Sylow subgroups, exponents, and character values
HTML articles powered by AMS MathViewer
- by Gabriel Navarro and Pham Huu Tiep PDF
- Trans. Amer. Math. Soc. 372 (2019), 4263-4291 Request permission
Abstract:
If $G$ is a finite group, $p$ is a prime, and $P$ is a Sylow $p$-subgroup of $G$, we study how the exponent of the abelian group $P/P’$ is affected and how it affects the values of the complex characters of $G$. This is related to Brauer’s Problem $12$. Exactly how this is done is one of the last unsolved consequences of the McKay–Galois conjecture.References
- Antonio Beltrán, María José Felipe, Gunter Malle, Alexander Moretó, Gabriel Navarro, Lucia Sanus, Ronald Solomon, and Pham Huu Tiep, Nilpotent and abelian Hall subgroups in finite groups, Trans. Amer. Math. Soc. 368 (2016), no. 4, 2497–2513. MR 3449246, DOI 10.1090/tran/6381
- Richard Brauer, Representations of finite groups, Lectures on Modern Mathematics, Vol. I, Wiley, New York, 1963, pp. 133–175. MR 0178056
- Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
- François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR 1118841, DOI 10.1017/CBO9781139172417
- GAP Group, GAP—Groups, algorithms, and programming, version 4.4, 2004, http://www.gap-system.org.
- Daniel Gorenstein and Richard Lyons, The local structure of finite groups of characteristic $2$ type, Mem. Amer. Math. Soc. 42 (1983), no. 276, vii+731. MR 690900, DOI 10.1090/memo/0276
- Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups. Number 3. Part I. Chapter A, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998. Almost simple $K$-groups. MR 1490581, DOI 10.1090/surv/040.3
- P. N. Hoffman and J. F. Humphreys, Projective representations of the symmetric groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992. $Q$-functions and shifted tableaux; Oxford Science Publications. MR 1205350
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3
- I. Martin Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423]. MR 2270898, DOI 10.1090/chel/359
- I. M. Isaacs and Gabriel Navarro, Characters of $p’$-degree of $p$-solvable groups, J. Algebra 246 (2001), no. 1, 394–413. MR 1872628, DOI 10.1006/jabr.2001.8985
- Martin W. Liebeck, Jan Saxl, and Gary M. Seitz, Subgroups of maximal rank in finite exceptional groups of Lie type, Proc. London Math. Soc. (3) 65 (1992), no. 2, 297–325. MR 1168190, DOI 10.1112/plms/s3-65.2.297
- G. Malle, The Navarro–Tiep Galois conjecture for $p=2$, Archiv der Math. (to appear).
- Gunter Malle and Donna Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, vol. 133, Cambridge University Press, Cambridge, 2011. MR 2850737, DOI 10.1017/CBO9780511994777
- G. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, vol. 250, Cambridge University Press, Cambridge, 1998. MR 1632299, DOI 10.1017/CBO9780511526015
- Gabriel Navarro, The McKay conjecture and Galois automorphisms, Ann. of Math. (2) 160 (2004), no. 3, 1129–1140. MR 2144975, DOI 10.4007/annals.2004.160.1129
- Gabriel Navarro, Character theory and the McKay conjecture, Cambridge Studies in Advanced Mathematics, vol. 175, Cambridge University Press, Cambridge, 2018. MR 3753712, DOI 10.1017/9781108552790
- G. Navarro, B. Späth, and C. Vallejo, A reduction theorem for the Galois–McKay conjecture (in preparation).
- Gabriel Navarro and Pham Huu Tiep, Real groups and Sylow 2-subgroups, Adv. Math. 299 (2016), 331–360. MR 3519471, DOI 10.1016/j.aim.2016.05.013
- Gabriel Navarro and Pham Huu Tiep, Irreducible representations of odd degree, Math. Ann. 365 (2016), no. 3-4, 1155–1185. MR 3521086, DOI 10.1007/s00208-015-1334-5
- Gabriel Navarro, Pham Huu Tiep, and Alexandre Turull, $p$-rational characters and self-normalizing Sylow $p$-subgroups, Represent. Theory 11 (2007), 84–94. MR 2306612, DOI 10.1090/S1088-4165-07-00263-4
- Gabriel Navarro, Pham Huu Tiep, and Alexandre Turull, Brauer characters with cyclotomic field of values, J. Pure Appl. Algebra 212 (2008), no. 3, 628–635. MR 2365337, DOI 10.1016/j.jpaa.2007.06.019
- A. A. Schaeffer-Fry, Actions of Galois automorphisms on Harish-Chandra series and Navarro’s self-normalizing Sylow $2$-subgroup conjecture, https://arxiv.org/pdf/1707.03923.pdf (2017). Trans. Amer. Math. Soc. (to appear).
- Pham Huu Tiep and A. E. Zalesskiĭ, Unipotent elements of finite groups of Lie type and realization fields of their complex representations, J. Algebra 271 (2004), no. 1, 327–390. MR 2022486, DOI 10.1016/S0021-8693(03)00174-1
Additional Information
- Gabriel Navarro
- Affiliation: Departament de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain
- MR Author ID: 129760
- Email: gabriel@uv.es
- Pham Huu Tiep
- Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
- MR Author ID: 230310
- Email: tiep@math.rutgers.edu
- Received by editor(s): May 18, 2018
- Received by editor(s) in revised form: May 19, 2018, and August 15, 2018
- Published electronically: April 4, 2019
- Additional Notes: The research of the first author is supported by the Prometeo/Generalitat Valenciana, Proyectos MTM2016-76196-P, and FEDER
The second author gratefully acknowledges the support of the NSF (grants DMS-1839351 and DMS-1840702).
The paper is partially based upon work supported by the NSF under grant DMS-1440140 while the authors were in residence at MSRI (Berkeley, California), during the Spring 2018 semester. We thank the Institute for the hospitality and support. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 4263-4291
- MSC (2010): Primary 20C15; Secondary 20C33, 20D06, 20D20
- DOI: https://doi.org/10.1090/tran/7816
- MathSciNet review: 4009430