Galois automorphisms on Harish-Chandra series and Navarro’s self-normalizing Sylow $2$-subgroup conjecture
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Abstract:
Navarro has conjectured a necessary and sufficient condition for a finite group $G$ to have a self-normalizing Sylow $2$-subgroup, which is given in terms of the ordinary irreducible characters of $G$. In a previous article, Schaeffer Fry has reduced the proof of this conjecture to showing that certain related statements hold for simple groups. In this article, we describe the action of Galois automorphisms on the Howlett–Lehrer parametrization of Harish-Chandra induced characters. We use this to complete the proof of the conjecture by showing that the remaining simple groups satisfy the required conditions.References
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Additional Information
- A. A. Schaeffer Fry
- Affiliation: Department of Mathematical and Computer Sciences, Metropolitan State University of Denver, Denver, Colorado 80217
- MR Author ID: 899206
- Email: aschaef6@msudenver.edu
- Received by editor(s): July 17, 2017
- Received by editor(s) in revised form: January 31, 2018, and March 12, 2018
- Published electronically: October 2, 2018
- Additional Notes: The author was supported in part by a grant through MSU Denver’s Faculty Scholars Program, a grant from the Simons Foundation (Award No. 351233), and an NSF-AWM Mentoring Travel Grant. She would like to thank G. Malle and B. Späth for their hospitality and helpful discussions during the research visit to TU Kaiserslautern supported by the latter grant, during which the majority of the work for this article was accomplished.
Part of this was completed while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester program on Group Representation Theory and Applications, supported by the National Science Foundation under Grant No. DMS-1440140. She thanks the institute and the organizers of the program for making her stay possible and providing a collaborative and productive work environment. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 457-483
- MSC (2010): Primary 20C15; Secondary 20C33
- DOI: https://doi.org/10.1090/tran/7590
- MathSciNet review: 3968776