Sylow subgroups, exponents, and character values

Navarro, Gabriel; Tiep, Pham

doi:10.1090/tran/7816

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