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Finite groups with odd Sylow normalizers


Authors: Robert M. Guralnick, Gabriel Navarro and Pham Huu Tiep
Journal: Proc. Amer. Math. Soc. 144 (2016), 5129-5139
MSC (2010): Primary 20D06; Secondary 20D20
DOI: https://doi.org/10.1090/proc/13223
Published electronically: June 10, 2016
MathSciNet review: 3556259
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Abstract: We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight conjecture for these groups at these primes.


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  • [Hall] 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, https://doi.org/10.1090/tran/6381
  • [GL] 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, https://doi.org/10.1090/memo/0276
  • [GLS] Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR 1303592
  • [GMN] Robert M. Guralnick, Gunter Malle, and Gabriel Navarro, Self-normalizing Sylow subgroups, Proc. Amer. Math. Soc. 132 (2004), no. 4, 973-979 (electronic). MR 2045411, https://doi.org/10.1090/S0002-9939-03-07161-2
  • [Is] I. Martin Isaacs, Character theory of finite groups, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. MR 0460423
  • [IMN1] I. M. Isaacs, Gunter Malle, and Gabriel Navarro, Real characters of $ p'$-degree, J. Algebra 278 (2004), no. 2, 611-620. MR 2071655, https://doi.org/10.1016/j.jalgebra.2003.09.032
  • [IMN2] I. M. Isaacs, Gunter Malle, and Gabriel Navarro, A reduction theorem for the McKay conjecture, Invent. Math. 170 (2007), no. 1, 33-101. MR 2336079, https://doi.org/10.1007/s00222-007-0057-y
  • [KL] Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341
  • [KS] S. Koshitani and B. Späth,
    The inductive Alperin-McKay and blockwise Alperin weight conditions for blocks with cyclic defect groups and odd primes, J. Group Theory, DOI: 10.1515/jgth-2016-0006, Feb. 2016.
  • [LSS] 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, https://doi.org/10.1112/plms/s3-65.2.297
  • [MN] Gunter Malle and Gabriel Navarro, Extending characters from Hall subgroups, Doc. Math. 16 (2011), 901-919. MR 2880677
  • [MS] G. Malle and B. Späth, Characters of odd degree, arXiv:1506.07690v1.
  • [MT] 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
  • [N] Gabriel Navarro, The McKay conjecture and Galois automorphisms, Ann. of Math. (2) 160 (2004), no. 3, 1129-1140. MR 2144975, https://doi.org/10.4007/annals.2004.160.1129
  • [NS] Gabriel Navarro and Britta Späth, On Brauer's height zero conjecture, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 4, 695-747. MR 3191974, https://doi.org/10.4171/JEMS/444
  • [NTV] Gabriel Navarro, Pham Huu Tiep, and Carolina Vallejo, McKay natural correspondences on characters, Algebra Number Theory 8 (2014), no. 8, 1839-1856. MR 3285617, https://doi.org/10.2140/ant.2014.8.1839
  • [NTT] Gabriel Navarro, Pham Huu Tiep, and Alexandre Turull, $ p$-rational characters and self-normalizing Sylow $ p$-subgroups, Represent. Theory 11 (2007), 84-94 (electronic). MR 2306612, https://doi.org/10.1090/S1088-4165-07-00263-4
  • [S1] Britta Späth, A reduction theorem for the blockwise Alperin weight conjecture, J. Group Theory 16 (2013), no. 2, 159-220. MR 3031870, https://doi.org/10.1515/jgt-2012-0032
  • [S2] Britta Späth, A reduction theorem for the Alperin-McKay conjecture, J. Reine Angew. Math. 680 (2013), 153-189. MR 3100954, https://doi.org/10.1515/crelle.2012.035
  • [TZ] Pham Huu Tiep and A. E. Zalesski, Real conjugacy classes in algebraic groups and finite groups of Lie type, J. Group Theory 8 (2005), no. 3, 291-315. MR 2137972, https://doi.org/10.1515/jgth.2005.8.3.291
  • [T] Alexandre Turull, Character correspondences in solvable groups, J. Algebra 295 (2006), no. 1, 157-178. MR 2188855, https://doi.org/10.1016/j.jalgebra.2005.01.028
  • [W] Robert A. Wilson, The McKay conjecture is true for the sporadic simple groups, J. Algebra 207 (1998), no. 1, 294-305. MR 1643110, https://doi.org/10.1006/jabr.1998.7450

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Additional Information

Robert M. Guralnick
Affiliation: Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089
Email: guralnic@usc.edu

Gabriel Navarro
Affiliation: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain
Email: gabriel.navarro@uv.es

Pham Huu Tiep
Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
Email: tiep@math.arizona.edu

DOI: https://doi.org/10.1090/proc/13223
Keywords: McKay conjecture, Alperin's weight conjecture, Sylow normalizers
Received by editor(s): February 24, 2016
Published electronically: June 10, 2016
Additional Notes: The first author gratefully acknowledges the support of the NSF grant DMS-1302886.
The research of the second author was supported by the Prometeo/Generalitat Valenciana, Proyectos MTM2013-40464-P. He would like to express his gratitude to the Mathematics Department of the University of Southern California where part of the present work was completed for its warm hospitality.
The third author was partially supported by the NSF grant DMS-1201374 and the Simons Foundation Fellowship 305247. Part of this work was done while the third author visited the Institute for Advanced Study (Princeton, NJ). It is a pleasure to thank Peter Sarnak and the Institute for their generous hospitality and a stimulating environment.
The authors are grateful to Gunter Malle for helpful comments on the paper, and to the referee for a careful reading of the paper.
Communicated by: Ken Ono
Article copyright: © Copyright 2016 American Mathematical Society

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