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Arithmetic results on orbits of linear groups


Authors: Michael Giudici, Martin W. Liebeck, Cheryl E. Praeger, Jan Saxl and Pham Huu Tiep
Journal: Trans. Amer. Math. Soc. 368 (2016), 2415-2467
MSC (2010): Primary 20B05, 20H30
DOI: https://doi.org/10.1090/tran/6373
Published electronically: August 19, 2015
MathSciNet review: 3449244
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Abstract: Let $ p$ be a prime and $ G$ a subgroup of $ GL_d(p)$. We define $ G$ to be $ p$-exceptional if it has order divisible by $ p$, but all its orbits on vectors have size coprime to $ p$. We obtain a classification of $ p$-exceptional linear groups. This has consequences for a well-known conjecture in representation theory, and also for a longstanding question concerning $ \frac {1}{2}$-transitive linear groups (i.e. those having all orbits on nonzero vectors of equal length), classifying those of order divisible by $ p$.


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Michael Giudici
Affiliation: Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009
Email: michael.giudici@uwa.edu.au

Martin W. Liebeck
Affiliation: Department of Mathematics, Imperial College, London SW7 2BZ, England
Email: m.liebeck@imperial.ac.uk

Cheryl E. Praeger
Affiliation: Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009
Email: Cheryl.Praeger@uwa.edu.au

Jan Saxl
Affiliation: Department of Pure Mathematics and Mathematical Statistics, CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England
Email: saxl@dpmms.cam.ac.uk

Pham Huu Tiep
Affiliation: Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, Arizona 85721-0089
Email: tiep@math.arizona.edu

DOI: https://doi.org/10.1090/tran/6373
Received by editor(s): March 12, 2012
Received by editor(s) in revised form: January 14, 2014
Published electronically: August 19, 2015
Article copyright: © Copyright 2015 American Mathematical Society