998.
Isaacs, I.M., Malle, G., Navarro, G.: A reduction theorem for the McKay conjecture. Invent. Math. 170, 33–101 (2007)
P. B. Kleidman, M. W. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Ser., no. 129, Cambridge University Press, Cambridge, 1990.
Kleshchev, A.S., Tiep, P.H.: Representations of finite special linear groups in nondefining characteristic. Adv. Math. 220, 478–504 (2009)
А. С. Кондратьев, Нормализаторы силовских 2-подгрупп в конечных простых группах, Матем. заметки 78 (2005), вып. 3, 368-376. Engl. transl.: A. S. Kondratiev, Normalizers of the Sylow 2-subgroups in finite simple groups, Mat. Notes 78 (2005), 338-346.
Lewis, M.L., H. P.: Tong Viet, Brauer characters of q′-degree. Proc. Amer. Math. Soc. 145, 1891–1898 (2017)
Li, C.: The inductive blockwise Alperin weight condition for PSp2n(q) and odd primes. J. Algebra. 567, 582–612 (2021)
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., Späth, B.: Characters of odd degree. Ann. of Math. 184(2), 869–908 (2016)
Manz, O., Wolf, T.R.: Brauer characters of q′-degree in p-solvable groups. J. Algebra. 115, 75–91 (1988)
Manz, O., Wolf, T.R.: Representations of Solvable Groups. Cambridge University Press, Cambridge (1993)
Navarro, G.: Characters and Blocks of Finite Groups. Cambridge University Press, Cambridge (1998)
Navarro, G., Späth, B., Tiep, P.H.: Coprime actions and correspondences of Brauer characters. Proc. Lond. Math. Soc. 114(3), 589–613 (2017)
Navarro, G.: Pham Huu Tiep, A reduction theorem for the Alperin weight conjecture. Invent. Math. 184, 529–565 (2011)
Navarro, G., Tiep, P.H.: Real groups and Sylow 2-subgroups. Adv. Math. 299, 331–360 (2016)
Navarro, G., Tiep, P.H.: On 2-Brauer characters of odd degree. Math. Z. 290, 469–483 (2018)
Späth, B.: A reduction theorem for the blockwise Alperin weight conjecture. J. Group Theory. 16, 159–220 (2013)
Taylor, J.: The structure of root data and smooth regular embeddings of reductive groups. Proc. Edin. Math. Soc. 62, 523–552 (2019)
Willems, W.: Blocks of defect zero in finite simple groups of Lie type. J. Algebra. 113, 511–522 (1988)
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
To the memory of Jim Humphreys
Gunter Malle is supported by Deutsche Forschungsgemeinschaft-Project-ID 286237555-TRR 195.
Gabriel Navarro is supported by Ministerio de Ciencia e Innovación PID2019-103854GB-I00.
Pham Huu Tiep is supported by the NSF (grant DMS-1840702), the Joshua Barlaz Chair in Mathematics, and the Charles Simonyi Endowment at the Institute for Advanced Study (Princeton).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
MALLE, G., NAVARRO, G. & TIEP, P.H. ON ALPERIN’S LOWER BOUND FOR THE NUMBER OF BRAUER CHARACTERS. Transformation Groups 28, 1205–1220 (2023). https://doi.org/10.1007/s00031-022-09734-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-022-09734-8