$p$-rational characters and self-normalizing Sylow $p$-subgroups
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- by Gabriel Navarro, Pham Huu Tiep and Alexandre Turull
- Represent. Theory 11 (2007), 84-94
- DOI: https://doi.org/10.1090/S1088-4165-07-00263-4
- Published electronically: April 19, 2007
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Abstract:
Let $G$ be a finite group, $p$ a prime, and $P$ a Sylow $p$-subgroup of $G$. Several recent refinements of the McKay conjecture suggest that there should exist a bijection between the irreducible characters of $p’$-degree of $G$ and the irreducible characters of $p’$-degree of $\mathbf {N}_G(P)$, which preserves field of values of correspondent characters (over the $p$-adics). This strengthening of the McKay conjecture has several consequences. In this paper we prove one of these consequences: If $p>2$, then $G$ has no non-trivial $p’$-degree $p$-rational irreducible characters if and only if $\mathbf {N}_G(P)=P$.References
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
- 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
- Larry Dornhoff, Group representation theory. Part B: Modular representation theory, Pure and Applied Mathematics, vol. 7, Marcel Dekker, Inc., New York, 1972. MR 0347960
- 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, DOI 10.1090/surv/040.1
- Robert M. Guralnick, Gunter Malle, and Gabriel Navarro, Self-normalizing Sylow subgroups, Proc. Amer. Math. Soc. 132 (2004), no. 4, 973–979. MR 2045411, DOI 10.1090/S0002-9939-03-07161-2
- B. Huppert, Endliche Gruppen, Springer-Verlag, Berlin, 1964.
- I. M. Isaacs, Gunter Malle, and Gabriel Navarro, Real characters of $p’$-degree, J. Algebra 278 (2004), no. 2, 611–620. MR 2071655, DOI 10.1016/j.jalgebra.2003.09.032
- I. Martin Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR 1280461
- G. Lusztig, On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 10, 157–166. Orbites unipotentes et représentations, I. MR 1021495
- Gabriel Navarro, Linear characters of Sylow subgroups, J. Algebra 269 (2003), no. 2, 589–598. MR 2015855, DOI 10.1016/S0021-8693(03)00391-0
- 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
- Robert Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92. MR 354892, DOI 10.1016/0001-8708(75)90125-5
- A. Turull, Strengthening the McKay Conjecture to include local fields and local Schur indices, J. Algebra (to appear).
- Alexandre Turull, Character correspondences in solvable groups, J. Algebra 295 (2006), no. 1, 157–178. MR 2188855, DOI 10.1016/j.jalgebra.2005.01.028
Bibliographic Information
- Gabriel Navarro
- Affiliation: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, Spain
- MR Author ID: 129760
- Email: gabriel@uv.es
- Pham Huu Tiep
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
- MR Author ID: 230310
- Email: tiep@math.ufl.edu
- Alexandre Turull
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
- Email: turull@math.ufl.edu
- Received by editor(s): November 23, 2004
- Published electronically: April 19, 2007
- Additional Notes: The first author was partially supported by the Ministerio de Educación y Ciencia
The second author acknowledges the support of the NSA (grant H98230-04-0066) and the NSF (grant DMS-0600967)
The third author acknowledges the support of the NSA - © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory 11 (2007), 84-94
- MSC (2000): Primary 20C15; Secondary 20C33
- DOI: https://doi.org/10.1090/S1088-4165-07-00263-4
- MathSciNet review: 2306612